SAMSO-TR-68- 90 THE UNIX 8525-1-F Copy - - LSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGIEERING Radiation Laboratory 8525-1-F = RL-2179 Investigation of Coated Re-entry Vehicle Cross Sctior (U) )i z o t Final Report o ~ 18 December 1966 - 1 December 1967 rO, By ~ R. F. GOODRICH, J. J. BOWMAN, B. A. HARRISON, a J E. F. KNOTT, T. B. A. SENIOR, T. M. SMITH, H. WEIL 0 and V. H. WESTON I >. z Z - o January 1968 "t t W 6 ', Su..iorl2z a u isciosure SubLec to Cron g g Contract F 04694-67-C-c0,Ss". Distribution Statement: In addition to security requirements which apply to this document and must be met, it may be further distributed by the holder only with specific prior approval of SAMSO, SMSD, Air Force Station, Los Angeles, CA 90045 Contract With: Hq. Spa Air Fore Norton Administered through: OFFICE OF RE E ACCH GROUP DOWNGRADED AT 3-YEA INTERVALS; DECLASSIFIED AF ER YEARS ice and Missile Systems Organization:e Systems Command Air Force Base, California 92409 UMINISTRATION. ANN ARIOR '*,This dcant com LC- in ormation affecting tlh national defens e RIl&t1 tates whtlin tUw meanin of d the ZKp oj Laws, Ttle 18 U. B. C. sections 7TM and 7 / transaissiofn or the unauthorised per6 i pribited by Law.

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UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F FOREWORD (U) This report, 'SAMSO-TR-68-90 was prepared by the Radiation Laboratory of the Department of Electrical Engineering of The University of Michigan under the direction of Dr. Raymond F. Goodrich, Principal Investigator and Burton A. Harrison, Contract Manager. The work was performed under Contract F 04694-67-C-0055, "Investigation of Re-entry Vehicle Surface Fields (SURF)". The work was administered under the direction of the Air Force Headquarters, Space and Missile Systems Organization, Norton Air Force Base, California 92409, by Capt. J. Wheatley, SMYSP, and was monitored by Mr. H.J. Katzman of the Aerospace Corporation. (U) The studies presented herein cover the period 18 December 1966 through 18 December 1967. (U) In addition to security requirements which must be met, this document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of SAMSO, SMSDI, Air Force Station, Los Angeles, CA 90045. (U) Information in this report is embargoed under the Department of State International Traffic in Arms Regulations. This report may be released to Foreign governments by departments or agencies of the U.S. Government subject to approval of Hq. Space and Missile Systems Organization (SMSDI), Air Forse Station, Los Angeles, Calif., 90045 or higher authority within the Department of the Air Force. Private individuals or firms require a Department of State export license. (U) The publication of this report does not constitute Air Force approval of the report's findings or conclusions. It is published only for the exchange and stimulation of ideas. SAMSO Approving Authority William J. Schlerf BSYDR Contracting Officer U NCLASSdFIED

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F ABSTRACT (S) This is the final report on Contract F 04694-67-C-0055, an investigation of re-entry vehicle radar cross section, the third phase of a program designated Project SURF. The objective of the SURF program is (1) to achieve the capability to determine the radar cross section of metallic and coated reentry vehicles which are sphere-capped-cones in shape, or modifications of that basic shape, (2) to determine the effect on radar cross section of the plasma re-entry environment and (3) to study methods for countering short pulse discrimination of these re-entry shapes. Parts (1) and (2) of this program are based upon the interpretation of surface field data obtained on models illuminated by radar in a specially designed facility. Radar backscatter measurements and computer programs are used to check theoretical conclusions This final report discusses the work carried out in the fourth quarter of this contract and such formulas for radar cross section as were developed and which extend the results previously reported. SECRET

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F TABLE OF CONTENTS FOREWORD iii ABSTRACT iv I INTRODUCTION 1 II EXPERIMENTAL INVESTIGATION 4 2.1 Introduction and Review of Program 4 2.2 The Influence of Absorber on Cone Spheres 7 2.3 Coated Perturbed Models 15 2.4 Indented Base Models 32 2.5 Flat-Based Models 36 III INTERPRETATION OF EXPERIMENTAL DATA 45 3.1 Introduction 45 3.2 Agreement Item (Task 3.1.1) 46 3.2.1 Introduction 46 3.2.2 Bare Cone-Sphere 50 3.2.3 Indented (ID) Models 60 3.2.4 Nose Tip Antenna (LSP) Model 74 3.2.5 Coated Cone-Spheres 80 3.3 Backscattering Behavior of Model LSP 96 3.4 Effective Estimates for the Nose-on Backscattering of Flat-Back (FB) Models 107 3.5 A Quantitative Failure of a Scattering Estimate 141 IV THEORETICAL STUDIES (SURF) 153 4. 1 Introduction 153 4.2 An Empirical Correction to the Estimated Creeping Wave Contribution for a Non-Spherical Body 153 4.3 Various Approximations for the Creeping Wave Contribution of a Sphere 168 4.4 Surface Current in the Illuminated Region on a Parabolic Cylinder 179 4.4.1 Introduction 179 4.4.2 Integral Representation for the Surface Current 179 4.4.3 The Method of Geometrical Optics 183 4.4.4 Reflected Creeping Waves 189 4.4.5 The Surface Current in the Region of Penumbra 193 4.5 Zeros of Parabolic Cylinder Functions 196 V RADAR CROSS SECTION IN THE PLASMA RE-ENTRY ENVIRONMENT 218 5.1 Introduction 218 UNCLASSIFIED

UNCLASSIFIED I THE UNIVERSITY OF 8525-1-F MICHIGAN Table of Contents (Cont'd) 5.2 Anisotropic Impedance Boundary Condition Approach to the Base Return for a Thin Plasma Sheath 5. 3 Re-entry Plasma Experiment (Task 2.1.5) 5. 3. 1 Introduction 5.3.2 The Plasma Covered Flat Plate 5. 3. 3 Plasma Coated Flat Back Cone 5.3.4 Backscattering by a Current Sheet 5. 3.5 Conclusion VI SHORT PULSE INVESTIGATION-TASK 4.0 6. 1 Introduction 6.2 Transient Surface Fields 6. 3 Integral Equation Formulation of Time Dependant Scattering Problem 6.3.1 Addendum on the Evaluation of Qjk. 6.4 Causality and Reality in the Synthesis of Pulse Responses 6.5 Pulse Scattering from a Perfectly Conducting Flat Back Cone. 6.6 Pulse Scattering from a Perfectly Conducting Cone-Sphere 6.7 Bandwidth Effects on Pulse Return 6.7.1 General 6.7.2 Examples: Sphere Computations VII HANDBOOK OF RADAR CROSS SECTION FORMULAS 7. 1 Introduction 7.2 Indented Cone Sphere (revised) 7.3 Flat Backed Cone 7.4 Coated Cone Sphere with Indented Base 7.5 Coated Cone Sphere, LSP with Slotted Coating 7.6 Coated Cone Sphere, LSH with Slotted Coating 7.7 Coated Cone Sphere with Longitudinal Slots VIII COMPUTER PROGRAM FOR A ROTATIONALLY SYMMETRIC METALLIC BODY 8. 1 Introduction 8.2 Description of the Numerical Analysis 8.2.1 Introduction 220 240 240 242 245 246 268 269 269 271 276 292 296 300 306 312 312 314 318 318 318 321 322 323 326 328 329 329 331 331 vil U NCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F Table of Contents (Cont'd) 8.2.2 The Evaluation of G 8.2.3 The Evaluation of m T. 8.2.4 The Evaluation of dm.j j1 8.3 Description of Computer Programming IX ACKNOWLEDGEMENTS X SUPPORTING TECHNICAL STUDIES REFERENCES DISTRIBUTION LIST DD 1473 MICHIGAN 332 337 347 348 382 384 385 vii UNCLASSIFIED

I SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F I INTRODUCTION (S) This is the Final Report under Contract F 04694-67-C-0055. It reports the results of the third phase of Project SURF, a complex program to determine the radar cross section of cone-sphere-like re-entry vehicles under exoatmospheric and re-entry conditions. The investigation is part of the ABRES program. It was carried out under the auspices of Mace and Missile Systems Organization (SAMSO) at the Radiation Laboratory of the Department of Electrical Engineering of The University of Michigan. The SAMSO project officer was Captain James Wheatley. The work was monitored for the Air Force by the Aerospace Corporation under the direction of Mr. H.J. Katzman. (S) SURF made use of experimental methods of measuring the surface currents induced on metallic and coated models of re-entry shapes to obtain a basis for an analytical synthesis of radar cross section formulas. Inasmuch as the re-entry shapes being considered were of extremely low cross section, it would have been difficult to study them solely with radar range techniques or using full-scale models and it would have been impossible using these techniques to isolate in a practical or economical way, the effect on radar cross section of perturbations, coatings and antennas. (S) The objectives of the SURF investigation were to achieve a capability to determine the radar cross section of metallic and coated re-entry vehicles taking into account variations in shape, particularly in the generally spherical termination of the conical portion of the body and the effect on radar cross section of slot and annular (ring) antennas. The study included a determination of the radar cross section of these bodies in tie plasma re-entry environment and a determination of methods for countering short pulse discrimination techniques. S

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) The groundwork for meeting these objectives was laid in studies under contracts AF 04 (694)-683 and AF 04 (694)-834 under which radar reflectivity of metallic bodies was investigated and the studies to determine the effect of coatings, perturbations of shape and the effect of antennas were initiated. Included in the Final Reports on these earlier phases (Goodrich et al, 1965 and 1967), was a "Handbook" of radar cross section formulas for computation of the reflectivity of the shapes which were studied in those periods. The present report supplements that "handbook" in Section VII. (U) Many of the radar cross section formulas which were constructed were checked with experimental measurements. The comparisons are described in Section 3. 2 and were undertaken as a result of recommendations made by the Aerospace Corporation in a Technical Discussion meeting, one of the several technical meetings held by SAMSO, Aerospace Corporation and Radiation Laboratory personnel to review the progress of the program. However, the most recent results of the SURF programs embodied in the radar cross section formulas of Section VII have not been compared with experimental data and it would be desirable to do so in order to obtain a figure of merit and a range of validity for them. (U) During the course of the SURF investigation, many computer programs were written for the machine calculation of the radar cross section of the various shapes which were studied. An independent programming effort was also undertaken at the Aerospace Corporation by Dr. Fred Meyers who produced programs for the radar cross section formulas of the first two "handbooks." However, it was believed to be desirable to have a very useful computational tool in the form of a general computer program for metallic and coated rotationally symmetric shapes. An attempt has been made to construct such a computer program and the results of this work are described UNCLASSIFIED

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F in Section VIII. The problem proved to be mucn more difficult than had been anticipated at the onset of the analysis. More time and effort have been put into it than had been initially planned and although at many times it seemed that the programming was completed, difficulties in numerical analysis continue to preclude a satisfactory solution to the problem. (S) The plasma re-entry study was started during the second phase of SURF under Contract AF 04 (694)-834. It has as its objective the formation of a theoretical foundation to explain the data which was being obtained during flights carried out under other Air Force programs and to facilitate the calculation of the radar cross section of plasma embedded bodies. During this year, an experimental program was set up here to investigate the soundness of the assumptions on which the theory was based. As will be seen from the disucssions in Section V, the course of the experiments, more questions were raised than were answered. It had been expected that this would be a continuing study but with the termination of this program, the solution of some of these problems will have to be deferred. If a resumption of this work is undertaken in the future, it would be desirable to have more empirical data on the nature of flow fields. (S) The short pulse study described in Section VI was based upon the-need to investigate the feasibility of countering short pulse discrimination techniques which might be used distinguish between the radar echoes of a re-entry vehicle and decoy bodies by scrutiny of their short pulse radar returns. A method for studying perfectly conducting bodies has been developed and is described in Section VI. It can be made applicable to coated bodies for which impedance boundary conditions hold. (U) A list of technical reports, in addition to the quarterly reports written under AF 04694-67-C-0055 is given in Section X. 3S R

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F II EXPERIMENTAL INVESTIGATION 2.1 Introduction and Review of Program (S) The experimental basis for the SURF program rests on two experimen tal techniques. One technique, radar backscatter measurements, is well-known and anechoic facilities for making these measurements were in operation at the Radiation Laboratory at the start of the investigation. The other technique, surface current measurements, was a relatively new technique. It was completely new in this application. A small facility has been constructed and operated before the SURF program began in order to test the feasibility of using this technique to obtain information which would be useful in devising methods for computing the radar cross section of extremely low cross section shapes. The technique was shown to be a valuable tool and upon the initiation of the SURF program, a working facility and special electromagnetic and electrostatic probes were designed and constructed so that it could be applied to models of the re-entry shape under investigation. The particular shape, of course, was that of the cone-sphere re-entry vehicle typified by the LORV series of missiles and perturbations of cone-sphere typified by the indented rear termination of the Mark-12 re-entry vehicle and by the antenna and rocket nozzle perturbations of the LORV and Mark-12 vehicles. It was necessary that the surface current measurement technique be applicable to both the metallic and the absorber coated models under study. (U) The surface current facility which evolved is described in Knott (1965). A brief summary of the experimental program is given in the following pages along with typical results for the increasingly complicated experimental situations which were studied. Detailed analysis of the experimental results are given in Section III of this report. For added coherence, the re-entry plasma experimental work is described in Section V in conjunction with the theory and analysis of the plasma embedded re-entry vehicle. SECRET SECRET

SEC RET THE UNIVERSITY OF MICHIGAN 8525 -1 -F (S) In retrospect we can enumerate items of special interest. For one, the "indentedness" of the base cannot be discerned from a flat back and this is because the radius of curvature immediately aft of the join, and not that of the indentation, is the most important contributor to the radar cross section. We also learned that in general sheets of absorbent material markedly decreased the effects from perturbed models. The absorbing material acts as a shield and hides the underlying model from the incident field. However, perturbations in the form of annular slots can be large and frequency dependent, because the impedance looking into a typical slot depends strongly on frequency. (U) Probably the most important thing we learned during the last year was the importance of maintaining sets of experimental measurements which can be readily compared with each other. This is not always easy to plan in the beginning of a long program because one cannot always see the direction the research is going to take. Early in the program we began measuring surface fields on objects and recorded the data by hand from observations of meter readings. We plotted the data manually. We soon discovered how to do this efficiently and during the second year made a great deal of measurements this way by the third year it was apparent that linear, not logarithmic, plots of amplitude vs distance along the surface were more convenient. We had been measuring amplitude in decibels, which is a logarthmic system, and a hand conversion was required to compare data. Therefore, early in the third year we designed and built a semi-automatic recording system, constructed to index and probe, in small steps. The system permitted the probe to stop wobbling or dancing after indexing and after a short interval of about 3 seconds, the pen on the recorder was commanded to drop down onto the re 5

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F corder chart paper to record a datum point. After the point had been recorded the system told the probe to index one more interval and the process was repeated. In addition, the recorded data was plotted linearly and not in decibels. The purpose was, of course, to provide plots that could be observed directly without the laborious hand conversion process from the logarthmic system. The system worked well but used a great deal of chart paper and later, to solve this problem, we halved the chart scale. This had the effect of putting the data on half as much paper and therefore consumed less space. The upshot of all these changes was that we had a series of hand-recorded data in decibels, some linear recordings at one distance scale and some linear recordings with half that distance scale. This made comparison of many sets of early and late measurements difficult. (U) These variations in the recording format show up in this report. The reader will observe as we progress through some experimental results that the scales are sometimes logarthmic and sometimes linear and that the distance scales along the surface are not always the same. This is because we are comparing data taken over many months of the contract period and have had some trouble comparing all the data on this basis. (S) In the remainder of the experimental section we will review the effects of coatings on cone spheres, work that was actually carried out late in 1966; the behavior of surface fields on coated cone spheres obliquely illuminated; show how coatings suppress the effects of perturbations on cone spheres, such as the annular tip antenna simulations and join antenna simulations; show that the slots in the absorber jacket of a cone can seriously perturb surface fields by exposure of the underlying structure and by the formation of surface discontinuities in the absorbent material itself. We will also show that the indentation of concave-capped models is not important if the object is coated __ 6 SECRET II I

SEC RET THE UNIVERSITY OF MICHIGAN 8525-1-F with absorber and, finally, will present surface field and backscatter data for flat-backed models which show that the radius of curvature at the join is a dominant influence on backscatter. 2.2 The Influence of Absorber on Cone Spheres (S) The experimental results previously obtained are documented in quarterly reports and in the final reports for Phase I and Phase II of SURF. In this section we intend only to hit highlights and to point out major influences of coatings on cone spheres. It will be recalled that several coatings were examined by means of transmission line experiments and some of these coatings were applied to cone spheres. Thin, heavy resonant coatings, flexible urethane foam coatings, a thick, heavy, lossy coating called "Eccosorb CR!? were used. In Fig. 2-1 is a photograph of a 3-inch diameter, 15o total angle cone sphere covered with Eccosorb CR material and is probably the best model that we produced with a coating on it. The observer can see that the tip is a sharp one, that the surface is smooth, and that auxiliary distance scales appear on the side of the model to augment reading out probe position. This coating had magnetic, as well as dielectric, loss and the coating was about 3/8?? thick. In Fig. 2-2 is a photograph of a model coated with LS-26 material. Observe that the coating was sewn on with thread and that we also have a scale on this model to indicate probe position. Note that near the tip the threads cause irregular, but unavoidable bumps on the surface; because of the flexible nature of the material and because of its thickness, the coating could not be wrapped very accurately around the tip. The model therefore is somewhat shorter than it would have been, had the coating been of uniform thickness all the way out to a sharp tip. This fact will be evident as we progress through the experimental plots. The coating on the model in Fig. 2-1 was not removable, but the coating jackets seen in Fig. 2-2 could be removed and re placed with other materials because of the sutures. 7 h e

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F FIG. 2-1: OUR BEST ABSORBER-CLAD CONE-SPHERE HAD A MACHINED OUTER SURFACE; CONE HALF-ANGLE IS 7.5~, BASE DIAMETER IS 3. 000", COATING THICKNESS IS 3/8". 8 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F FIG. 2-2: FLEXIBLE ABSORBERS HAD SUTURED SEAMS; CONE HALF-ANGLE IS 7.50, BASE DIAMETER IS 3.0001", COATING IS APPROXIMATELY 1/4"1 THICK. 9 UNCLASSIFIED

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F (S) In Fig. 2-3 we present the results of surface field measurements on three models for ka = 1, where a is the radius of metallic spherical base. The fields in the plot have been normalized with respect to the incident magnetic intensity, H. The underlying model in each case was a 3" diameter cone sphere with a 15 total angle at the apex. We present amplitude in db and thus have a logarithmic scale. The results for three coatings are shown and it will be observed that two of the coatings behave nearly identically and, in fact, the curves tracing out the response for LS-22 and LS-26 coatings are nearly the same as that for a base model: this is because the coating is nearly transparent at this frequency. The Eccosorb CR coating, on the other hand, had a slightly better effect but not much near the side. The field intensity for the Eccosorb CR coating extends much further to the left than does the LS coating because the Eccosorb coating was more perfect and formed a more perfect, hence longer, tip. There is a sharp dip shortly aft of the tip but the intensity builds up to a value of about 2 db. The amplitude remains constant at about 2 db until one reaches the join and at the very antipode the intensity is 0 db. The marked oscillations for the LS coatings are spaced very nearly a half wavelength apart. (S) Turning now to Fig. 2-4 we see the effect of much increased frequence upon the same models. Again we plot amplitude in db as a function of distance traversed along the surface aft from the tip and for this figure, ka = 5. Note that although the fields attained a maximum intensity of about 8 db as shown in Fig. 2-3, they fall to about - 5 db in Fig. 2-4. This is because the coatings were much more effective in attenuating the wave traveling along the conical portion. One can also see that the perturbations are much faster along the side, evidence that a higher frequency was being used. Note that once the join has been passed the fields decay quite rapidly until the anti pode reached. The Eccosorb absorber is much more efficient in suppressing the wave near the tip than the LS materials, but after about 2 wavelengths 10 SECRET

in IV I - -p 51 - m rr z (A Cr) -n rn (fD or 0 O o -4 0 cM -5 - Eccosorb Coating LS-22 Coating LS-26 Coating I I I I I I I I I I I I I co CA I 0 0 P11 z C) II r~f C/) C,) -n -10I -15 I Distance Along Surface Join FIG. 2-3: SURFACE FIELDS ON CONE-SPHERES SUPPRESSED BUT SLIGHTLY AT LOW FREQUENCIES BY COATINGS.

I 0 40 % 44% qkft**A 44% *% sf 14 "* *% N4%4** 0I. 4%I 4%I 44I IlL I -5 ~0 bf) 0 -lok Eccosorb Coating - - -LS-22 Coating I I I I I I co Ul ND Ul 1 -4 1 It Hz m~ z CA CA -n ni ~ ~ ~~~LS-26 Coating -20 I I I f &d I Distance Along Surface Join FIG. 2 -4: COATINGS HAVE A BETTER CHANCE OF REDUCING FIELD~ INTEN',,TTIES AT HIGHER FREQUENCIES:

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F aft of the tip the LS coatings proved to be more effective. It was once thought that the LS-22 coating was more efficient than the LS-26 but the data in 'Fig. 2-4 suggested this is not true. It may be possible that the technicians interchanged coatings and failed to note the change in the data. We note that both Figs. 2-3 and 2-4 are for nose-on incidence. (S) In Fig. 2-5 we see the effe.ct of changing the angle of incidence. There are four curves displayed here representing angles of incidence running from 0, which is nose-on, to 82. 5 which is broadside to the conical surface. The model in this case is coated with Eccosorb CR and ka = 5. Observe that we have a symmetrical pattern, with the tip lying near the left and right extremes of the figure and the antipode near the center. For nose-on incidence we would expect a symmetrical pattern and indeed we do get one as evidenced by the solid trace. If now the angle of incidence is changed to 7. 5 we see that the fields on the lit side (the left side) of the figure are more intense than on the shadowed side, which is the right side of the figure. If we change the angle of incidence to 37. 5~ the fields on the lit side increase even more, approximately 5 db above the nose-on case, while on the right side the fields are reduced of the order of 15 db. Finally, as we go out to 82. 5, the fields attain their maximum intensities on the lit side and their minimum on the shadowed side. There are small perturbations near the tip on the lit side which are about half a wavelength apart and they are evidence of waves traveling back and forth along the illuminated surface, despite the fact that the incident wave has no component in this direction. (S) Note that on the illuminated side for the 82. 5 incidence angle, the field intensity is nearly constant at a level of about + 3 db. Ordinarily if the model has not been coated the intensity would have been very close to 6 db. 1 3SECRET SECRET

U NCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 5 OL 0 0 bfM 0 -5 - -10 - 00 Incidence -__ 7.5 ~Incidence - _ _ 37. 5 Incidence —. --- 82. 5 Incidence -20 - f / / I I I I -25 - -30 Distance Along Surface FIG. 2-5: FOR OBLIQUE INCIDENCE, FIELD INTENSITIES ON THE LIT SIDE ARE MUCH STRONGER THAN ON THE SHADOWED SIDE. 14 UNCLASSIFIED

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F This suggests that the absorber is nominally at 3 db absorber. The data very graphically show that the absorber, although it may be a poor one for normal incidence, is a good one for attenuating waves traveling along the long slanted surface. One would expect that even though an absorber would be very poor, say only 2 or 3 db, it would be a reasonably good one if the body is a long one viewed nose-on. 2. 3 Coated Perturbed Models. (S) In this section we will show that if a smooth body, like a cone sphere, is perturbed by either tip antennas or antennas situated near the join, these can be large contributors to surface field perturbations and backscatter measurements. Then if these perturbed bodies are sheathed in an absorber jacket the perturbations will practically disappear and the cross sections of the bodies will be much reduced. Finally, we will show that if slots are cut in the coating just above the antenna simulations, that the perturbations will again arise and be relatively large contributors to the backscatter. Figures 2-6 and 2-7 are photographs of the perturbed models that we built in 1966 to aid surface field studies. The model in Fig. 2-6 is called LSP (Lucite Spacer, Point) while the model in Fig. 2-7 is called LSH (Lucite Spacer, Hemisphere). The antenna simulations were provided by lucite wafers, 1/4?? thick. These two models form the models for which a great deal of experimental work was done. (S) In Fig. 2-8 we present a composite trace of the surface field intensities for the two models and we compare them with a plain cone-sphere for which ka = 5. Observe that we again plot amplitude in db as a function of distance aft from the tip. The tip lies on the left side of the figure, the antipode lies at the right, and the join is clearly marked by a dashed line. The solid trace, the behavior of the surface fields on a plain cone-sphere, be gins near 0 db at the tip and rises gradually to about 5 db two wavelengths aft, 15 SECRET

FJ(;. 21-6: MODEL LSP IhAS — A LUCITF SPACERl NIEAR TillF TPm, ()'fEBFIN wTSL 41lif V F 4~. ISA7. S hAL ANGL BXSE DIVMETER N TL~~

c: z -"CO -n rr -1 -'- i ". il -I'T I.ed n4 > Z FIG. 2-7: MODEL LSH HAS A LUCITE SPACER JUST BEHIND THE JOIN, ()THFERWISE IS A CONE-SPHERE OF THE SAME SIZE AS MODEL LSP.

- 10 I!. I -3 wr,r\ \ -5 /! z C) rr C,) Cl) '1 ni /! I I I I oa ~00 bD FC4 1 I I -0' — 5 I V! I I 'i 1 1 -Plain Cone-Sphere z Co4 w en t\^H Ul I A b 1 Im 0 r4 To 0 zf P-4 a z C) 'ni - -- Model LSP --- Model LSH ka = 5.0 -10 Distance Along Surface FIG. 2-8: THE DISCONTINUITIES OF MODELS LSP AND LSH HAVE DIFFERENT EFFECTS Z ON THE SURFACE FIELDS.

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F and finally near the join attains a value of about 6 db. If we now examine the trace comprized of the long dashes, we see that the fields begin at the tip slightly below that of the plain cone-sphere but have a series of very strong oscillations amounting to more than 10 db in amplitude. These oscillations are due to the spacer near the tip and once the spacer has been traversed, the trace is much like that of an ordinary cone-sphere. We conclude that the effects of the Lucite Spacer Point are felt only forward of the spacer and not aft. Finally, considering the trace comprised of the small short dashes, we see that the intensity begins again at 0 db and builds up to about 6 db near the join, but that the intensity has a series of quite regular perturbations amounting to about 2 db in amplitude. Since the spacer for this model was near the join and since the perturbations are largely along the conical surface, we can see that the LSH model perturbs the entire surface field structure. Near the spacer we can see the strong rise in intensity and the fields around the back are not the same as they were for a plane cone sphere. Bearing these characteristics in mind we now turn to Fig. 2-9 through 2-12 which illustrate the effects of covering the model with an absorber layer. (S) In Fig. 2-9 we plot the results for ka = 1. Note that we plot the linear field intensity normalized to the incident field, not decibels, as a function of distance along the surface. All three models show very nearly the same behavior and we are led to conclude that the coating shields the spacers from the incident field. The fields on model LSP show a very small dip slightly inward from the tip and the fields from model LSH show another small dip very near the join. Aside from these tiny perturbations, the structures along all three models are practically identical. Figure 2-10 shows that for a ka = 3 the model LSH produces a phase inversion of the SECRET SECRET

2. r a - I I I: I I. I -.. I I. I.. I I L.# -1 IT- HHHI+i tFFFFV'pi liT11 - pi I 11 l - I r' I I I I I I I I I I I I I I I 1 1 T I I I I I I I I T I I I I I T.. I I I I I I V- 1 I.. - I I I I.,,. I I... - Milililill T I-..!, I... I i I i,,I7, I.. I I 2. C) z~ C) -n ni 1. (HI IHI 0 T TIl ii - T rd I L I T ~ ~i ~PanCoe perL 'iii~ Moel LP T 4 ij I Fr, Moel LSHI tit th0K I —3 -, --- —-- - z C,) z U) '1 1.1 0.. I I LLL I I tttl -t1:. I I I I I I I i trITITY. I i IITT" I I'll I I I I I F-17 W ~ I -AI-+-1.~ vi -: 1 1 1 f 1 1 i 1 1 0 1 1 1 1 1 1 1 1 1: 1 a 1 1 1 4 1 0 1 1 i 1 1...... T -r nA - i:-' 4 ir ' I l I+ 4,1 '4 4-.R UILLU!I ILLL ffH iLiIFT 3___6_____ I I a v Distance Along Surface FIG. 2-9: EVEN FOR LOW FREQUENCIES., THE DISCONTINUITIES OF MODELS LSH AND LSP ARE HIDDEN WITH AN ABSORBER SHEATH, ka = 1. 1.

2. 51 T I 1 i I I T I! 1 i. I '1 7 - 0 1f iI I I I i 7 1 1 r 1!- F I I 1 1 I I I I I! I I- -..... I: II. I I;: I C) Cl) Cl) -n ni 1. IHI 0 Iz 01 -0 Plain Cone-Sphere._.._.... Model LISP.....-.. Model L SH _ ka = 3. 0.......... 5........ - All Models Coated with LS-26 *4 1 1111 II1 IL12 4I 12 I __ _ __ _ __ _ ___ 36___ _ __ _ 72~ (17! 0-4 -e4 m Pd CO) 00 P-4 C.n N. .q C.Tl I 0. -A I Ind 0 It CA -n ni HzT 1. 0. 0 -z I DISTANCE ALONG SURFACE FIG. 2-10: FOR ka = 3. 0, THE PERTURBING EFFECTS OF MODELS LSP AND LSH ARE NOT COMPLETELY -SHIELDED.

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F surface fields; i.e., the reflections seem to peak up where previously there were nulls and to have nulls where there were previously peaks. Note that beyond the Lucite spacer the fields for the plain cone-sphere and for model LSP are practically identical. For this ka we see the strongest effect for the LSH model and the fields around the rear are some 2 db lower than for model LSP or for the plain cone-sphere. (S) In Fig. 2-11 the fields are plotted for ka = 5. Notice that model LSP shows some perturbations but that the plain cone-sphere and model LSH also have these perturbations. We ascribe the similarity to the roughness of the tip because of the way the coating was sewn on. The fields for all three models decay to about 0. 6 near the join and between the join and the tip the fields are about the same for all three models. Around the back there are small differences but again we ascribe these to experimental errors. In Fig. 2-12 appear the results taken for the highest ka (ka = 8). Again there are perturbations on both the LSH and LSP models due to irregularities in the coating near the tip and all three traces are very nearly coincident from tip to antipode. In reviewing Figs. 2-9 through 2-12 we conclude that the coating markedly suppresses the effects of the perturbations of the underlying model. If the model is long enough in terms of wavelengths, surface field intensities near the join will be very much those that would be found on an infinitely long cone, as will be brought out in later sections of this report. (S) The above results show that an absorbent coating can reduce the surface fields, and therefore the backscatter, from long targets provided the coating covers the entire model. But if there are antennas on the object which must radiate, the coating will interfere or reduce the efficiency of that radiation. We therefore undertook to expose the underlying perturbations (the SECRET SECRET

U N CLASSFE THE UNIVERSITY OF MICHIGAN 8525-1-F co 4. - -1, +-7 -_-_ - -,,. i. -1 -— l - I (1r... I ( IP N ISSWS i 3AI I 1 LI I -4- - 777 - - - - CI -1 -- ---- - -~ -1-1-4 - a)------- ---- - T II CI) bf) 0 P-4 0 4J LO LO CN113 'I -fl_ 23 UNC~LASSFE

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 00 0 qp NOISSIWSN'VNI gAlivi9b a i 0 - - O o * I — L i.., -i:1~ I I= -. > 141 * - -* 4-* -- / P- f Cl)d * ~ ~ g ~ g o 0. ~ -r- - 04 -- -4 —tI v14 C) 0 Cl -r4 Ss mo U 0 0 r-4 C) *-1 m rA 0 z UIz CO 0 E-4 o f g0 CMi C —4 CMI 0 0 1-1 '-4 -la0 2UNCLASSIFIED UNCLASSIFIED

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F Lucite spacers), by cutting slots in the absorber coating directly above them. Two jackets of LS-26 were fabricated, one with the slot aligned at a point directly above the point where the spacer would be for the LSP model, and one having a slot in the coating directly above the point where the spacer for the LSH model would be. This is illustrated graphically in Fig. 2-13. (S) To fully assess the effects of model and coating combinations, we alternately left the slots air-filled or we loaded them with Lucite. We inserted model LSP, as well as the plain cone-sphere, into the upper left hand jacket of Fig. 2-13 and used the air and Lucite loading in the slot. This had the effect of producing four different measurement situations. We also inserted model LSH and the plain cone-sphere in the absorber sheath shown in the upper right hand figure of Fig. 2-13. With the slot loaded with air or Lucite we produced four more model combinations. The total effect was 8 models which we examined for four ka values: we will not present the results for the ka values but instead have selected typical ones for ka's of 3, 5, and 8. (S) Turning now to Fig. 2-14, there are four traces showing effects of the surface fields for the four models mentioned above for ka = 3. These curves were traced directly from the raw data and the quantity displayed is the amplitude of the surface field intensity as a function of distance along the surface, not db. Notice that the air and Lucite perturb the fields about equally, independent of which model is used inside the coating. On the other hand, the reader will note that the perturbations are much stronger if the underlying model is the LSP model and the perturbations seem to be largely on the forward side of the spacer. Aft of the spacer the field intensities are practically identical for all four model configurations, and although around the back there seems to be one trace that is much different from the other three, we ascribe it to experimental error. 25 _ SECRET

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F Absorbent Shea Model LSP Plain Cone-Sph Model LSH FIG. 2-13: VARIOUS COMBINATIONS OF MODELS AND SLOTTED ABSORBER SHEATHS PRODUCED EIGHT SEPARATE MEASUREMENT SITUATIONS. 26 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN 2.1 - I 0 - - - - w E I - SlotI' I I " I;., I I, I 1. 7 I T i i j I T i I T i I I i i I! 1 - r - I I I I I - F 1i i I I - I I I I I Surface Fields on Models with Slotted Coating I: I i I - 1. 1. 0. '- - ' ": i i ' ' ii, ' ' ' Coating: LS-26 ' '! ~ ' 2 Incidence: Nose-on 'i: i '/'Z: -Slot Air-Filled, Model Plain ' k = \ i: --- ~ ------— Slot Lucite-Filled, Model Plain '/ \ -l -- --- Slot Air-Filled, Model LSP - - --,,-.......................ur.ac e -....._..___.. __.... __....... I FIG. 2-14: A FORWARD SLOT IN THE COATING AFFECTS ONLY THE INTENSITIES IN FRONT OF THE SLOT. 27 UNCLASSIFIED

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F (S) Turning now to Fig. 2-15, which is for the same ka as Fig. 2-14, we have shown model LSH and a plain cone-sphere inside the absorber coating. Again, notice that the air and -Lucite seem to perturb the fields about the same but that if the underlying model is the LSH model, the perturbations are much stronger than if the model underneath is a plain one. Again, we see a phase inversion due to the presence of the LSH model, similar to that seen in an earlier figure. Observe that the fields around the back of the model are markedly different and that one would therefore have to know the impedance looking into the slot in order to predict the fields around the back. (S) In passing to higher ka, Fig. 2-16, we have the results for a ka = 5. For this figure we have again studied the model LSP where the slot in the coating beam forward, and this time the air seems to perturb the fields more than the Lucite. In fact, the greatest perturbation occurs when the model underneath was model LSP and when the slot in the coating was filled with Lucite. The amplitude of the fields along the conical part between the spacer and the join decays in nearly the same behavior but with slightly different phases. Around the back of the model, from the join to the antipode, there is hardly any difference between the four models. It appears that some perturbing effects can be obtained even with a plain cone-sphere underneath, but if the underlying model is LSP, then the effects are much greater and they always appear forward of the spacer, and not aft. In Fig. 2-17 are displayed the effects of inserting model LSH inside the sheath. From the tip back very nearly to the join the fields are nearly the same, which we ascribe to the fact that the absorber is attenuating the incident wave much more than when the spacer is near the front. Careful examination shows the model LSH perturbs the field slightly more, especially aft of the spacer. SECRET SECRET

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN 2. 2.( 1.5 IH( IHI 1. 0. 5 FIG. 2-15: A REAR SLOT AFFECTS THE FIELD INTENSITIES ON BOTH SIDES OF THE SLOT. ka = 3, Coating is LS-26. 29 UNCLASSIFIED

UNCLASI)iE THE UNIVERSITY OF MICHIGAN 8525-1-F 2.; 2.1 1. IHI IHI 0 1. ITT 4 Surf ace Fields on Models ie with Slotted Coating 8 -7 Slot Incidence: No se -on Tip p0 - - So ArfildMde S o5- ~ 3 — Slot Luite-fill ed Model PlSn V T_ H <I 7 zon 76 1< Slo2 Air-2iDistancedeloLSP o 1w 2 1f iH~b > LI;2; h8 61 0. FIG. 2-16: STRONG PERTURBATIONS OCCUR IN FRONT OF A FORWARD SLOT FOR ka = 5.0. Coating: LS-26. 30 UNCLASIFIE

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN 2. 2. 1. 5 r <I......... Surface Fields on Models __ with Slotted Coating -..... -. - -. 0: l0... Incidence: Nose-on..; ' - Tip. 4...,.....- - - -............... t- - --- ----- - -8 --- ------- - - --- —..-.-6 — -..r.:-- Slot Air-filled, Model Plain j 5 --- Slot Lucite-filled, Model Plain v G -- - -- Slot Air-filled, Model LSH __ ____ ---- Slot Lucite-filled, Model LSH - - -,...... Distance Along Surface.., 0 _...1......... -....... 1. IHI |HJ 0. FIG. 2-17: A REAR SLOT SEEMS TO HAVE LESS PERTURBING EFFECT FOR ka = 5.0 THAN ka = 3.0. Coating: LS-26. 31 UNCLASSIFIED

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F (S) Finally, turning to Fig. 2-18 we caution the reader that the scale has been expanded because the field intensities were very small. The frequency for this figure was such that ka = 8 and in this figure model LSH was slipped inside the sheath. Note that the fields are not perturbed on the front nor around the back but that a small local perturbation is centered near the slot. The perturbation seems to be about the same whether the model underneath it is LSH or a plain one, and it seems to be about as strong whether the slot is filled with air or filled with Lucite. 2. 4 Indented Base Models (S) Early in the contract we constructed a series of indented base models and measured the surface fields on them. Results of these measurements show that the surface fields were more or less independent of the depth of indentation on the back. In Fig. 2-19 is a sketch of thie geometry of the terminating base and it can be seen that the base was formed from two circles, one near the join and another whose center lay on the axis of the model. Thes circles were tangent to each other and to the sides of the cone. The radius C was selected to give the variation in the depth of indentation while the radius just after the join was held fixed at 0. 553". These models were given the names ID-1, ID-2 and ID-3 with the depth of indentation increasing with increasing model number. The models were coated with an absorbent jacket, as shown in Fig. 2-20, and because of the geometry of the base we found it difficult to glue the absorbent material onto the indentation. We therefore stretched the absorbing material directly across the back forming a void between the material itself and tie surface at the indentation. We measured these models for four values of ka, but will present the results for only three ka's. 32

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F I I 1. 0. t I l t I IHI IHol \ I I FIG. 2-18: FOR ka = 8.0, A REAR SLOT PRODUCED RELATIVELY LOCAL PERTURBATIONS. Coating: LS-26. 33 UNCLASSIFIED

UNCL~ASSFE THE UNIVERSITY OF 8525-1-F MICHIGAN / Hi Q PC1 0 1L4 VI) z vi -s 0 IQ la4 N MM N I a 34 UNCL~ASSFE

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F COATING VOID VOID I MICHIGAN ~-ID MODEL DENTED BASE NE FIG. 2-20: THE ABSORBER JACKET WAS STRETCHED ACROSS THE INDENTED BASE AND CREATED A VOID. 35 UNCLASSIFIED

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F (U) The results of the measurements are summarized in Fig. 2-21. There are three families of curves displayed, one for each ka value used: these were ka = 3, 5 and 8. The curves for each of the ID models are clearly identified as solid, alternate solid and dashed, and dashed traces. The coating was LS26 and the angle of incidence was held at nose-on. (S) Notice that for all three values of ka the field at the tip is the same. Considering firstly the uppermost curve, ka = 3, the three traces are nearly coincident. The incident field begins near unity, builds up to a value of 1. 8, and then is nearly constant aft to the join. A series of small perturbations are spaced about half a wavelength apart and the intensities drop off beyond the join and around the back. For ka = 5, the center group of traces, the fields barely reach a value of 1.4 about a wavelength behind the tip and at the join they attain the value of about 0. 6. For ka = 8, the lower group of traces, the fields drop off exponentially attaining a value of about 0.4 at the join, and the traces are substantially coincident. (S) From Fig. 2-21 we can see two things. One is that, independent of ka, surface fields on all three models behave practically identically. Secondly, as ka increases the mean value of the intensity greatly decreases, showing that the absorber was more effective at higher frequencies. Since the three models had varying degrees of indentation and since the families of curves are very nearly coincident,we conclude that the nature of indentation is not important. We will see in a later presentation of data that it is not the depth of indentation, but the radius of the curvature near the join, that influences the backscatter. 2. 5 Flat-Based Models (S) In the foregoing description, we discovered that the radius of curvature near the join did not influence the measurements because that radius was 36

2. 5 E l I I I I I I I I I I I I I I I I I I I I I I f I I I I I I I I I I I I I I I T I I I I I I I I I I I I..II I I I I I I I I I I I I I1 IIIIIII II IIII 1111 III II I 1 1111111111 111111II 111111 I III I 1 I 1 I 1 I 1 I 1 I 1 t-tt 1 1 1 t 1 t t 1 I 1 I f t I 1 I 1 I 1 I 1 I 1 I 1 I I I 1 1!-+-H 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i i —H Surface Fields onI Coated ID Models T 2 -0O -I 'Coating: LS-26 oGd, '~jtiIl i i iiiiiiiiiiiiiii iliil i i i I111iiii i ii 'i i I4,111 I 1 1 1 11! iii II ii il i 111 111 ft I I IIIII!IIII 1 1!IIIIIIIII I I - I I I I I I I I I I 1 211 1 1 1111 I I II I II I 1 I~ I II ~ idence::Nose-on U-~-I-4-4 —I —f4 --- —-4-f ---I II I I I I III It11t 11 It — ----- -i — 111 f 1 H-ti 1 4 ---t —1 ----t1-I — I-+-+-4 —+-+-I-+A4 I I 1 I 1 1 I1 I 1 1, 11!1~ ~ ~ I I IIlot CA Clf) ~n 11111111111"IFT 1. -i f; rl l i l l 1 1 1 1!!! 1 I t* 501 1 1 1 1 I 1 1 1 1 1 1 1! m l 1 1 1 1 1 1 1 1 1M 1 1 I I I 1 1 1, I I I H H i H i i i i! I i i!z lw I.,, -4 1 1 1 1 1 1 1 I I I I I-j " I I I I i IHI H I 0 lJH-H 1- -Jfl 'o Oft uj > I I I I I P&I'm I I I I I I I., I I I I I I- 1. Til~+tTiII ID-2 1 1 1 1 1 M i l l U — i i mo ui l 4414 ID-3 - - - - 01 01 tyl z (12 H 0 P) C) z C~ -n mD l. t I I I $4L111111111 - ++++1 1 1 1 1 1 '-H-H+14+44-44 1 1 1 1 144+4+i i i i i! i i i i i i i 11; i i! i i i i i i I i i i i i IM I I! I I! I I I 1.0 [I II IIII I 111 K WiI.i III IIII II I II III it I 1~ 1 i.. 4..1t.. II I II II I IIII 1 1 I IIt II I I II I III; I I III I I fI l I 11 2 I IIl l i-H I —I I I I II I I I I I I 4-44-4-I4-4-k-I-A-I-4-I-I — l —4-141 t TI 1-J44 11 - II-4-Ld!42L2L1diLLLLLLLLLLLL llff1 LI -Join - - -. I~.~.. II II1 111 ka11115.0 Tipnge I I I 0.1,- I jill I IiIlIlilt i l'' I ''Igi 11= I I ItIII I I I1I I 1 liii ITTIIILI IIHI I~IIN'1 4I1Il J1 i v I I I. I. illH I 1 rTI I II II 2j I I IIIIIII I'' I I I I i I I I I:~ I I4T II III III I 1- -.1 IIIIIIIII!I- I 1111 II 1 111 1 [ I I I I I I T-1 - FT -F-F = 8. 0 — I I I I I I F 11 W-!Ad I I ll il lI 1 1 1 1 1 1 1 11 11 1 1 1 11 1 11 1,1 1 11 1,I ' I 11141 1111111 1111 1 1 11111 1 I ' i I 1 l l 1 1 1 1.1 1 1; — I I I I I I UI I I I I I I I1! I I!I IIII II II I I I IlI I f I11 I I I I I I I I1 I I I I I I I I I I I I I I I I 1 I I i I H i i i "! i i i I i i i i ii i i i I F!k''V'L&# IN VIA I 1 11 11 Il u i I I I II IQ1 11 1 1 111111 II 11 111 1..I.. i It-, -I. i 61. 41111 l111~1111l~1I1l1llI Distance Along Surface - 2j I II II II II III II II 1 1 1 1 I I I I I I III III ljjtjj II] I. I. I,,. III III III III II 11-1 III 41TI i ii I t 41 - I?01 I qWrl41 2' 'fl'fjtj tflt f ~ f i l~- I 11 1 11 1 Il.2' 1 II I & I I IIIIII I I II I II7 21' 1 50 U FIG. 2 -2 1: MEASUREMENTS OF SURFACE FIELDS ON ABSORBER-CLAD INDENTED-BASE MODELS SHOW THE DEPTH OF INDENTATION HAS LITTLE OR NO EFFECT.

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F not changed. The depth of indentation was the only variable and we found that the indentation is not important. Therefore, we constructed a series of flatbacked models whose radius of curvature near the join varied from model to model. These models are shown in the photograph of Fig. 2-22. Notice that two conical front portions were used in conjunction with any of four caps to produce the four models. These models were labelled the FB series, FB standing for flat-backed. In Fig. 2-23 is a sketch of the models and we prudently selected the radius of the base of the cone to be 1. 878" so that ka would be precisely the frequency expressed in GHz. Depending on the radius of curvature just after the join, the maximum model diameter could exceed 1. 878" by a small amount, but this was of the order of half percent, at most. The small radius b was selected to provide ratios of b/a to 0. 1, 0.2, 0. 3 and 0. 4. (S) We measured the surface fields on the flat-based models for four values of ka, but will present only the data for ka's of 3 and 8 for illustration. The results for all four models are summarized on a single plot (Fig. 2-24) for ka = 3. Observe that the smallest radius of curvature, that of FB-1, gives rise to the largest perturbation on the side of the cone, and that the model with the largest radius of curvature, FB-4, has the smallest perturbation. In general the normalized fields attain a value of about 2. 0 near the join, and the curve for model FB-4 is displaced downwards somewhat from the other three. This, we believe, is due to experimental error in the calibration of the data. Figure 2-25 shows the surface fields for these four models for ka = 8. Again, the strongest perturbation occurs for FB-1, and the slightest perturbation for FB-4. It seems that the strong effects of the smallest radius of curvature continue for relatively high frequencies. (S) In examining Figs. 2-24 and 2-25, one notices that the curves for the larger radius of curvature extend farther to the right than the ones for the smaller radii. This is because the distance around the back of these models 38 __________________SECRET 38 ________________________ SEC RET

C CUn > FIG. 2-22: THE FB MODELS CAN BE ASSEMBLED FROM EITHER OF TWO Z IDENTICAL CONES AND ANY OF FOUR DIFFERENT REAR CAPS.

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN b MOD CL FB-I FB-Z F FB-3 FB -4 b, INCHES 0.188 Ob ' O.376 O.Z 0.5 7.T51 FIG. 2-23: THE FLAT BASED MODELS WERE BASICALLY 9 (HALF-ANGLE) CONES. Sketch is not to scale. 40 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 2.5 2.0 1. 5 IHI IHol 1.0 0. 5 0:I::!:'::: —"B l1t 12 l 0 i 1-74T4Tim FB-1 iFB -2 li 11 i2 t II 1 It i I + ------- FB-1 16 t I III Distance Along Surface FIG. 2-24: SURFACE FIELDS SHOW STRONGER PERTURBATIONS FOR SMALLER RADII OF CURVATURE FOR ka = 3.0. 41 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 2. 5 2.0 LT 111111 1111 Hil 1111111111111 HillT111l I~~ ~ ~ I III II T 1 11 11 11.... - 11111 I r WI Vrrr T Vl r UMY ILk k. u 1 ^ l I T1 1 1 1 1 1 1 I L I H * i i H i, - - - - - * " i i - * - IF - -,I,,,, - TT 11F113r ~.t...I..a... I::::: lllii 11 I - - i I m I I I I I I I I I I I 0. I Il- I ----— ITFB-4 III I 0.0 - IIIIIIOIA 1111T 1111111111111 1E 2!171 Z 1 11 Distance Along Surface FIG. 2-25 PERTURBATIONS IN SURFACE FIELDS PERSIST FOR ka = 8.0. 42 UNCLASSIFIED U r11-TT -— 111T T 11111111t 11l111111L

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F was larger, and therefore, extends farther to the right by virtue of the increased radius of curvature. As the base of the model becomes flatter by virtue of sharper corners, the reflections due to the corner radii will be greater and the radar backscatter will be greater. As a confirmation of this we turn to Fig. 2-26. Here is plotted the radar cross section of the four FB models and we display, in addition, the results of a plain cone-sphere for comparison. The measurements were all made for nose-on incidence and although complete azimuthal scattering patterns were obtained, only the nose-on aspect was read from the patterns and plotted. Observe in Fig. 2-26 that radar cross section in db relative to a square meter, is plotted and if we are to strike averages through these curves we must do so cautiously because of the logarthmic scale. (S) The data span quite a large difference in ka, running from about 1 on the left to 6 on the right. This required the use of both L-band and an S-band scattering systems. Note that there is a relatively shallow null for FB-1 and that this null increases in depth as we go to models FB-2, 3 and 4, and reach a maximum depth with model FB-4. Observe, also, that the position of the null has a tendency to shift gradually to the left, toward lower ka. If one were to carefully strike average values through each of these four patterns, one would see that the average cross section is creeping upwards slightly with decreasing radius. Notice that the average return is from 2 to 10 db higher than for a cone sphere, depending on the sharpness of the radius and upon ka. 43

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F in LV FB-1 - FB-2 m V, *< O O - Cone-Sphere -10 - I -20 - v\ v FB-4 II v1 yr -30 I I I I I I 1 2 3 ka 4 5 6 FIG. 2-26: NOSE-ON BACKSCATTER MEASUREMENTS SEEM TO QUALITATIVELY VERIFY PREDICTIONS BASED ON SURFACE FIELD DATA: Experiment datum points have been deleted and replaced with smooth curves. 44 U NCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F In INTERPRETATION OF EXPERIMENTAL DATA 3. 1 Introduction (U) This section contains an interpretation of the experimental data and gives comparisons between the computed (theoretical) and measured (experimental) values for the surface current and for the backscattering cross section for a variety of cone-sphere-like objects. Some of this interpretation and comparison has been grouped under the heading, "Agreement Item". This is the term given to work initiated and carried out as a result of recommendations made by Aerospace Corporation personnel at Technical Discussion meetings and agreed to by SAMSO and Radiation Laboratory representatives. The Agreement Item, Section 3.2 of this report, describes comparison between computed and measured values for the cone-sphere with a concavity at the back of the sphere (the ID models), the cone-sphere with a dielectric insert representing a ring antenna near the tip of the cone (the LSP models) and coated cone-spheres. In Section 3.3 is a further discussion of the backscattering behavior of the LSP type of vehicle. (U) A study had been made of the effect on radar cross section of variation of the radius of curvature near the cone-sphere-join. Experimental data was obtained on metallic models with different radii of curvature. The conical portion of the models was terminated with a flat back (the FB models). An interpretation of the data on the FB models is given in Section 3.4. (U) It is important to know the limitations of the theoretical approach. Although effective formulas for computation have been derived and comparisons with measured data show excellent agreement between theory and experiment, the formulas cannot be applied to all situations without understanding their limitations. Section 3. 5 is a discussion of such limitations which we call a "failure" of a scattering estimate. 45 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) The experimental investigation which accompanied the study of the radar cross section of a re-entry vehicle in a plasma environment is reported in Section IV which also includes a report on the theoretical investigations of the re-entry plasma. (U) The formulas for the computation of radar cross section which were developed under SURF III are given in Handbook form in Section VI although some of them appear prior to Section VI in the various technical discussions, e.g., in the report of the Agreement Item. 3.2 Agreement Item (Task 3.1.1) 3.2.1 Introduction (U) At the SURF Technical Direction Meeting held on 14 and 15 March 1967, it was agreed that a comparison would be made between predicted values for the backscattering cross section based on formulas originated under the SURF program and the results of experimental measurements that we have carried out. The bodies listed for this study were: I A conducting shape (such as a cone-sphere) (a) with concavity (b) with ring-type antenna II A coated shape (such as a cone-sphere) (a) with lossless coating (b) with lossy coating and it was requested that the comparison be performed for all angles of incidence out to the specular glint, with (preferably) the same ka value (or frequency) employed in each case. _______46 UNCLASSIFIED

UN CLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) Implicit in this agreement was that we would use experimental data already acquired (and reported) under the SURF program. As of March 1967 data was available for a bare (metallic) cone-sphere at 10 values of ka (15 patterns); the bare ID-1 model (indented base) at 4 values of ka; the bare LSP model (Lucite spacer point, simulating a nose-tip antenna) at four values of ka, and models coated with LS-22, LS-24, or LS-26 at seven values of ka each. In every case the polarization used was horizontal, * and the half-cone angle, a, was 7.50. Table Il-1 gives a listing of the available data, and shows the variety of shapes and parameters for which the theoretical comparison could be made. (U) In the following we present comparisons of the theoretically-predicted cross sections with a selection of the measured patterns, together with certain additional curves aimed at indicating the degree of correspondence of theory with experiment in those cases where no complete comparison is included. Although there was no express requirement for a consideration of the bare metallic body per se, it was felt desirable to make the comparison for this body also. In all cases the experimental curves are direct tracings of the measured pen recordings with the vertical scale expressed in db relative to a square wavelength. To convert this to dbsm it is merely necessary to add a constant to the ordinate, which constant depends of the frequency used. For four typical frequencies the constants are given on page 50. This negated the original intention that the theoretical comparison be carried out for vertical polarization, and because of the relatively low and irregular nature of the backscattering cross section for horizontal polarization in the aspect range between the backward cone and the specular glint, the resulting tests of the theory were more stringent than they would otherwise have been. 47 - UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN TABLE III-l: Available Backscatter Patterns Base Cone-Sphere ka Freq. (GHz) Radius, a(inches) Pattern No. 2.976 2.53 2.21 3905 2.976 2.53 2.21 3970 3.010 3.77 1.50 3946 3.010 3.77 1.50 3948 3.494 2.97 2.21 4086 3.965 3.37 2.21 3898 4.506 3.83 2.21 3903 4.506 3.83 2.21 3904 4.655 5.83 1.50 3944 5.212 4.43 2.21 4090 6.318 5.37 2.21 4088 6.318 5.37 2.21 4089 6.732 8.43 1.50 3942 6.741 5.73 2.21 3891 6.741 5.73 2.21 3892 Base ID-1 ka Freq. (GHz) Radius, a(inches) Pattern No. 2.976 2.53 2.21 3969 3.965 3.37 2.21 3978 4.5059 3.83 2.21 3980 6.741 5.73 2.21 3963 48 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F Base LSP Model ka Freq. (GHz) Radius, a(inches) Pattern No. 2.976 2.53 2.21 3967 3.965 3.37 2.21 3976 4.5059 3.83 2.21 3983 6.741 5.73 2.21 3965 Coated Cone-Spheres 49 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN Frequency (GHz) Additive Constant (db) 2.53 -18.5 3.37 -21.0 3.83 -22.1 5.73 -25.6 3.2.2 Bare Cone-Sphere (U) For a metallic cone-sphere of radius a and half-angle a viewed at an angle 0 to nose-on with horizontal polarization, the expression for the theoretically predicted backscattering cross section given in Goodrich et al, (1967) is: [A] within the backward cone (0 < 0 < a): a 1 X2 X7 A S1 +S2 +S3 (3. 1) where S1 is the tip contribution 2 -i tan a -2ika cosec a cos 0 S S'tip 4 (1 - sin2 2 3/2 e S1 =i4(1-sin 0sec a) (3.2) S2 is the join contribution i 2 -2ika sin a cos e S S = o-u a J (2 ka cos a sin )e 2 join 4 o and S3 is related to the sphere creeping wave contribution __ 50 UNCLASSIFIED (3.3) -

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN I 8525-1-F k1 4/3 ir /3 ei~r/3 ir/3 S =2/ er/31+ 2 60/3( -ier/63 (ka13 exp irka-e 7r/31 -2 1 2a1/3 3 +9)22/3 1 i /6r 3 9) (1 (3 - e 60 - 9) (3 (31_9.4) in which Ai(-x) is the Airy function with 11 = 1. 018793 and Ai(-131) = 0. 5356567. The phase associated with each term in (3. 1) is that appropriate to an origin at the shadow boundary. (U) The contribution S3 is related to S as follows: 3 cw S 'yS 3 cw (3.5) where y is an enhancement factor. Based on surface field measurements, an empirical curve showing the dependence of y on ka for a = 7.5 has been given by Senior and Zukowski (1965, Fig. 2-10). A theoretical (asymptotic) approximation to -y, is - / 1 +| A/ |1+ 1 (a / 3 2 2 -i7r/3 'Y 2 1+ Ai(-x) 1 + -- a e + 0 2/3 /3 c2e i /3Ai(/3 e1 ka 1/3 -i/6 ka, a e /Ai(-j3) - exp - -2 a — e +/_) 1 (3.6) (Hong and Weston, 1965), and it will be noticed that in this approximation y is complex, albeit with only small imaginary part (phase of order 50). Eq. (3.6) fails when ka is amll, a fact which is evident from the comparison given in Fig. 3-1. Nevertheless, for all ka > 3 it may be adequate (and is computationally covenient) to use Eq. (3. 6) in the theoreticaly prescription of the scattering. 51 UNCLASSIFIED

1.6 Approximation (Eq. 3. 1. C) z rr bU (f) t,.3 -n mD Il4 1.2 1.0 Empirical - Z PC) C4} r^ -- 10 n l H 10 C( 10 O z P-4 z.... a I I I I I 0 FIG. 3-1:! 2 2 4 4 I I. 8 8 ka 6 MODULUS OF CREEPING ENHANCEMENT FACTOR FOR a = 7.50: Empirical (Based on Surface Field Data) and Asymptotic Approximation (Eq. 3. 6).

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) To obtain an appreciation of the magnitude of the various contributors to the scattering, we show in Fig. 3-2 a plot of Sti |, | Sj i and S | for J tipJ, J Sjoinj a cwf a = 7. 5 and 0 = 0. It will be observed that S | is somewhat greater than Sjoin throughout the range of ka considered, and that by comparison with these two contributors, JStip is negligible. The theoretical nose-on cross sections computed from Eq. (3.1) using the empirical and asymptotic enhancement factors for the creeping wave contribution are plotted as functions of ka in Figs. 3-3 and 3-4. The experimental values are those of Blore (1964), together with the one derived from the measured patterns listed in Table ILI-1. The agreement is excellent, bearing in mind the scatter of the experimental data particularly for the larger ka values and for ka < 3 (say) the superiority of the empirical enhancement factor over its asymptotic approximation (3.6) is clearly evident from Fig. 3-4. [B] between the backward cone and the specular flash (ac < < < r/2 - a): 2 1 S (3. 7) 2 r spec with S il4r/4 ka cos ac 2ikacotacos(a + 0) Sspec = 1/4 e I sIin - tan(a + 0)e spec 7 T sin 0 x 1 - F 2 kacot a cos(a + 0) (3.8) where.2 ( -i 2 e it2 F(T) = e dt (3.9) T 0 _______53 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 0. 5 0. 1 iS) 0.01 0.002 I I I I I Creeping Wave Join -/ E Tip - Tip 1 I I I 1 0 2 4 6 8 10 12 ka FIG. 3-2: COMPARISON OF TIP, JOIN AND UNENHANCED CREEPING WAVE SCATTERING AMPLITUDES FOR NOSE-ON INCIDENCE ON A CONE-SPHERE, a = 7. 5 54 UNCLASSIFIED

cn cr CO CI m Cl a cn 2 01 X -0.2 0.1. 1 0 00.n I I 13 C to H C) tC) P C) z rr C) -n *I a I a a * * i FIG. 3-3: THEORETICAL NOSE-ON BACKSCATTERING CROSS SECTION FOR A CONE-SPHERE WITH = 7.50, COMPUTED USING THE EMPIRICAL ENHANCEMENT FACTOR (-) AND ITS ASYMPTOTIC APPROXIMATION ( ---).

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN 0C 2 7T a FIG. 3-4: THEORETICAL NOSE-ON BACKSCATTERING CROSS SECTION FOR A CONE-SPHERE WITH a = 7.50, COMPUTED USING THE EMPIRICAL ENHANCEMENT FACTOR (-) AND ITS ASYMPTOTIC APPROXIMATION ( ---). 56 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F is related to the Fresnel integral. In the direction 0 = 7r/2 - a of the specular flash itself, Eq. (3.7) reduced to a cosec a sec a3 (3.10) (ka) (3. 10) X 92T The cross section computed from this is plotted as a function of ka in Fig. 3-5, along with the experimental data points obtained from Blore (1964) and from our own measured patterns. The agreement is relatively good, though we do observe a tendency for the measured values (particularly those of Blore) to fall below the theoretical estimate by an amount which increases with ka. At least part of this seems attributable to near-zone effects in the measurements. [C] Comments: (U) When the backscattering cross section provided by Eqs. (3.1) and (3. 7) is computed as a function of 0 for given ka, the resulting curve has an abrupt discontinuity at the edge, 0 = a, of the backward cone for all except the very largest values of ka. The discontinuity arises partly from the assumption of a reduced join contribution for 0 > a, but more particularly from the assumed absence of any creeping wave return outside the backward cone. Inasmuch as the creeping wave contributor is the dominant contributor for ka = 0(10) or less, the net effect is a jump of order 10 db or more for small ka, and only for ka > 100 (say) is the discontinuity * of no concern. It is clearly desirable that we attempt to bridge it. (U) To this end, the first (and obvious) modification of the theoretical prescription is to retain the expressions for Sti and Sj oin even outside the backward cone, and only transfer the formula (3. 7) at such a value of 0 as * Note that the discontinuity does not appear with vertical polarization. 57 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F - 25 -20 - 15 - 10 5 0 0 0 0 0 4 X2 0 'X e 0 i/X2 (db) X x -5 0 2 4 ka 6 I a I i I 8 I * I --- FIG. 3-5: THEORETICAL SPECULAR FLASH CROSS SECTIONS FOR A CONESPHERE WITH a = 7.5~, COMPUTED USING EQ. (3.10). Experimental Data, ooo Keys and Primich (1959), xxx Radiation Laboratory. 58 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F the use of the wide angle formula for the Bessel function is truly valid. Since the join continues to be excited outside the backward cone, the modification is theoretically defensible, and leads to some (but not a complete) reduction in the magnitude of the jump at 0 = a. (U) The second modification is more empirical in nature and is promoted partly from a consideration of the nature of the shadow boundary excitation (the origin of the creeping waves), but more from an examination of the measured patterns. It is found that (i) the width of the nose-on lobe in the patterns is in no degree equal to a, but does appear to decrease with increasing ka for given a; (ii) marked (single) side-lobes are apparent between the noseon lobe and the commencement of the build-up to the specular flash; (iii) the widths and center (0) values of these side lobes tend to decrease with increasing ka; and (iv) the heights of the side lobes relative to the nose-on lobe do not appear to depend on ka, and average -8 db or less. These facts are suggestive of a J dependence for the creeping wave return as well as for the join contribution, and by likening the shadow boundary to a (pseudo) ring singularity, it is apparent that the multiplicative factor should be J (2kasin 0). Note that within the backward cone (0 < 0 < a), these modifications have only an infinitesimal effect on the cross section provided by Eq. (3.1). (U) The resulting prescription for the backscattering cross section is now 0 < 0 < 3: 1 2 2 S + S + S J (2 ka sin 0) (3.11) 2 7r 1 3 o o where S1 S,2 and S3 are as shown in Eqs. (3.2), (3.3) and (3.5) respectively. ________________________ 59 _______-____ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 3 < 0 < Ir /2 - a: 2= S sp (3. 12) x2 spec where S is as shown in Eq. (3.8). The angle /3 at which the transition spec between the two formulae is made should be beyond the value of 0 corresponding to the first zero of J (i.e. beyond sin 2 405) and at the first o 2 ka subsequent intersection of (3. 11) with (3. 12). (U) Comparisons between the measured and predicted values of the cross section for ka = 2.98 and 4.51 are given'' in Figs. 3-6 and 3-7 respectively. The agreement is less good at the higher frequency, though we remark that the measured flash values are lower than for other (similar) models at the same frequency, and that the nose-on lobe width is atypically large.. 3.2.3 Indented (ID) Models (U) The basic ID Model is a metallic cone of half-angle 7.50 terminated in a toroidal ring of radius 0.533 inches, which ring is itself joined to a concave spherical surface of radius c. The profile of the rear portion is shown below (Fig. 3-8.) At each junction in the profile the tangents are continuous, and the termination is therefore a smooth one simulating an indented base. (U) Three ID models were constructed differing only in the values of the parameter c. In particular, the maximum radius, a, was the same for each and was insignificantly different (by less than one percent) from the value 2.210 inches appropriate to the corresponding cone-sphere. For the three bodies the parameter c was as follows: Here, and subsequently, all wide angle computations based on Eq. (3.12) were carried out on a digital computer, and the rest by hand. 60 UNCLASSIFIED

12 8 4 0 -4 BARE CONE-SPHERE f = 2.53 GHz ka = 2.98 -d /X 2 (db) x x V -8 x x ( x z cn T2 C.) 0 z C z C) -n rn bn ~l -12 -16 -20 x * 0 I I 72 36 0 36 72 Solid FIG. 3-6: COMPARISON BETWEEN THEORY AND EXPERIMENT. line is experimental data.

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F x x 16. 12. 8 4 -0 - BARE CONE-SPHERE f = 3.83 GHz ka = 4.51 x x -8 XX X C X x C S 0 * * S 0 0 * -12 II * -16 0 0 72 36 0 36 72 FIG. 3-7: COMPARISON BETWEEN THEORY AND EXPERIMENT. Solid line is experimental data. 62 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F J // ID-1, c = 4.558 inches ID-2, c = 2.212 inches ID-3, c = 1.519 inches implying that the indentation was greatest for model ID-3. Surface field data was obtained for each of these models, but as of March 1967, backscatter patterns had been produced only at four isolated values of ka for model ID-1. Since that time, however, a much greater volume of more accurate data has been accumulated not only for model ID-2 but also for other shapes related to the ID series, and it is convenient to refer to (and reproduce some of) this later data in support of the theoretical estimates of the cross section. Nevertheless, in accordance with the ground rules, the oblique incidence comparison will be carried out using two of the initial (inferior) patterns for model ID-1. _____ 63 _ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) For incidence at or near to nose-on the contributors to the back scattering cross section are again the tip, the join and the creeping wave, but the last two are vitally affected by the nature of the termination. Taking first the creeping wave, this is born in a region which is non-spherical and, for nose-on incidence, starts out along a geodesic whose longitudinal radius of curvature is b. According to our philosophy under which the creeping wave is primarily determined by the local geometry, and leaks off energy in the tangential direction at all points of its path, the way in which the wave crosses the indentation is by means of this leaked energy and any contribution from energy which has followed the surface is negligible by comparison. Because of this, the backscattering cross sections of all three ID models should be indistinguishable, and this is confirmed by the results for models ID-1 and ID-2 shown in Fig. 3-9. The theory also implies that were we to cover the indentation with a (flat) metal disc the cross section would be unchanged, and this was verified by probe measurements following first a free space trajector across the indentation. In consequence, the scattering is determined only by the parameters a and b, and is independent of c, and in judging the efficacy of our theory we can therefore make use of the quite complete data for the FB models (having b/a varying from 0.4 to 0. 1, and otherwise flat back, and a = 9 ) in addition to that obtained explicitly with the ID models. (U) The far field prescription for a body having half angle a and radii a and b as previously defined is as follows: 0 < 0 < 2 1 7 S1 + S2 + J (2 ka sin 0) (3. 13) 2 1 2 3 o where S1 is the tip contribution shown in Eq. (3.2); S2 is the join contribution whose specification is (see Section 3.4): 64UNCLASSIFIED U NCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN x - 0 0 0 * 0x 0 0e \* 0 0.0 0 00 - 10 a/X (db) i 0 0 0 0 0 0 -qn I 2 A 4 6 I 1 I1 I -"Vu a FIG. 3-9: MEASURED NOSE-ON BACKSCATTERING CROSS SECTIONS FOR MODELS ID-1 (x x x) AND ID-2 (...). 65 UNCLASSIFI ED

UNCLASSIFE I THE UNIVERSITY OF MICHIGAN 8525-1-F j 2 a S = -sec a -B(kb) J 2 4 b o (2ka cos a sin 0) e -2ikb sin a cos 09 (3. 14) for kb K< kb< Oo, where kb = - 3n sin 2if /n. l 6 cos~a with n= 3 /2 + a/ir, and (3. 15) with B (kb) = 1 - C = 5/64 C (kb)06 (3. 16) 6n s in 27r~I 2 Cos a )0 06 (3. 17) and S2 = - aU ycose - 7 J (2 ka cos a 2 2n n 0 sine)e 2 ikbsin a cos & (3. 18) for kb < kb;and S is the creeping wave contribution 1' 3 1/32ia-) P~r - y(ka) B(kb) a~ e i~b+12 32b \2/ kab x 1 s( 1)(kb) (3. 19) where 66 UNCLASIFIE

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F s(1)(b kb 4/3 ir/3 e2 ( /3 3 1 S(kb) =e 1 e/3 1 (32 + 9)1 2 60 123 1 )3Ai(-)2 Xexp{i)rkb-e-T/7r/ -ie 6 7 (13 - 9) (3.20) (11) with T = (kb/2) /. B(kb) is as defined in Eq. (3.16), and y(ka) is the enis the (unenhanced) creeping wave contribution for a sphere of radius b and, as such, its expressions s identical to that in Eq. (3.4) with a replaced by b.,3 -e 6. X/2 - 9: 2 = / |Sspee | (3.21) (see Eq. 3.12). (U) Plots of S computed from Eqs. (3. 14) and (3. 15) for = 0 and cr = 9 as functions of ka for various values of b/a are shown in Fig. 3-10. For this value of a, C = 0.2625 and kb = 0.2356. Note the smooth continuation into the flat-backed cone result. It is our belief that the expression for the join contribution given in Eqs. (3.14) and (3.18) is valid and numerically effective for all a, b and a, and, as such, its derivation is a notable achievement. In contrast, the expression for the creeping wave contribution (see Eq. 3.19) is not quite so accurate, and though the results obtained from it are, as we shall see, more than adequate for almost all ka and b/a, the transition to the flat-backed cone result (b -- 0) is not correctly reproduced for small ka, and the transition to the pure sphere result (b -- a) is discontinuous for all ka. 67 U NCLASSIFIED

ls21 0. 1 FIG. 3-10: o00 Ul Ul I I cj z -) 0 0 s P-4 z C) rr C,) C,) -n ni 1 ka 10 100 S FOR AN FB OR ID MODEL WITH a = 9~ 2 MODULUS OF JOIN CONTRIBUTION FOR VARIOUS b/a RATIOS.

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) A comparison between the predicted and measured nose-on back scattering cross sections of models ID-1 and ID-2 is shown in Fig. 3-11. For ease of computation we have here* omitted any tip contribution, and have neglected any creeping wave enhancement (i. e. taken y - 1). The fit is extremely good, and the slight discrepancy in the depths of the minima could almost certainly have been removed had we included a tip contribution. A similar comparison, but now for model FB-4 (b/a = 0.4, tu = 9 ) is given in Fig. 3-12, and hence again the fit is excellent. (U) The measured and theoretical backscatter patterns for model ID-1 at the frequencies 2.53 and 3.83 GHz, corresponding to ka = 2.98 and 4.51, are presented in Figs. 3-13 and 3-14 respectively. At the higher frequency the agreement is reasonably good, though there is some discrepancy in the structure of the pattern just outside the nose-on lobe. At the lower frequency, this discrepancy is much more marked, and there is not much agreement between theory and experiment in the aspect range between nose-on lobe and the side lobes of the specular flash. The discrepancy has its origin in the failure of the Bessel function factor J to adequately reproduce the creeping wave behavior at oblique incidence and, as noted elsewhere (see Section 3.5), the measured patterns for the ID and FB models in this region are, for small ka, in closer agreement with the patterns for a true flat-backed cone than for a cone-sphere-like object. We also observe a slight discrepancy in level in Fig. 3-13, but since this occurs equally at broad-side and nose-on, it is believed due to an error in experimental calibration. * As was done in all computations throughout this section. 69 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F X 0 0 0 (db) -10 -20 -3 -30 ID-1, 2 b/a = 0.25, a = 7.5~) 0 2 ka 4 6 FIG. 3-11: COMPARISON BETWEEN THEORY ( —) AND EXPERIMENT FOR NOSE-ON BACKSCATTERING FROM ID-1 (xxx) AND ID-2 (eee). 70 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F or -10 a/X2 (db) -20 x FB-4 (b/a = 0.4, a = 9~) x -30 - 0 I 2 I 4 I 6 I ka FIG. 3-12: COMPARISON BETWEEN THEORY ( —) AND EXPERIMENT (xxx) FOR NOSE-ON BACKSCATTERING FROM MODEL FB-4 (b/a = 0.4, a = 9~). 71 UNCLASSIFIED

16 12 8 ID-1 f = 2.53 GHz ka = 2.98 1* — Experimental Data 0. 7/X2 (db) -4 0o Cn I l-A -8 -12 -16 -20 x x xx x H m c( z H 0;>. M0 0 2;4 z rc Cll rn xx x 72 36 0 36 72 0 Degrees FIG. 3-13: COMPARISON BETWEEN THEORY AND EXPERIMENT.

20 - 16 12 ID-1 f = 3.83 GHz ka = 4.51 ---- Experimental Data C 8 Z Z Z,/X2 4 -I (db) C x OgH o ~ 0- 0 -4 -XXI -8 X x X~ '' ~. -4 \^ x f I ~ * X -12 X.. > 72 36 0 36 72 0 Degrees FIG. 3-14: COMPARISON BETWEEN THEORY AND EXPERIMENT.

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 3.2.4. Nose Tip Antenna (LSP) Model (U) The LSP (Lucite spacer point) model is a bare cone-sphere of half angle a = 7.50 and base radius a = 2.210 inches from which a segment 1/4 inch thick centered 3-3/8 inches from the tip has been removed and replaced by a lucite insert of the same size, so that the resulting model has the same dimensions as the original cone-sphere. The lucite disc is meant to simulate a nose-tip antenna. (U) In order to provide a rigorous derivation of the effect of the lucite on either the surface or far fields, it is necessary to measure or compute the complex far field amplitude of the spacer regarded as a ring antenna excited in those modes appropriate to the incident field, the radiation impedance of the antenna so excited, and the loading impedance provided by the lucite and measured at the surface. Such an analysis would be tedious, though relatively straightforward, and would be similar in all respects to the approach generally adopted in the consideration of problems involving the reactive loading technique; and in view of the extremely close agreement between theory and experiment in such reactive loading problems, the accuracy of the far field specification for the LSP model resulting from this analysis could be expected to be very high. (U) The purpose of "reactive loading' applied to scattering problems is the control of the scattered field by appropriate choice of the loading impedance, and a feature of this technique is that relatively small variations in the applied load can produce large changes in cross sections. Ergo, any practical antenna must be modeled most precisely, including simulations of the feed and line impedances, for the measured data to be truly indicative of the cross section that would be realized. In contrast, the LSP model is only a very crude simulation which can at most indicate the type of cross section changes 74 U N CLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F that could occur, and because of this there was neither reason nor justification for undertaking the detailed analysis that could have led to an entirely theoretical and rigorous prediction of the backscattered field. Rather was it our intent to investigate the modifications to the surface field which might result from the presence of the antenna (a full description of these can be found in Goodrich et al, 1967) and, in the present note, to compare a formula for the cross section with measured data. In such a formula there is one parameter, representing the excitation strength of the antenna, which is unspecified; and in the absence of a general analysis of the form described above, it is necessary to deduce this parameter as best we can from the surface field data. (U) The far field amplitude in the backscattering direction associated with the spacer alone can, for angles of incidence not greatly different from noseon, be written as, '.-2ik(a-a cos a)cosec acos 0 S = ka Pei J (2ka sin O)e (3.22) sp 9 s o s where a is the radius of the spacer at its mid-point, e0 is the complex excitation strength, and the phase is referred to an origin at the shadow boundary of the cone-sphere. Inasmuch as surface field measurements at points beyond the spacer show the field to be relatively unaffected by its presence, it is concluded that the join and shadow boundary contributions will remain those which are appropriate to a pure cone-sphere, and accordingly, Eq. (3.22) is but an additive effect. S must be included (and may be domsp inant) in the aspect range 0 < 0 < f3, where j3 is the limiting angle defined following Eq. (3.12). Depending on r and ka, it may still be significant outside this region but, for practical purposes (and computational convenience) can be neglected outside this range in view of the rapidly increasing side lobes of the specular flash. 75 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) The far field prescription is now as follows: 0 < 0 < 13: 2 2 = 1/ S + S2 + S3 J (2 ka sin 0) + S (3.23) 2 1 2 3 r Sp (see Eq. 3.11), where S1, S2, and S3 are as defined in Eqs. (3.2), (3.3) and (3.5) respectively, and S is given in Eq. (3.22); 3 < 0 < ir/2 - a sp 2 1/ = 1/ S (3.24) 2 spec (see Eq. 3.12). (U) Both F and p could be estimated from measured data for the amplitude and phase of the surface field on model LSP at the appropriate frequencies. Unfortunately, only amplitude data is available, but from this the curve of f versus ka shown in Fig. 3-15 has been constructed. We observe that for ka = 2.98, F = 1.0, whereas for ka = 4.51, 7 = 2.9. (U) Comparisons between the measured and computed values for the back scattering cross section of the LSP model at near nose-on aspects with frequencies 2.53 and 3.83 GHz corresponding to ka = 2.98 and 4.51 respectively, are shown in Figs. 3-16 and 3-17; and to illustrate the large variations that can result from changes in the phase p of the spacer excitation, we 2 have in each case computed a/k with p chosen such that the spacer contribution is in-phase or out-of-phase with the remaining contributors to the cross section. This entailed the choice 0 = -42. 846 (in-phase) and p = 137.154~ (out-of-phase) for ka = 2.98, and 0 = -108.952~ (in-phase) and 0 = 71.048~ (out-of-phase) for ka = 4.51. At each frequency, the two curves are the extreme ones as regards the nose-on lobe, and these bracket the 76 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F r 20 10 0 0.5 1.0 1.5 2.0 ka s FIG. 3-15: AMPLITUDE r OF EFFECTIVE RING EXCITATION FACTOR FOR LSP MODEL, INFERRED FROM SURFACE FIELD DATA. 77 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN Or XXX X -4 X X X x x X x -8k 0 -12 a/X2 (db) -16 FIG. 3-16: 36 18 0 18 36 0, Degrees COMPARISON BETWEEN THEORY AND EXPERIMENT (-) FOR THE LSP MODEL WITH ka = 2.8. The Theoretical Values are Computed from Eq. (3.23) with r = 10 and 0 = 42.846~ (inphase, xxx) and 0 = 42.846 + 180~ (Out-of-phase, * e). 78 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN 4 O - — 4 -8 (db) -12 — -16 -2C - 54 36 18 0 18 36 54 0, Degrees FIG. 3-17: COMPARISON BETWEEN THEORY AND EXPERIMENT (-) FOR THE LSP MODEL WITH ka = 4.51. The theoretical values are computed from Eq. (3.23) with F = 2.9 and 0 = 108.952 (in-phase, xxx) and 0 = - 108.952~ (out-of-phase, ***). 79 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F measured data. Using only the lobe shape and magnitude as a guide it would appear that to fit the data at the lower frequency, 0 should be chosen near to, but somewhat distinct from, its value corresponding to the out-of-phase condition, and for the purposes of the subsequent comparison we selected 0 = -42. 846 + 225. In contrast, at the higher frequency it would appear that we can do little better than to choose 0 corresponding to the in-phase condition, that is, 0 = -108.952. Note that these values are not necessarily the optimum, and since the nature of the side lobes in a scattering pattern is quite critically dependent on 0, it is entirely possible that other values of 0 would lead to a better overall fit to the measured data. (U) The theoretical backscattering patterns for ka = 2.98 and 4.51 computed from Eqs. (3.23) and (3.24), using the above values of 0, are shown along with the measured patterns in Figs. 3-18 and 3-19. The agreement is rather good, and though there are some discrepancies in the "sensitive" aspect ranges midway between nose-on and specular, it is probable that these are attributable partly to a less-than-optimum choice of 0 and partly to a failure of the Bessel function J ( 2 ka sin 0) to adequately diminish the creeping wave contribution at wide aspects. 3.2.5 Coated Cone-spheres (U) Although the scattering behavior of a metallic cone-sphere-like body is by no means simple, the presence of one or more non-metallic coatings complicates the problem by many orders of magnitude, and it is only in such trivial cases as the sphere or infinite circular cylinder that even the case of uniform homogeneous and isotropic coatings has received rigorous consideration. In order to make progress with the cross section prediction problem when coatings are present, it is therefore necessary to adopt some approach which builds on the information gained from the study of the corresponding 80 UNCLASSIFIED

12 7/Experimental Data 8 4 0 LSP 0-3 Z m f = 2. 53 GHz, ka = 2. 98 Cl) Cl) rn -4 (db) -8 0 0 0 *00....0 0 0 0 0 0 Ac 0 0 X z C,, Q1~ CT) 10. z U) CA) Cl) -n 0 -12 -16 -20 0, Degrees 72 36 0 36 72 FIG. 3 -18: FIG 3-8:COMPARISON BETWEEN THEORY AND EXPERIMENT USING fl = 1. Oe i(225 - 42. 845)0

16 12 8 4 00 tD A2 (db) -4 -8 -12 -16 -20 Experimental D x x XXX xM. LSP f = 3.83 GHz, ka = 4.51 tata X. x X * \ XX co C.n Qn I l H )z m *z 0-.en m Td C) H H< 0 -4 0 a 1-1 O > z C) ni m 0, Degrees 36 36 T 2 THEORY AND EXPERIMENT USING = 2.9e ( 80 108 952) FIG. 3-19: COMPARISON BETWEEN

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F metallic shape, but yet enables us to take the coating into account, and for this purpose the impedance boundary condition is almost ideal. Such a boundary condition simulates the effect of the entire coating (which, in practice, may be intentionally variable in its properties as a function of depth, and unintentionally variable in the transverse directions) by means of a single boundary condition, E - (n. E)n = r ~Zh x H (3.25) applied at the outer surface, where E, H are the total (incident plus scattered) fields, n is a unit vector normal in the outwards direction, and Z is the intrinsic impedance of free space. r] is the parameter representing the effective impedance of the coating relative to that of free space, and is zero for a perfect conductor -- in which case, the boundary condition (3.25) reduces to that used heretofore. (U) For a single homogeneous coating composed of a material of relative (complex) permeability p, and of depth large compared to the skin depth at the frequency of interest, rl is simply the bulk impedance of the material and is j7 1= '. (3.26) If the depth, 6, of the layer is not that large, so that a significant amount of energy returns to the surface after reflection at the (metallic) substrate, the expression for r) required for use in (3.25) becomes r = - i if tan ( feq k 6), (3.27) 83

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F where k is the free space propagation constant, and in the limit k6 -* oo, Eq. (3.27) reduces to the form shown in (3.26). It is convenient to refer to (3.27) as the tangent approximation. (U) The formal conditions under which the impedance boundary condition (3.25) can be justified are fairly restricted, but from an examination of surface field measurements for a variety of shapes and coatings, it is believed that the condition has a much wider practical applicability provided only that adequate estimates of the effective surface impedance can be arrived at. It certainly affords the only feasible method for estimating scattering when nonuniform coatings are present and/or when the body is not of trivial shape. It also enables multiple coatings to be taken into account via a simple (computable) modification to the postulated impedance and, last but not least, makes possible the computation of both the surface and far fields for a class of nontrivial shapes by a development of those same techniques that have proved efficacious for the corresponding metallic bodies. The condition (3.25) is therefore basic to all our analyses of coated bodies, and because of the generally small thickness of the coatings that have been considered, the required surface impedance must be computed from (3.27) using the bulk properties of the material determined from coaxial line measurements (see Goodrich, et al, 1966, Section 2.1.6). (U) From surface measurements on pointed non-metallic bodies it has been concluded that at points not too close to the tip the surface field can be estimated using the known fields on a infinitely-long circular cylinder of the appropriate radius (see Senior 1966) and composed of the same material; and as we recede still further from the tip, the field components reduce to what they would have been had the surface at that point been part of an infinte 84 _ UNCLASSIFIED

UNCLASSIFIED ----- THE UNIVERSITY OF MICHIGAN 8525-1-F tangent plane. In practice this limiting behavior is almost always* attained prior to the cone-sphere join being reached, and for near nose-on incidence it is therefore sufficient to modify the join contribution by the Fresnel reflection coefficient. Moreover, as we go to more oblique angles, for which the incidence is more nearly normal to the side of the cone, this limiting behavior is attained more early, and over an ever greater percentage of the illuminated side the field behavior is that obtained by invoking the Fresnel reflection coefficient in conjunction with the fields that would have existed had the cone been metallic. It is therefore trivial to derive the expression for wide-angle scattering from that which has already been given for a metallic body. (U) It will be recalled that the other main contributor to the scattering from a metallic cone-sphere is the creeping wave, and that its expression is derivable from a consideration of a sphere in isolation. The same is true when the surface has an impedance ri associated with it, but whereas the sphere creeping wave had a simple asymptotic expression which was accurate even down to very small values of ka (<< 1: see Section 3.4) this is unfortunately not true for rY = 0. Creeping waves of both the electric and magnetic type can now be excited, and even if we postulate ka > > 1 (as was done in the original derivation of the creeping wave expression for a metallic sphere), the computation of the corresponding result for a coated sphere still requires the solution of two transcendental equations. Thus, for a sphere coated with a material of surface impedance rl, the far field amplitude produced by the creeping waves in the backscattering direction is S S(e) + (m) (3.28) cw The exception is for very small values of ka. UNCLASSIFED UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F with where (e) _a4/3 ir mka-iTr/6 1 4 {i(s ) s Ai(- 3-e q q = (ka/2) 1/3 e3 (3.29) (3.30) and the j3 are the roots of s) Ai' (-3 ) - q Ai(-3 ) = 0; S S (3.31) and 2 (m) = (ka/2)4/3 i7r ka - ir/6 ___ s Ai(-cr ) e aC -q| s 1 s qm (3.32) where qm = - (ka/2)1/3 1 rl (3.33) and the a are the roots of s Ai' (-a ) - qm Ai (-a ) = 0 S s (3.34) Since r1 is, in general, complex, the roots of Eq. (3.31) and (3.34) are similarly complex. Although no difficult should arise the determination of these roots in any given case, the computation will almost certainly be tedious and has not been attempted. 86 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) The only other contributor to the scattering from a cone-sphere is the tip, and this is quite inconsequential for an infinitely sharp (ideal) conesphere. For completeness, however, its contribution will be included in the estimate of the scattering, and from a consideration of surface field data it is believed to be unaffected by the coating. The resulting prescription for the backscattering cross section of a cone-sphere or half-angle a, radius a and surface impedance r) is now as follows: 0 < 0 < |3: 2 2- = 1/7 S1 + S2 + S3J (2ka sin 20) (3.35) where S is the tip contribution given in Eq. (3.2), S2 is the join contribution, namely S = R (0) Sj (3.36) 2 join with R(0) = 1 -n sin (0 + a) (3.37) 1 + r7 sin (0 + a) and Sjoin as given in Eq. (3.3), and S3 is the creeping wave contribution shown in Eq. (3.28). 3 < 0 < ir/2 - a: 2 = 1/7r R(0) S (3.38) 2 spec where R(0) is as given in Eq. (3.36) and S as given in Eq. (3.8). spec (U) For a specified surface impedance rY, the computation of the expression (3.38) for the wide angle scattering is entirely trivial. But such is not the case for nose-on and near nose-on scattering to which the creeping waves ________87 UNCLASSIFIED

U NCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F form a dominant contributor. The expression for S given in Eq. (3.28) ow incorporates not only the influence of the magnetic waves, but also the change in the contribution of the (dominant) electric waves resulting from the changes in birth and launch weights, as well as in decay rates, attendant on the coating. Inasmuch as these effects can be rigorously determined only by laborious computation of Eq. (3.28), and the digital programming necessary for this has not been carried out, it is fortunate that alternative and physically-motivated approximations to S are possible to those limiting cases called for in this Agreement Item. (U) Since all coatings of interest are electrically thin, and since the tangent approximation which must then be used for r7 introduces a pseudo-attenuation effect which attributes a "loss" component rj asf a consequence of the thickness, it is unrealistic to label a coating?'lossless" or Itlossyl? from an examination only of the real and imaginary parts of the effective surface impedance. Indeed, were we to do so, it would be found (see Table III-2) that all of the coatings employed were lossy. In contrast, surface field measurements strongly support the contention that some of the coatings were lossless at least some of the frequencies considered, and to provide a more unambiguous categorization of the coatings it is necessary to examine the extent to which power incident on the coating is actually absorbed. A measure of this 2 effect is provided by the power reflection coefficient, R(0); if this is almost unity, the coating clearly behaves as an almost lossless one, but if it is much less than unity (< 0. 5, say) it would seem safe to regard the coating as a lossy one. Inasmuch as this distinction is maximized at normal incidence on the surface, the following definition will be adopted: lossless: i_ 2 1 I - ^^ 1 1 + j 88 __________________ 88 __________________A U NCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F lossy: 1 2 1 + <<1 (U) A listing of the cases for which coated cone-sphere backscatter patterns had been measured as of March 1967 is given in Table mI-2. It will be observed that three different coating materials were employed: LS-22, LS-24 and LS-26, and each is a lossy poly-urethane foam impregnated with graphite particles. The LS-22 coating averaged 0. 1875 inches in thickness, but the other two were somewhat thicker (6 = 0.25 inches). In general the effect of each coating was measured at 7 different frequencies (see Table III-1), but since the highest frequency (8.43 GHz) is considerably beyond the range fo which the bulk electrical properties of the coating materials were determined (Goodrich et al, 1966), it is necessary to confine ourselves to the lower six frequencies in the selection of patterns for the thoeretical comparison. (U) Table III-2 gives a listing of the bulk parameters of the coatings at these six frequencies deduced, where necessary, by interpolation from the coaxial line data, together with the corresponding values of the real and imaginary parts of the surface impedance rn computed using Eq. (3.27), and the power reflection coefficients at normal incidence. It will be observed that rhas a significant imaginary part in every case, and that the variation with frequency is more or less the same for each coating. Indeed, the general impression is that each coating appears relatively lossless at the lowest frequency, but as the frequency increases, leading to an increase in the electrica thickness of the layer, the loss increases substantially. (U) According to our previous criterion of "lossless", only the LS-22 coating at the lowest frequency (2.53 GHz) unquestionably fulfills it, and since the associated value of ka is 2.98, it is convenient to select this as the loss 89 - UNCLASSIFIED

UNCLASSIFIED T THE UNIVERSITY OF MICHIGAN 8525-1-F TABLE III-2: Coating Parameters. LS-22 ^16 arg 1i1 argpu Re. rl Im. rl |1_2 1+-7 LS-24 arge 141 argp Re. rl Im. rl 1-n 2 i+r7 LS-26 I I arg 1U1 argp Re. rl Im. rl tzad 2 Il+-l 2.53 GHz 3.11 1.05 1.00 0.0438 0.0273 -0.259 0.903 3.37GHz 2.42 0.948 1.04 0.0577 0. 0524 -0. 364 0.831 3. 77 GHz 2.43 0. 942 1.00 0. 0009 0.0410 -0.399 0.868 3.83 GHz 2.45 0.946 0. 985 -0. 0115 0.0372 -0.400 0.879 5. 73 GHz 2.01 0.649 0.998 0. 0942 0.185 -0. 643 0.593 5. 83 GHz 2.04 0.631 0.983 0.0698 0. 173 -0.658 0.617 2.53 GHz 3.37 GHz 3. 77 GHz 3. 83 GHz 5. 73 GHz 5. 83 GHz 7.81 1.65 1.00 0. 0191 0.0884 -0.298 0. 722 6.32 1.21 0.994 0.0716 0.213 -0. 388 0.474 6.01 1.04 1.10 0. 0113 0. 343 -0.482 0. 326 5.98 1.05 1.13 -0. 0073 0.361 -0.491 0.310 4.39 1.08 0. 952 0.0290 0.496 -0.429 0. 181 4.37 1.14 0.966 0.0156 0.483 -0.427 0.189 2. 53 GHz 3.37 GHz 3. 77 GHz 3.83 GHz 5. 73 GHz 5. 83 GHz 7.81 1.73 0.997 0. 0292 0.0858 -0.291 0. 729 5.77 1.41 1.05 0.0463 0. 187 -0.400 0.523 5.72 1.52 0. 965 -0. 0441 0. 168 -0.407 0. 561 5.76 1.60 0.950 - 0. 0665 0. 153 -0.401 0. 590 4.25 1.42 0. 9693 -0. 0029 0.353 -0.468 0.311 4.26 1.48 1.00 -0. 0216 0.353 -0.456 0.307 90 i UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F less case for the comparison of theory with experiment. To typify lossy behavior, either of the coatings LS-24 or LS-26 at any of the higher frequencies would suffice, but since the ka values 2.98 and 4.51 are the preferred ones, attention is naturally directed at the LS-24 coating at the frequency 3.83 GHz (for which the power reflection coefficients is 0.310). [A] Low Loss Coating (U) For any type of coating the theoretically correct prescription of the scattering is that given in Eq. (3. 35) (for 0 < 0 < _ ) or Eq. (3. 38) (for 3 <_ 0 < r /2 - a). As previously remarked, however, the computation of the creeping wave contribution presents some difficulty (or, at least, tedium) and here and in the next section we offer alternative, practically-motivated, estimates of this particular contribution which are appropriate to the cases of low- and high-loss coatings respectively. (U) For a coating which is essentially lossless, the fields induced on the surface of the body are of the same character as in the case of the metallic body, with only their magnitudes differing somewhat. In particular, the magnetic creeping waves are insignificantly excited (because q >> qe implying very rapid damping of the magnetic waves). Moreover, Iq e << 1 due to the smallness of jIr\ in most practical cases, and it therefore appears reasonable to postulate an electric creeping wave which is identical to that supported by the metallic body, which suffers no additional decay due to the coating, and which even has its amplitude unchanged since the field is incident at grazing angles at the shadow boundary. (U) The resulting prescription for the far field scattering is then as follows: 0 < 0 <: _______91 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F I 2 = S1 + S2 + S J (2kasin 0) (3. 39) 2 1 23 o (see Eq. 3.35), where S1 and S2 are as defined in Eqs. (3.2) and (3.36) respectively, and S.3 = S (3.40) 3 cw where S is as given in Eq. (3.4). 3 < 0 < 7T/2- a: 2 -2 = /r R(0) S (3.41) 2 = spec (see Eq. 3.38). (U) The best example of lossless behavior that is available is coating LS-22 at 2.53 GHz for which ka = 2.98. Even for normal incidence on the surface the power reflection coefficient is only - 0.45 db. This is the amount by which the peak of the specular lobe is reduced by the coating, with the reduction being less away from the specular angle. Near nose-on the reduction in the cross section is infinitesimal since only the join contribution is affected by the coating, and this is smaller than, and almost in-phase with, the creeping wave contribution at 2.53 GHz. The comparison of theory with experiment is shown in Fig. 3-20, and overall the agreement is quite good. [B] Lossy Coating (U) As the loss increases one must expect that the contribution of the magnetic creeping wave will increase relatively to that of the electric one and may, in fact, exceed it. But at the same time, however, the contribution of both is decreasing as a result of the higher decay rates associated with the coating and for coatings of sufficiently high loss the creeping wave contribution will be negli 92 _ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F )~x x A 4 x 04 CI 00 CNl II 00 * N @0. clq z 0 $- I - Cl Cl Co S x Cl1 - II IC 93 UNCLASIFIE

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F gible compared with the join contribution. The creeping waves can then be neglected in their entirety, and an examination of surface field data suggests that this is a reasonable approximation for many of the cases examined. (U) A rough (very rough) estimate of how much the creeping wave contribution is reduced by the coating can be had by likening the wave to a surface wave traveling over a plane surface characterized by a surface impedance q or qm depending on which type of creeping wave is considered. The net reduction obtained in this manner is then ei ka 1- which, for the LS-24 coating at 3. 83 GHz, amounts to -33.3 db. This certainly supports the contention that we may neglect the creeping wave, in which case the cross section prediction becomes: 0 < 0 < 3: 2 - =1/T S1 + (3.42) 2 1 2 where S and S2 are as defined in Eqs. (3.2) and (3.36) respectively. 1 2 13 < 0 < 7r /2 - c: 2 -2 = I/ R(0) S (3.43) 2 - spec (see Eq. 3.38). (U) The predicted cross section obtained in this manner is compared with the measured pattern for coating LS-24 at 3. 83 GHz in Fig. 3-21. Once again the agreement is quite good, and though there is a tendency for the theoretical formula to slightly underestimate the return at near nose-on aspects, the discrepancy is a good deal less than that between the two measured patterns for this one case. 94 UNCLASSIFIED

12 8 -LS-24 COATED CONE-SPHERE f= 3. 83 GHz, ka = 4. 51 4 Exp. -(3949) --- (3986) 0 -Z 4,2 xx I f)(db) 1 I Emmimix x X XI h igl xIx % x -16 -xX O I x II -20 Ii I I*IxI Ig 11 72 36 9 36 72 0, Degrees H z 0 -n 0 r 0A zC" r FIG. 3 -2 1: FIG 3-1:COMPARISON BETWEEN THEORY AND EXPERIMENT.

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 3. 3 Backscattering Behavior of Model LSP (U) The LSP (lucite spacer point) model discussed in Section 3.2.4 is a bare cone-sphere of half-angle 7.5 and base radius a = 2.210 inches from which a segment 1/4 inch thick centered 3-3/8 inches from the tip has been removed and replaced by a lucite insert of the same size, so that the resulting model has the same dimensions as the original cone-sphere. The lucite disc is meant to simulate a nose-tip antenna. (U) Complete backscattering patterns have been measured at a series of 43 S- and C-band frequencies spanning the range 2.5 < ka < 6.8, and from these patterns, values of the cross sections at nose-on, specular and rear-on incidence have been read. Corresponding patterns have been obtained for the pure cone-sphere of identical dimensions and it is the purpose of this Section to examine the effect of the spacer at these particular angles of incidence, an to compare the resulting cross sections with the theoretical estimates based on formulae given heretofore. (U) The measured values of the rear-on cross sections of the pure conesphere and LSP models are plotted as functions of ka in Fig. 3-22 where, for clarity of presentation, we have displaced the values for the latter model by 10 db. Taking first the cone-sphere data, the theoretical estimate of the cross section is based on the existence of only a specular contribution provided by the spherical rear, and using the formula (Senior, 1965) for the specular return from a sphere, we have - = 1/47 (ka)2 1 + 1 (3.44) 2 | (2 ka)2 ( Inasmuch as the second term in braces gives the correction of less than 0. 17 db over the range of ka covered by the measurements, we can neglect it for - 96 _ UNCLASSIFIED I

8-7 0 * 0.0.00.00oo0 0 0 Cone-Sphere 0 0 4-1 0 - 0 0 0 0 0 4 0 0 00 CA Cl) Cl) 0 H z "ii -4-~ LSP (-10db) u/X.2 (db) 01 I. C) ~n -8 - -12 - 'i if gi 'I 'I 0 I C)4 0 I z 6 -16 0 -T 2 ka 4 FIG. 3-22: MEASURED VALUES OF REAR-ON CROSS SECTIONS OF PURE CONE-SPHERE AND MODEL LSP.

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F all practical purposes and take 2 = i/42r (ka). (3.45) The curve computed from this equation has been included in Fig. 3-22, and is seen to be in excellent agreement with the measured data for ka less than (about) 4, but for larger values of ka the formula appears to underestimate the cross section by as much as 3 db. The nature of the discrepancy is mor clearly seen in Fig. 3-23 where we have plotted the ratio of measured and predicted cross sections. There is some evidence that the error is oscillato in character, with quite long period, and though a likely explanation is the existence of a traveling wave contribution associated with a reflection at the tip, the period is incompatible with this. We note in passing that the theoretical estimate of such a traveling wave return is based on the concept of a long thin wire and, in consequence, provides no effect at end-on incidence. Only away from, but near, end-on incidence is a contribution predicted, but because of the significant "thickness" of the present body, a return at rearon incidence would not be unexpected. (U) For the LSP model the specular return at rear-on incidence is the same as for the cone-sphere, and the corresponding curve has been superimposed on the LSP data in Fig. 3-22. Quite large discrepancies of as much as 7 db are evident, particularly for ka in the vicinity of 5, and to throw some light on their nature, we have plotted in Fig. 3-23 the ratios of the measured to the theoretical (specular) cross sections. The result is a high frequency oscillation or remarkably regular period constrained within a slowly varying envelope. There is no doubt but what this is the consequence of a traveling wave effect whose reflection is provided by the ring antenna. On 98L I UNCL~ASSFE

4-. 0.0.0 * 00 0.0 0* *e *- Cone-sphere 0 - 0 0 @ 0 0 0 00.0.00 0~ ~ H -4 - CZ C, I1 rn / cr cs (db) -12 - 0 A A I I 1 I 0I ts I I I ' I I I I I I I I I A I I I I I I I I I 1 I I I 1( I' I II I!I 0 1 I II I I0 I I, i; I I I I I I I I I I I' I I I II I iT I I I' 00 CJ1 tl. Ul I I It LSP (-10 db) I I I; 0 I I 11 I ~ I II II II I if iI II z 011 t21 0 CA z 04 O z: O6 C) z C) Cl) Cl) rn IID -16 - I I I. I-t 0 2 ka 4. I 6 * FIG. 3-23: RATIOS OF MEASURED TO THEORETICAL (OPTICS) CROSS SECTIONS OF PURE CONE-SPHERE AND MODEL LSP FOR REAR-ON INCIDENCE.

UNCLASSIFIED -- THE UNIVERSITY OF MICHIGAN 8525-1-F the assumption of a wave which proceeds directly the the shadow boundary and thereafter follows the surface back to the ring, traveling with the velocity of light throughout, the resulting interference with the specular returns leads to an oscillation in the cross section of (uniform) period 0.44 in ka, compared with the period 0.41 evident in the measured data. The difference could be attributable to an effective phase velocity of the traveling wave smaller than c by about 7 percent. Since the mean of the data for the LSP model in Fig. 3-23 is near unity, it is apparent that the other (specular) contribution is accurately predicted by theory. We therefore postulate 2 = 1 + A(ka) exp i(acka + ) (3. 46) where the exponential arises from the phase difference between the traveling wave and specular returns, and A(ka) is proportional to the amplitude of the former. Some of the values of A(ka) determined from the envelope of the oscillations in Fig. 3-23 are as follows: ka A(ka) ka A(ka) 3.5 0.10 5.5 1.04 4.0 0.30 6.0 0.68 4.5 0.66 6.5 0.32 5.0 1.09 7.0 0. 10 A(ka) is, of course, proportional to the amplitude of the (voltage) reflection coefficient at the ring antenna. We observe that it displays a characteristic resonance phenomena centered on the frequency for which the circumference of the ring is one wavelength. Such resonance was previously observed in studies of the cross section of the LSP model at nose-on incidence...00 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) Turning now to incidence in the specular direction, the measured data for the cross sections of the cone-sphere and LSP models are shown in Fig. 3-24, where for clarity, the values for the latter model have been reduced by 10 db. The differences between the cross sections are quite miniscule, implying that the ring spacer has no effect at this aspect. This is in accordance with theory in which the cross section is likened to that of a cylinder whose length is the slant length of the cone and whose radius is 4/9 aseca (Senior, 1967). Regardless of the presence of any ring, the theoretical estimate of the cross section is therefore 2 a cosec a sec a 3 - = - (ka) X 972 = 0. 6667 (ka)3 for a = 7. 5~, and the curve computed from this formula has been included in Fig. 3-24, The agreement is good, a fact which is more clearly shown in Fig. 3-25, where the ratios of the measured to the theoretical cross sections have been plotted, but even so we do notice a tendency for the measured values to fall below the theoretical ones for ka greater than (say) b, with the discrepancy increasing as ka increases. As remarked on previous occasions, such a discrepancy is often an experimental error attributable to a near-field effect and caused by the necessity of bringing the model closer to the antenna in order to measure accurately the nose-on cross section. (U) The measured data for nose-on incidence is given in Fig. 3-26, with the upper sequence of points showing the values for a cone-sphere, and the lower sequence showing the corresponding values for the LSP model. The latter have been reduced by 20 db to permit a clearer presentation of the data. Whereas the cone-sphere cross section displays the regular sinusoidal 101 UNCLASSIFIED

UNCLASSIFIED THE UNIVER SITY OF 8525-1-F MICHIGAN -24 -20 -16 - 12 * 0 *0. 0. I. 0 Cone-*here 0 Y 0 0 Oee 0 0 0 (db) 0 0 0 0* I 0 0, * 0 le 0 0 e 0~ 0*. 0 0 7^0.0 00 LSP (-10 db) 6 a 1 0 - 4 0 / *0 0* y 2 1 4 1 I I 0 ka FIG. 3-24: MEASURED VALUES OF SPECULAR FLASH CROSS SECTION OF PURE CONE-SPHERE AND MODEL LSP. 102 UNCLASSIFIED

U NCLASSIFIED THE UNIVERSITY OF z 8525-1-F MICHIGAN - 2 0 0 0 Cone-phere * 0@ 0 0 0 0 0 0 0 - 0 *0 0 0 0 0.0 0 0 0 * 0 @0 0 0 0 0.0 * 0.0 — 2 a (db) (db) - 2 0 * 0.0 0 0* * 0 LSP - 0 0 0 0 0 0 0 *.0 0 0 0 0 0 * 0 0 0 0 0 0 0 0 0 0 a 2 I 4 I 6 I ka FIG. 3-25: RATIOS OF MEASURED TO THEORETICAL (OPTICS) CROSS SECTIONS OF PURE CONE-SPHERE AND MODEL LSP FOR INCIDENCE AT THE SPECULAR ANGLE. 103 UNCLASSIFIED

UNCLASSIFIED THE UNIVER SITY OF 8525-1-F MICHIGAN Cone-Sphere @0 0 -7 0 S 0 0 0 0 0 0.0 0 0 0 0 0 0 0. 0 0 0 0 0 0 0 I I 1 I I I I I I 0 I A\/ II II J,.. I, I;,:f, T I 0 0 Pt -19 -/X2 (db) -23 --27. -31 - I '' I I Ir I I I l l l I I I I LSP (-20 db) I I I *I v1 ~., I.. I a. I 0 2 4k ka 6 FIG. 3-26: MEASURED VALUES OF NOSE-ON CROSS SECTIONS OF PURE CONE-SPHERE AND MODEL LSP. 104 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F oscillation that is expected, the values for the LSP model follow a much more irregular pattern composed of a high frequency oscillation of period approximately 0.4 in ka superimposed on a slowly varying curve. For ka < 3.5 this mean curve is indistinguishable from the cone-sphere curve, but for ka > 3.5 the mean LSP cross section departs considerably from its cone-sphere value, and exceeds the latter by almost 20 db for ka in the range 4.5 to 6.0. (U) A comparison between the measured data for the pure cone-sphere and the theoretical formula for the cross section is shown in Fig. 3-27. The computed curve has been taken directly from Fig. 3-3 of "The Agreement Item, " using the asymptotic formula for the creeping wave enhancement factor. We note in passing that the empirical factor (see Fig. 3-2 of Section 3.2) gives better agreement with the cross section data in the vicinity of the peak near ka = 3. 1, but leads to somewhat deeper minima near ka = 4.0 and 5.8. We also observe that a value for the enhancement factor even greater than that provided by the asymptotic formula would improve the agreement with experiment for ka > 4.0, but in spite of these comments, the agreement between theory and experiment evident in Fig. 3-27 is more than satisfactory. (U) According to the theory for the LSP model outlined in "The Agreement Item," the cross section for nose-on incidence is S2= 1/r Sc + S (3.47).2 cs sp where S is the far field amplitude for a pure cone-sphere and cs S Ti 10-2ik(a -a cosa) cosec (3.48) S = 23- ka re sp 9 s is the far field amplitude of the ring (spacer) antenna. Here, a (=a/5) is the s 105.. UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F - -5 *0 0 0 0 (dt _-7 - -9 L-11 — 13 0 0. i 0 — 15 2 I 4 I 6 ka FIG. 3-27: COMPARISON OF MEASURED AND THEORETICAL NOSE-ON CROSS SECTIONS FOR A PURE CONE-SPHERE. 106 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (center) radius of the ring and Fe i is the excitation strength. Assuming r and 0 are, at most, slowly varying functions of ka, S will beat in and sp out of phase with Ss as a consequence of the large phase possessed by Ss and the resulting period of oscilllation is in excellent agreement with that of the high frequency oscillation displayed by the cross section of model LSP. If, therefore we average out these oscillations to obtain a mean cross section, a, for model LSP, it now follws from Eq. (3.47) that 2 2 c - +2 Ik H' (3.49) =2 2 a +l/ p 2 27 enabling us to deduce P as a function of ka from the measured data. The resulting curve for r is shown in Fig. 3-28 along with the curve previously deduced from four isolated sets of measurements of the surface field behavior. Using the curve now obtained, the measured data for the LSP cross section can be reproduced precisely, and though the curve for r does differ in several particulars from that used in "The Agreement Item," the two are reasonably close at the two frequencies (corresponding to ka = 2.98 and 4.51) which were considered in detail there. 3.4 Effective Estimates for the Nose-on Backscattering of Flat-Back (FB) Models. (U) In the light of the analyses given in Section 3.2 of Goodrich et al, 1967b, the expression for the nose on backscattering cross section of an FB (or ID) model is now as follows: 2 = 1/ S8 + S3 (3.50) x where __ 107 UNCLASSIFIED

UNCLASSIFIED THE UNIVER SITY OF MICHIGAN 8525-1-F 8 - 6. 4 r 2 -0. /.~/ ----- J — Je J0 0 2 Ika 4 6 0 2 ka 4 6 N..ff. I 0 0.25 I 0.5 ka s 0.75 I 1.0 I 1.25 FIG. 3-28: EXCITATION INCIDENCE. ( ---) from STRENGTH K OF RING SPACER FOR NOSE-ON Deduced (-) from present far field data; postulated isolated surface field measurements. 108 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F S - i/4 tan2 -2ik (a cot a cos a + b sin a) (3.51) S= i/4 tan e (3.51) is the tip contribution; 2 -2ikb sin a S = i/4 sec a e A(kb) (3.52) is the join contribution, with the factor A(kb) as given in Section 3. 1.3 of Goodrich et al, 1967b: and S3 is the net creeping wave contribution which differs from the function discussed and computed in Section 5.2 of the present report only in an enhancement factor, viz. S3 = yS (3.53) a kb 1/3 e2ik(a-b)+ir/12 1 (1) Scw 2b 2/ 2 S (3. 54) cw 2b 2k (a-b) Ai(-1) S )is here the creeping wave contribution for a sphere of radius b, and is computed from the formula (1) 4 i7r/3 e Dr /3 3 1 S+ 2 2 (323 + 9)1 60T2 1 1 Ai(_-jl) exp iTrkb -e - T7r/6 T7 /6 (31 - 9)t (3.55) 60ir kb 1-e 1/3 (1) (see Senior 1965), where T = (kb/2)1/3 Extensive tables of S()(x) have been given in the above-mentioned report and in Section 5.3, and have since been extended down to x = 0.05. (U) It would seem natural to take the creeping wave enhancement factor Y to be the same as for a cone-sphere, and that is what is advocated. How ever, the purpose of the present Section is merely to investigate the general ___ 109 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F character of the cross section predicted by (3. 50) to determine the effectiveness of this formula. To these ends we shall: (a) Ignore any creeping wave enhancement, i.e. take -y = 1; and (b) omit the tip contribution S1. From an examination of the magnitudes of the terms in (3. 50), it is clear that the effect of S1 would be noticable only near deep minima in the cross section. (U) The join factor A(kb) was discussed in Section 3. 1.3 of Goodrich et al, 1967b, and asympotic expansions for large and small kb were there presented. There is the question as to which of these to employ for the values of kb near unity that are of most interest to us; and as to how many terms to retain in the expansion in order to achieve the most effective numerical values. From the initial computations that have been performed it appears probable that the best (numerical) extimates are obtained by using only the leading term in the high frequency (large kb) expansion, and to begin with we shall therefore take A(kb) = a/b. (3.56) The resulting formula for the nose-on cross section is now 2 2 = 1/ S + S (3.57) 2 - 1 2 3+ 3 with - i a 2 -2 ikb sin a(3.58) S sec a e (3.58) and 110 _______. UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F a /kb) /3 e2ik(a -b)+ir/12 (kb). (3.59) 3 2b v/2k (a - b)' 1lAi(-_) k 2 (U) The computation of the above expression for cr/X is quite straightforward, and is facilitated using the values for S (kb) computed from Eq. (3.55) and listed in Table I11-3. Observe that the modulus and phase maintain their smooth character even down to ka = 0.1 and beyond. S (1)(kb) is graphed in Fig. 3-29 and in Fig. 3-30 IS3 is plotted as a function of ka for b/a = 0.5, 0.25, and 0.1. Formally at least, these three curves are identical to the corresponding ones in Fig. 4-7 of Section 4.2 over the ranges of ka for which they are shown in the latter Figure, but in contrast to the roundabout method of computation there adopted, the present curves have been obtained directly from Eq. (3.59). Notice that for small ka the moduli tend to increase with decreasing ka, with the up-swing setting in at a larger value of ka as b/a decreases. It becomes apparent at about kb = 0.1 regardless of b/a. The lowest curve in Fig. 3-30 is for b/a = 1, and is simply S ()(ka) transcribed from Fig. 3-29. (U) Even at this stage and without more detailed computations it is possible to get a general impression about the behavior of the cross section U/Xk for different b/a, and for this purpose we shall henceforth take a = 90 as appropriate to the FB (but not the ID) models. We then have S2 = 0.25627 -, (3.60) 2 b which is independent of ka for fixed b/a.. The values corresponding to b/a = 1, 0.5, 0.25 and 0.1 are indicated by the horizontal lines in Fig. 3-30, and whereas for b/a = 0. 1 the join contribution always exceeds the creeping 111 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F TABLE III-3 (1) (1) kb S (kb) arg S (kb), degrees 0.05 0.32065 190.017 0.10 0.33389 194.682 0.15 0.34615 202. 183 0.20 0.35629 210.555 0.25 0.36488 219. 320 0.30 0.37230 228.296 0.40 0.38462 246.592 0. 50 0.39459 265. 128 0. 60 0.40286 283.796 0.70 0.40988 302.537 0.80 0.41588 321.321 0.90 0.42107 340.136 1.00 0.42558 358.963 1.10 0.42950 377.802 1.20 0.43291 396.645 1.30 0.43590 415.490 1.40 0.43849 434.333 1.50 0.44075 453.178 1.60 0.44268 472.009 1.70 0.44435 490.837 1.80 0.44576 509.663 1.90 0.44696 528.481 2.00 0.44794 547.295 _______112 UNCLASSIFIED

U N CLASSFE THE UNIVERSITY OF MICHIGAN 8525-1-F H 01. 0 -0 H 1-4 wZ V-4 '-4 113 UNCLASSIIE

0.1 1 5; rTn 0.25 0. 1 -. - M. b/a = 0 25 * 1.0 \b/aa. 1. - = 0b/a\=0. 0. 10b/a = 1.0 0\: Cen c-l It H z cT Cz, 0 --e Pd 0 z C z Cn 5D Cl Crl m 0~ 0. 1 a a A. I.. a a a A i I a A A A A - -& -A 0.01' 1 0.1 1 FIG. 3-30: |S FOR VARIOUS b/a COMPUTED APPROPRIATE TO A SPHERE, AND EQ. (3.58). ka 10 100 FROM EQ. (3.59); THE LOWEST CURVE IS S (1) THE LINES AT LEFT SHOW S COMPUTED FROM 2j

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F wave one, and does so by almost a factor 2 in the range 1 < ka < 10 (implying relatively shallow minima), the two contributions are almost identical in this range (implying very deep minima) for b/a = 0.25. If b/a is increased still further, the creeping wave contribution dominates until such time as tie exponential decay characteristic of the larger ka has reduced it magnitude to that of the (constant) join return. (U) The locations of the maxima and minima can be determined using the almost-linear phase variation of S(1)(kb) as a function of kb. Thus, for 0.3 < kb < 3.0 arg S()(kb) = 187.916kb + 171.312 (degrees) with a maximum error of 0.6 degrees. The formula is also effective, with only slightly greater error, for kb outside this range, and using it we have arg S3 = (114.592 + 73.424 b/a) ka + 186.312 (degrees). Hence arg S3 - arg S2 = (114.592 + 91.350 b/a) ka + 96.312 (degrees), and the maxima and minima in the cross section occur at those values of ka for which this is an even or odd multiple of 180 (degrees) respectively. Some locations are as follows: b/a = 0.5 b/a = 0.25 b/a = 0.1 max. ka = 1.677 ka = 1.919 ka = 2. 131 min. = 2. 821 = 3.228 = 3.586 max. = 3.966 = 4.538 = 5.041 min. = 5.110 = 5.848 = 6.496 max. = 6.255 = 7. 158 = 7. 950 min. = 7. 399 = 8.467 = 9.405 (U) By direct computation of the expression given in Eq. (3. 57) at a sequence of values of ka, and using also our knowledge of the precise locations of the maxima and minima, the cross sections shown in Figs. 3-31 115 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 0 --10 - b/a = 0.5 a/ 2 (db) -20 --30 - ) r 2 ka 4... -T I FIG. 3-31: 2 a/,X FOR FB MODEL WITH b/a = 0.5, COMPUTED FROM EQ: (3.57). 116 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F through 3-33 are obtained. The associated values of b/a are here 0.5, 0.25 and 0.1 respectively. The curves certainly have the character necessary to fit the measured cross section data for the FB models (see Section 3.2.2 of Goodrich et al, 1967b), and both the positions and depths of the minima vary with b/a in the manner displayed by that data. But there is some discrepancy in the height of the first one or two peaks, particularly for small b/a, and to illustrate this fact we show in Figs. 3-34 and 3-36 respectively the theoretical curves for b/a = 0.4 (corresponding to model FB-4) and b/a = 0.1 (corresponding to model FB-1), along with the measured data for the cross sections. In Fig. 3-35 the theoretical curve of Fig. 3-32 for an FB model with b/a = 0.25 is compared with the measured data for ID-1 and ID-2. Both of these experimental models have b/a = 0.25 and, in consequence, the formula should give a valid* estimate for these. For purposes of comparison, we have also included in each figure and theoretical curve for a flat-backed (right circular) cone. (U) Taking first the comparison between the theoretical and experimental values for model FB-4 shown in Fig. 3-34, we observe the fact noted above, namely, that the theoretical curve overestimates the cross section, and does so to a degree which decreases with increasing ka. Thus, to fit the first maximum and minimum, a reduction of approximately 3 db is called for, but at the second maximum the required reduction has decreased to about 1.5 db. Such a changing reduction of the theoretical values would, indeed, lead to a better fit to the experimental data at all values of ka, not merely those corresponding to the maxima and minima, and we further observe that for ka less than that of the first maxima, the data points are closer to the curve of the flat backed cone than they are to the FB model curve. The slight displacement of the minima is undoubtedly due to the fact that we have used a =9 in the computation, rather than the value 7. 5 appropriate to ID-1 and ID-2. ___ 117 _ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F b/a = 0.25 0 --10 -/X2 (db) - -20 --30 - -40.2 db at 3.226 2 4 ka 6 8 FIG. 3-32: c/Xk FOR FB MODEL WITH b/a = 0.25, COMPUTED FROM EQ. (3.57). 118 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F b/a = 0.1 0 -2 Jr a/) (db) -10 --20 - 0 0 I 2 I I 4 ka I I 6! 8 FIG. 3-33: 2 /kX FOR FB MODEL WITH b/a = 0.1, COMPUTED FROM EQ: (3.57). 119 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN 10 104 --- Flat-Back - FB(b/a = 0.4) a Xt XX FB-4 Exp. /I \ I/ I % I % I 0 *0 a/x2 (db) -10 --20 - I I x x 0 2 ka 4 6 8 FIG. 3-34: MEASURED DATA (xxx) FOR THE BACKSCATTERING CROSS SECTION OF MODEL FB-4, COMPARED WITH THE THEORETICAL PREDICTION (EQ. 3.57) FOR THIS MODEL ( -) AND FOR A FLAT-BACKED CONE ( ----). 120 UNCLASSIFIED

U NCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN 10 -0 -- - Flat-Back - FB (b/a = 0.25) * ~ * ~ ID-2 Exp. X X X ID-1 Exp. / r a1 2 (db) 0 I I I.. -10' -20' I 0 2 ka 4 6 8 FIG. 3-35: MEASURED DATA FOR THE BACKSCATTERING CROSS SECTION OF MODELS ID-1 (xxx) AND ID-2 (e~~), COMPARED WITH THE THEORETICAL PREDICTION (EQ. 3.57) FOR AN FB MODEL WITH b/a = 0.25 (-) AND FOR A FLAT-BACKED CONE ( ---). 121 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN 10 - / I 0 x F xx x 11 x,,~O( ft I I I X XX 2/ (db) AI I I I I 'A X \ xI x I ~x xx I I I I I I I 'I V --- Flat-Back FB (b/a = 0. 1) xx X FB-1 Exp. I.... * - I 0 I 2 2 4 4 I I 6 I 8 8 ka FIG. 3-36: MEASURED DATA (xxx) FOR THE BACKSCATTERIT' CROSS SECTION OF MODEL FB-1, COMPARED WITH THE THEORETICAL PREDICTION (EQ. 3.57) FOR THIS MODEL (-) AND FOR A FLAT-BACKED CONE ( ---). 122 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) The same conclusions apply to the comparison shown in Fig. 3-35, but because of the unusually deep theoretical first minimum, whose depth would undoubtedly be changed were the tip contribution to be considered, we can only derive reduction factors based on the first two maxima. Note, however, the tendency of the experimental data points to follow the curve for the flat-backed cone, and this is even more clearly seen in Fig. 3-36, where the comparison for model FB-1 is given. Substantial reductions of the theoretical FB values are now required throughout the range of ka covered by the experimental data, reaching as much as 10 db in the vicinity of the first maximum. (U) Similar comparisons of the predicted and measured values of the cross sections at the maxima and minima have been carried out for models FB-2 and FB-3, and when the required reduction factors thus obtained are examined in toto, it is found that these factors are functions of kb alone, and are more or less independent of ka. From the levels of the maxima and minima for all the bodies for which experimental data is available, and excluding only those few minima where it is clear that the full depth was not plumbed experimentally or where the theoretical depth would be markedly changed were the tip scattering to be included, we obtain the voltage (or amplitude) reduction factors shown in Fig. 3-37. The quantity plotted here is the factor that must be applied to the net (join plus creeping wave) scattering amplitude in order to best fit the experimental data. Denoting this factor by B(kb) and re-plotting log (1 - B) versus log kb, we find only a small amount of scatter about a linear variation. A visual fit to the points leads to the empirical curve B(kb) = 1 - 0.25 (3.61) (kb)0. and this has been included in Fig. 3-37.....,123 - UNCLASSIFIED

1 I I - I l e I 0. 0. 0. Eq. (3.63) o x x x x Q 1 CA O m 0.5 1.0 kb 1.5 2.0 6 ) 2 EXPERIMENTALLY DEDUCED (xxx) VOLTAGE REDUCTION FACTORS FOR NET FAR FIELD 0 AMPLITUDE. The solid line is a functional fit to the data points (see Eq. (3.61) of text), whilst the broken line is the function defined in Eq. (3.63). Z 0. 0. 1071 0. 099: FIG. 3-37:

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) Since B(kb) < 0 for kb < 0.9921, it is legitimate to ask what happens to the join contribution, for example, as kb approaches (and goes beyond) this value. To answer this, we show in Fig. 3-38 the moduli of the join contributions for bodies with b/a = 0.1, 0.2, 0.5 and 1.0 as modified by the reduction factor B(kb). It is observed that each curve, as a function of ka, approaches the line corresponding to the join contribution for a flat-backed cone, namely S. = - - cosec (n - + J 2n n 2 = 0.40793 for a = 9~, (3.62) becomes almost tangent to this line, and then falls away from it. This is just the sort of behavior that we were seeking in Section 3. 5, and clearly the required join contribution is that obtained by following each curve down to its first intercept with the flat-backed value, and then following the latter line thereafter. This first intercept occurs at kb = 0. 312 regardless of b/a. (U) The prescription would, of course, be tidier if the reduced join contribution curves were truly tangent to the flat-backed one, and we note that merely by changing the formula for B(kb) from that given in Eq. (3. 61) to B(kb) = 1 -0.2625 (3.63) (kb)0 6 the tangency is assured, with the point of tangency occuring at kb = 0.236. Such a change in B(kb) in no way affects the quality of the fit shown in Fig. 3-37, but to some extent the aesthestic improvement obtained thereby is illussory. Although the particular formulae (3.61) and (3.63) were deduced from an examination of measured data for bodies having a = 9, both the formulae 125 __ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 0) 0 e-ol CD 0 z H0) 0 0 0d m CY) Co rL4 0 —.14 0; 126 UNCL~ASSFE

UNCLASSIFIED - THE UNIVERSITY OF MICHIGAN 8525-1-F themselves and the technique by which they were derived are believed valid for all pointed objects of small included angle. Since the proportionality factor in the expression (3.62) for S. is a function of a, albeit a very slowly varying one, the above expression for B(kb) will produce tangency only when a is 9. We can, however, generalize it as follows: B(kb) = 1 - C - (3.64) (kb) 0.6 where 6n sin 2 -n C 2 64 (3.65) cos a and with this value of C, the curve for the join contribution modified using the factor B given in Eq. (3.64) is tangent to the curve for the flat-backed join contribution at kb = kb1 with 21 3n sin 2 kb - - (3.66) 16 cos a for all a. Typical values are: a C kb1 7.5~ 0.2631 0.2366 9~ 0.2625 0. 2356 15~ 0.2615 0.2341 For a = 90 the resulting formula for B(kb) is, of course, identical to that in Eq. (3.63), and computed data based on this expression are given in Table m-4. 127 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F TABLE 111-4 MICHIGAN kb 0. 1 0.15 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.5 1.8 2.0 2.5 3.0 4.0 5.0 8.0 10.0 B(kb) -0. 0451 0. 1806 0. 3105 0. 3968 0.4594 0. 5451 0. 6020 0. 6433 0. 6748 0. 6999 0. 7203 0. 7375 0. 7647 0. 7942 0.8155 0. 8268 0. 8485 0. 8642 0. 8857 0. 9000 0.9246 0. 9341 (db) -14.86 -10.16 -8.03 -6.76 -5.27 -4.41 -3.83 -3.42 -3. 10 -2. 85 -2.64 -2.33 -2.00 -1.77 -1.65 -1.43 -1.27 -1.05 -0.92 -0.69 -0.59 128. UNCLASSIFIED

UNCLASSIFIED - THE UNIVERSITY OF MICHIGAN 8525-1-F (U) As a result of these studies, the prescription for the join contribution is kb kb < oo: i 2 -2 ikb sin a a S. sec a e - B(kb) (3.67) J 4 b where B(kb) is given in Eq. (3.64). ka 27r -2 ikb sin a kb < kb: S. - i cosec - e (3.68) 1 j n n Observe the phase factor ie that has been inserted, at least temporarily, into Eq. (3.68). This has been done to preserve continuity of phase through the point kb = kbl, but the same effect could have been achieved as regards the total scattering by having the creeping wave contribution discontinuous in phase at kb = kb in such a way as to counterbalance a discontinuity in S.. (U) If the reduction factor B(kb) as given in Eq. (3.64) were to be applied only to the join contribution, with the creeping wave contribution S3 (see Eq. 3. 59) left unchanged, the predicted cross sections would still depart somewhat from the measured data. This is illustrated by Figs. 3-39 and 3-40 for b/a = 0.4 and 0.1 respectively in which the original predictions (with no reduction factor included) are shown as solid lines, and the predictions with only the join contribution reduced as broken lines. (U) Nevertheless, the factor B(kb) was originally derived on the premise that it would be applied to both the join and creeping wave contributions, and when this is done we arrive at the predictions shown as broken lines in _______129 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN 0 --10 -U/X2 (db) -20 - b/a = 0.4 x XX x x v I 0 II 2 4 ka 6 8 FIG. 3-39: PREDICTED CROSS SECTIONS FOR b/a = 0.4 WITHOUT REDUCTION FACTOR (-), AND WITH REDUCTION FACTOR APPLIED TO JOIN CONTRIBUTION ONLY ( ---), COMPARED WITH EXPERIMENTAL DATA (xxx). 130 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F in Figs. 3-41 through 3-45 for b/a = 0.4, 0.3, 0.25, 0.2 and 0.1 respectively.* The agreement with the measured data is hearteningly close in each case and for all values of ka. The solid lines again show the original predictions with no reduction factor applied, and in the case of the broken lines we have, for convenience only, terminated each at the value of ka corresponding to the transition value of kb = kb, namely at ka = kb. (U) The factor B(kb) applied to the join contribution was found to have the desirable effect of smoothing the transition from the values appropriate to a smoothly-terminated cap to that corresponding to a flat-backed cone, but when we come to examine the creeping wave contribution, with or without the factor B(kb) incorporated, we find a peculiar and somewhat distressing behavior. When the original (unreduced) creeping wave contribution is plotted as a function of k for fixed b/a, we obtain the moduli shown in Fig. 3-30 and each curve intersects the straight line representing the creeping wave modulus for a flat-backed cone at an acute angle. Application of the factor B(kb) to these curves has the effect of bending each one over the lower end of the ka range, with the bending being greatest the smaller b/a is, and the resulting curves no longer reach the flat-backed one. Undesirable as this is, the trouble may not seem unsurmountable, but a far more graphic illustration of the shortcomings of the creeping wave expression is obtained by plotting S3 as a function of b/a for fixed ka. The results are shown in Fig. 3-46 for the case in which no reduction is applied, and in Fig. 3-47 for the case in which the factor B(kb) is incorporated. On the left of each graph are the values appropriate to a flat-backed cone, namely 0.61569 ka, and on the right are the values of S( appropriate to a sphere, Taking * 0 In Fig. 3-43 the prediction has again been based on the choice a = 9 rather than the value 7. 5 appropriate to the ID models. 131 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN 10 - b/a = 0.1 / / \ \ / x \ x A xx W I 1 I I I I 1 2/X2 (db) xl ix x41 I I I I I I I I \ vi \ I I I I I I I I I I -20'.. I I I. I 0 2 2 II I 9 ka 4 6 6 8 FIG. 3-40: PREDICTED CROSS SECTIONS FOR b/a = 0.1 WITHOUT REDUCTION FACTOR (-), AND WITH REDUCTION FACTOR APPLIED TO JOIN CONTRIBUTION ONLY ( ---), COMPARED WITH EXPERIMENTAL DATA (xxx). 132 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN 0 --10 -2 (db) b/a = 0.4 I I V x x -30 - U 0!-v 2 ka 4! 6 8 FIG. 3-41: PREDICTED CROSS SECTIONS FOR b/a = 0.4 WITHOUT (-) AND WITH REDUCTION FACTOR B(kb) FULLY INCORPORATED, COMPARED WITH EXPERIMENTAL DATA (xxx). 133 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F b/a = 0.3 0 --10 - /2d (db) I I I I I I I I I I I I I I -20 --30 - 0 2 ka 4 6 8 FIG. 3-42: PREDICTED CROSS SECTIONS FOR b/a = 0.3 WITHOUT (-) AND WITH ( ---) REDUCTION FACTOR B(kb) FULLY INCORPORATED WITH EXPERIMENTAL DATA (xxx). 134 UNCLASSIFIED

U NCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN b/a = 0.25 0 - /1 I r I I I -10 - I I I I (ri2 (/X2 (db) iI v -30 - I I ~, I I 0 9 I2 2 ka I 6 8 4 FIG. 3-43: PREDICTED CROSS SECTIONS FOR b/a = 0.25 WITHOUT ( —) AND WITH ( ---) REDUCTION FACTOR B(kb) FULLY INCORPORATED, COMPARED WITH EXPERIMENTAL DATA FOR MODEL ID-1 (xxx) AND ID-2 (eee). 135 UNCLASSIFIED

UNCLASSIFIED THE UNIVERS ITY OF 8525-1-F MICHIGAN b/a = 0.2 I " I x I I I -10 - a/X2 (db) -20 --30 - \I v N 9 N ff I N 9 I I 0 I 2 4 ka 6 8 FIG. 3-44: PREDICTED CROSS SECTIONS FOR b/a = 0.2 WITHOUT (-) AND WITH ( ---) REDUCTION FACTOR B(kb) FULLY INCORPORATED, COMPATED WITH EXPERIMENTAL DATA (xxx). 136 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN 10 -0 -a/X2 (db) -10 b/a = 0. 1 / / / / I / I I I / / xx XX X ^x X ) X>l xx - 0 l 2 ka 4 ka 4! 6 l I 8 FIG. 3-45: PREDICTED CROSS SECTIONS FOR b/a = 0.1 WITHOUT ( —) AND WITH ( ---) REDUCTION FACTOR B(kb) FULLY INCORPORATED, COMPARED WITH EXPERIMENTAL DATA (xxx). 137 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 10 -1 -/s31 0. 1 - 0.01 - 4-a) sD Q) 0 ka=2 ka=5 ' \ \' ka=10 ' ' - - - ^ ka = 50 k ka= 100 0 0.25 0.5 b/a 0.75 1.0 FIG. 3-46: THE MODULUS OF THE CREEPING WAVE WITH NO REDUCTION; COMPUTED FROM CONTRIBUTION EQ. (3.59). 138 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN I, IU1 - If 4) 0 0 rd BS3 O, 1" 0.1 -a 0.011 4.) *S c; a.94 O Cd I T3 v P4 ka =50 ka = 100.. Mm% 0 0.25 05 0.5 75 0.75 I 1.0 FIG. 3-47: b/a THE MODULUS OF THE CREEPING WAVE CONTRIBUTION WITH THE REDUCTION FACTOR B(kb) INCORPORATED FOR 0 < b/a < 1. 139 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F first Fig. 3-46 we observe that only for the very largest ka (> 50 say) is there any evidence of a natural continuation into the flat-backed cone value. But much more distressing is the upswing in each curve that occurs as b/a approaches unity, and that takes place for b/a as low as 0. 5 for ka as small as 2. Such an upswing occurs no matter how large ka is, and in view of the factor /2k(a- b) in the denominator of the expression for S3, it was predictable that it would do so. Not expected, however, was the fact that it would be significant at such low values of b/a. It occurs, for example, at b/a - 0.84 for ka = 100, and here 2k(a - b) - 32. This is certainly large compared with unity. (U) Incorporation of the reduction factor B(kb) (see Fig. 3-47) does nothing to diminish these undesirable trends in behavior and does, in fact, accentuate them. The curves for ka = 5 and 2 now show a reverse turn-over for small b/a which further complicates the task of arriving at a smooth transition to the flat-backed cone results. Indeed, the entire curve for ka = 2 is now most peculiar, and is almost the mirror image of what one would prefer. (U) In spite of all these difficulties, the theoretical cross section prediction that we have given is numerically acceptable providing b/a < 0.5 (as it was in all cases that we exam.;ed), and the peculiarities that are evident in the behavior of the creeping wave estimates for small b/a are entirely masked by the large join contributions that then occur. It is, however, incumbent upon us to produce a theoretically tenable estimate of the creeping wave contribution, not only to permit cross section estimates for 0. 5 < b/a < 1. 0, but also to explain (and almost certainly remove) the pecularities evident in Figs. 3-46 and 3-47. Indeed, these figures cast some doubt even on the validity of Eq. (3.59) on an asymptotic basis for large ka, kb and k(a - b), and using the numerical 'feel' obtained from them, one is tempted to believe UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F that a viable expression for the creeping wave modulus could be obtained using smooth parabolic-shaped lines joining the extreme values representing the results for the flat-backed cone and the cone-sphere. 3.5 A Quantitative Failure of a Scattering Estimate (U) In Section 3.1.4 of Goodrich et al, 1967b we examined the nose-on scattering from models ID-1 and -2 and derived an expression for the backscattering cross section based on the 'corrected'formulae for the join and creeping wave contributions given in Sections 3. 1.3 and 3. 1.2 (of the same report) respectively. As is true with most of our results, however, this expression is an asymptotic one, valid only for large kL where L is the smallest 'effectivet dimension of the body, and in order to justify its use, it was conceded that kL may have to exceed (say) 5. Nevertheless, we often have to stretch the limit with asymptotic formulae, and in some cases at least (e.g. a sphere, or a cone-sphere) it has been found possible to obtain results which are quantitively accurate for values of kL as small as unity (or even less). (U) The ID models are characterised by two effective radii of curvature: the longitudinal radius, b, at the shadow boundary, and the transverse radius, a (which is tantamount to the maximum radius). For both ID models, a = 4b. We note in passing that for the FB models subsequently measured, a/b varies from 2.5 for model FB-4 to 10 for model FB-1. (U) Since b < a, kb is the parameter that can be expected to limit the validity of the asymptotic expansion for the join and creeping wave contributions, but if we were to demand that kb > 5, this would, for the ID models, exclude all of the range out to ka = 20, and thereby exclude the range of most practical interest, Indeed, for the ID models, the available experimental data for the nose-on cross section is confined to the range 1.2 < ka < 4.5, corresponding to kb satisfying 0.3 < kb < 1. 125, and to believe 141 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F that the formulae will provide accurate estimates in the range may be no more than a pious hope. This is particularly true of the join contribution, which undergoes a marked change of character as kb approaches 3(a-b) 3 a -(=- for a = 4b) but the absence of any rapid change in the creeping wave contribution is no criterion for judging the accuracy of its formula as kb approaches unity or less. (U) These facts notwithstanding, computations of the nose-on cross section of the ID model for 0.25 < kb < 1. 25 were performed and, as shown in Section 3. 1. 4 of Goodrich et al, 1967b the results are in tolerable (but by no means good) agreement with the experimental data. In brief, our feeling was that all qualitative features of the experimental data were predicted by the formula, and that any quantitative shortcomings in this range of kb would rapidly disappear with increasing kb. However, when we came to apply the same analytical prescription to the estimation of oblique angle scattering from an ID model, even the qualitative features of the measured pattern were not reproduced, and it appeared that any agreement with experiment for this range of kb was little more than coincidental. And when the more extensive measured data for the FB models became available, it was found that the formula which, for given ka and b/a, is the same regardless of whether the back is flat or indented, was quite inadequate for the estimation of the nose-on behavior even for kb as large as 2 with a = 2. 5 b. (U) In order to try to pinpoint the reasons for these shortcomings, let us survey some of the successes and failures of the theoretical prescription for backscattering by a body with smooth non-spherical termination. In the first place, the theory predicts that for nose-on and near nose-on incidence on 142 UNCLASSIFIED

U NCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F an indented-back model, the scattering is a function of ka and a/b, and is independent of c. This is confirmed by the identity of the measured nose-on cross sections of models ID-1 and 2 (see Section 3. 1. 3 "f Goodrich et al, 1967b) Since the parameter c does not enter into the formula, L. -ross section should remain unchanged if we allow c to become infinite whilst keeping a and a/b fixed. The resulting model is of the FB type, and to show that the scattering is still the same, we show in Fig. 3-48 the measured nose-on cross sections of models FB-2 and 3 (for which a/b = 5 and 3. 33 respectively) and of model ID-2 (for which a/b = 4). As required, the measured values for the FB models effectively bracket the data for ID-2. (U) The nature of the formula for the creeping wave contribution S cw for a body with non-spherical rear is such that over the range 1 < ka < 10 the magnitude of S is almost independent of a/b and smaller than the cw analogous quantity for a flat-backed cone by a factor 4 or more. The consequences of this are two-fold: (a) the character of the predicted oblique angle scattering more closely resemble the behavior of a cone-sphere rather than a flat-backed cone over this range of ka. In contrast, the measured data for the ID models, for example, is more closely akin to the data for a flatbacked cone, as is evident from Fig. 3-49 where we show the measured patterns for ID-2, a flat-backed cone* (Keys and Primich, 1959) and a conesphere for ka = 2.98 and 4. 51; (b) the formula is unable to reproduce the correct magnitudes for the peaks in the nose-on backscattering cross section of the FB models as a function of ka, and, in particular, fails completely to predicted the increasing depth of the minimum near ka = 5. 5 as b/a decreases from 10 to 2. 5. The fact that the formula did lead to tolerable agreement with the nose-on data for ID-2 in the vicinity of the minimum near ka = 3. 3 was entirely a consequence of the rapid variation in the join contribution taking * The actual ka - values for the flat-backed cone are 3. 08 and 4. 56. _________ 143 ___ U NCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F Or 0.0 0. 00 0 0 0 0 -5s 0 x x x x x* x 0 -10 - /X2 (db) x 0 x x x x x x 15r x x xx I I I I I I! -4 0 2 ka 4 6 FIG. 3-48: COMPARISON OF MEASURED DATA FOR MODELS ID-2, (a/b = 4), FB-2 o*o (a/b = 5) AND FB-3, xxx (a/b = 3.330). 144 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN ka = 2.98 I I I I I I I o/ 2 (db) 54 36 18 18 I 0 II 18 36 54 ka = 4.51 Flat-backed I II FIG. 3-49: COMPARISON OF MEASURED SCATTERING PATTERNS FOR MODEL ID-2 WITH THOSE OF FLAT-BACKED CONES AND CONE-SPHERES. 145 UNCLASSIFIED

: U NCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F place near kb = 0. 8, and the more one studies the measured data (see Fig. 3-50), the more one is led to believe that the agreement was merely fortuitious. (U) Because of the tendency for the measured data to approach the flatbacked cone behavior as a/b increases, and to approach it quite closely for even small values of a/b ( > 1) for ka in the range 1 to 10, it is of interest to examine the formula for nose-on scattering from a flat-backed cone. It is known (Kleinman and Senior, 1963) that the formula based on Keller's second order theory is in close agreement with measured data even for ka as small as unity (or less), particularly for small values of the half-cone angle a. The resulting expression for the far field amplitude in the backscattering direction for nose-on incidence is wr 37r 2ika - cos- - cos- 4 ka 27r ka wr n n e S = - -cosec - --- sec - 2n n 2 n w 3wT 2 (3.69) 4n cos- cos 2 n ka with n = 3/2 + a/7r. The first term on the right hand side of (3.69) is clearl the degenerate form of the join contribution as b - 0, where as the second term is the degenerate form of the creeping wave return. Denoting these by S. and S respectively, we have, for a = 90 S. = 0.40793, arg S. = 0 Sw = 0.61569 rka, arg Sw = 2ka 4 3. (3.70) cw ' ' cw 4 * Whereas a = 7. 5 for most of our previous models, the FB series have a = 9. The change in the numerical constants in (3. 70) on going to a = 7. 5 is small, with the first being replaced by 0.40339 and the second 0. 63348. 146 _ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 5 - -#0ae&** it / / / / / Flat-backed - 0 / a/X2 -(db) -5 -i -10 --15 -20 /. / / /. /0 I - r 0 \ \ \ \ / 1%' \ I I I * X P. I I ~ x I I Computed / \ I I \ I \ I <y wi / \ 0 I I I I 1' \ I I ' I I \ I \l v V / / / I i I I 1 ConeSphere I I V 0 0 0 -1 F 2! 4 ka 4 ka I 6 - FIG. 3-50: COMPARISON OF MEASURED DATA FOR NOSE-ON BACKSCATTERING OF MODEL ID-1 (xxx) AND ID-2 (ooo) WITH COMPUTED CURVES FOR ID MODEL ( —), FLAT-BACKED CONE ( ----) AND CONESPHERE ( — -). 147 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) The modulus of the degenerate creeping wave contribution for a flat-backed cone (b/a = 0) is plotted as a function of ka in Fig. 3-51, along with the corresponding quantity for a cone-sphere (b/a = 1). For simplicity, we have omitted any enhancement factor from the latter, and have taken the results directly from Senior (1967b), supplemented by new computations for ka > 10. It will be observed that the sphere curve is almost asymptotic to the curve for a flat-backed cone for ka small, and the latter provides a natural continuation of the former for ka < 0.3, say. (U) It is a most reasonable assumption that for any body having b/a < 1 the curve of the creeping wave modulus as a function of ka will lie between the bounds established by the flat-backed cone and sphere, and this is regardless of whether we insist that b remain constant, or allow b/a to be held fixed; and we further expect that as b/a decreases, the value of ka at which the curve for the creeping wave modulus 'breaks away' from the curve for a flat-backed cone will increase. (U) Using asymptotic analyses, a formula for the creeping wave contribution associated with an ID or FB model has been derived. This was presented in Section 3. 1.2 of Goodrich et al, 1967b, and has been computed as a function of ka for b = 0.25 a (appropriate to the ID models). From an inspection of the formula it can be seen that, for a body having b = c a, the creeping wave contribution is ka 1 3 i(1-4o)2 r S (ka/4a) = i4 I 1-4 e S (3. 71) cw 4 c(l-) M.)e aLc-Ja = 0.25 where the last factor on the right hand side is the quantity already computed for the body having b = 0.25 a, and this enables us to trivially deduce the creeping wave return for bodies having other (fixed) ratios of b to a. In _ 148 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN 1( I sl 0.1 0.01 0.1 1 ka 10 100 FIG. 3-51: COMPUTED MODULI FOR CREEPING WAVE CONTRIBUTION FOR BACKSCATTERING AT NOSE-ON INCIDENCE. 149 UNCLASSIFIED

U NCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F this manner we have computed the modulus of the creeping wave contribution for bodies having a = 0. 1, 0.2, 0.3 and 0.4, and two of the resulting curves are included in Fig. 3-51. They do not in any manner conform to the guidelines given above, and, for ka less than (about) 9, the moduli are even less than for a cone-sphere (for which a = 1). What is more, for ka < 9, the moduli are relatively independent of a, and this has been found true for all the values of a considered. It is clear that such values for the creeping wave contribution are not likely to be consistent with the marked dependence on a of the nose-on cross sections of the FB models. (U) In attempting to improve the theoretical estimate of the creeping wave contribution for a body with non-spherical rear, particularly for values of kb not much greater than unity, it should be borne in mind that the formula is, without doubt, correct for sufficiently large kb. For kb ~> 1, we can replace the function Aq() appearing in Eq. (3.1) of Goodrich et al, 1967b, by the leading term in its asymptotic expansions for large argument, viz. *7T 5, 1l Si 6 1 il3e q() - e 1 (3.72) 21 A(i) 2)} e and thereby obtain ka (kb 2/3 e2 ik(a - b) + i kb s 4 \2! ^ i7r kb l)1/3 3 1 i231ej) e 2 (3.73) 2 4^ 150 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F which is in accordance with the result provided by the geometrical theory of diffreaction. Using this asymptotic approximation, we have S(ka, a1 ka) 2/3 t S(ka. a2 ka) (a/a2) 1- a exp I 2 r,8 (ka/2) 1/3 (a1/3 - a1/3) '2 1 e - 2(/2) ( -a (3.74) with b = a a, and for sufficiently large ka, the right hand side of (3.74) is less than unity for a2 < a. Thus, at high enough frequencies, the magnitude of the creeping wave contribution increases with decreasing a, as expected. This is otherwise evident from the computations of the original formula for S: for ka large (near, say, 100), the curves correctly ordered as a function of a, being closest to the sphere curve for a near unity, and tending upwards towards the flat-backed cone curve as a decreases. Note that the relation (3. 74) does not, unfortunately, enable us to deduce the creeping wave amplitude for general a from its known values for either of the limiting cases a = 0 or 1. (U) Even though the formula for S is asymptotically correct for large kb, and can therefore be expected to yield an accurate numerical estimate of the creeping wave return for kb large enough, inspection of Fig. 3-51 suggests that for ka as large as 50 with a = 0.4 (implying kb = 20) the estimated modulus is beginning to depart from the behavior expected of it. The slope of the curve here is somewhat different from that desired, and this is certainly true of the curve for a = 0. 1 when ka = 100. It seems probable that we are, in fact, seeking two refinements of theory: one which is significant even for kb as large as 10 or 20, and which is doubtless associated with the higher order attenuation effects that have been neglected in the deri ' ' 151, UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F vation of the creeping wave expression: and one which is only of concern for kb quite small (less than 2, say), and has as its origin the rapid transition in the birth and launch weights as the radius of curvature of the shadow boundary decreases. The investigation of the latter is a relatively basic study, and is already in progress using a parabolic cylinder as the model. However, the first refinement may be the more significant one for practical purposes, and though a rigorous derivation of the correction would be a lengthy task, it may still be possible to arrive at an empirical and numerically effective correction. __ 152 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN - 8525-1-F IV THEORETICAL STUDIES (SURF) 4. 1 Introduction (U) In analyzing the surface field data and the backscatter data obtained experimentally, it was necessary to carry out theoretical studies of the physical factors which contribute to the observed effects. Where an exact theoretical description could not be constructed and where approximate methods did not give sufficiently accurate results, empirical modifications of formulas were obtained from the experimental data itself. Where a synthesis of a cross section formula for a specific shape could not be developed by a direct solution of the scattering problem for that shape, the formula was obtained by the extension of the solution for similar shape. This section reports on studies of this nature. 4.2 An Empirical Correction to the Estimated Creeping Wave Contribution for a Non-Spherical Body. (U) In Section 3.1.2 of Goodrich et al 1967b, the 'corrected' asymptotic expression* was given for the far field amplitude attributable to the creeping wave excited on a non-spherical body at symmetrical incidence. Specifically, for a body whose transverse radius of curvature at the shadow boundary is a, and whose longitudinal radius is b, with the latter radius remaining the same up to or beyond the point at which the profile is perpendicular to the axis of symmetry, the expression quoted there is* ir kb e2ik(a-b)-i/4 kb 2/3 [ ei/6 21/3 Sc =-r ka e1 e_ + 2- W/3^-^-T cw 2k (a - b) 2 307/ (kb) / 2 131 -i~r/6l E 3 A ---)} 2 ) exp,- e -6) (4.1) There is one additional correction over and above those previously described. This amounts to the removal of a factor ei /2 from the space factor and, hence, from the overall expression for Scw, and is necessitated by the fact that we are dealing with the (electric) field component nortnal to the surface. --- -153 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F where = r/2 (kb/2)1/3 (4.2) the js are the zeros of the Airy function derivative, and q(9) = q( )() is s the function tabulated on pp. 8-18 and 8-19 of Logan (1959). Detailed computations have been made as a function of ka, 1 < ka < 100, for a body having b = 0.25a, and using the fact that for a body having b = aa, 1 3 i-a (1 -4a)a Sc (ka/4a) = - e 2a Lsc, (4.3) cw 4(l-) e J -a c o5 where the last factor is the quantity already computed, it is a trivial matter to deduce the creeping wave contribution for a body having a L 0.25. (U) The above expression for S was derived under the assumption that kb >> 1 and, as such, is not valid in the limiting case of a flat-backed cone (kb- 0). It also breaks down in the limiting case of a sphere (b - a) due to the 'space' factor involving a - b in the denominator, and we therefore cannot investigate either of these cases directly using (4. 1). However, it is believed that these cases, whose creeping wave effects can be determined by alternative methods, provide effective bounds on the creeping wave contribution that the present' class of bodies can produce; and as shown in Section 3. 5 the results computed using (4. 1) violate these bounds, and do not lead to good agreement with the measured data for backscattering from cones with non-spherical terminations (e.g. the ID and FB models). (U) There are probably two main sources of these numerical discrepancies: firstly, the inaccuracy of the high frequency expression for the birth and launch weights of the creeping waves when kb is not large, and secondly, the failure of the expression (4. 1) to take into account the higher order atten uation factors which are known to be important' even at relatively high fre1___ 154 __ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F quencies (Senior, 1965). In this present section we shall endeavor to provide an empirical (or numerical) correction to the formula for S sufficient to cw remove some of the numerical inadequacies, and in so doing will concentrate on the higher order attenuation factors as the main source of the correction. (U) Figure 4-1 shows the modulus of the creeping wave contribution for a = 0. 1, 0.25 and 0.5 computed from Eq. (4.1). The results for a = 0. 1 and 0. 5 have been deduced from those for a = 0.25 using (4. 3), and this accounts for the reduced spans of the curves in these cases. Note the close agreement between the computed values for all three values of a when ka is less than (about) 9. The broken curve in Fig. 4-1 is for a sphere (a = 1), and will be discussed in a moment. (U) The effect of omitting the infinite sum from the right hand side of Eq. (4.1), i. e., replacing the interior of the square bracket by q(5) alone, is illustrated in Fig. 4-2 for the same three values of a. It will be observed that the main result is an increase in the level of each curve, with the bend that formerly characterized each curve for small ka being somewhat reduced. The broken curve is again that for a sphere. (U) To emphasize the increase that occurs with the deletion of the series, the results obtained with and without the series included for a = 0.25 are replotted in Fig. 4-3. It is seen that the increase can be as much as 33. percent (for ka in the vicinity of 2), and is still in excess of 10 percent at ka = 20 (implying kb = 5). This change is surprising as much in its direction as in its magnitude. The origin (and intent) of the series is a correction to take into account the finite transverse radius of curvature of the body and, as such, would be expected to include, at least in part, the effect of higher order attenuation factors. Certainly the expression with the series included should be more accurate than with the series deleted. As a result of our previous work, however, it was concluded (see Section 3. 5) that Eq. 155 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F LO0 r - I cZO 0 Lo0 0 LOCO 0) & 0 ~ I II to Lf'. 14 Co; * * 4 d r,4 0 co 156 UNCLASSIFIED

0.6 0.5 0.4 js 00 Cn Ul I ^ 0.3 0.3 p1 t1 Z C PT1 0 Pt Pf 0 z z C) rr '1 mn 0.1 O L 1 5 10 ka 50 100 FIG. 4-2: MODULI OF THE CREEPING WAVE CONTRIBUTION FOR BODIES HAVING b = aa COMPUTED FROM EQ. (4.1) WITH THE SERIES DELETED, COMPARED WITH A SPHERE CURVE ( ---) COMPUTED FROM EQ. (4.7).

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN 0.45 0. 35 Is cw 0.25 0.15 - N -\ / / / / / / / / / / / / / / / / / / / / / / / I a I I I I I ~ ~ I I ~ r * a.. I 5 0 ka 10 ka 50 100 FIG. 4-3: MODULUS OF CREEPING WAVE CONTRIBUTION FOR BODY HAVING b = 0.25a, COMPUTED FROM EQ. (4.1) WITH ( —) AND WITHOUT ( ---) THE SUMMATION INCLUDED. 158 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (4. 1) under-estimated the true magnitude of the creeping wave effect, particularly for ka < 10, and it would therefore seem most natural were the deletion of the series to reduce (rather than increase) the magnitude. (U) Although the creeping wave contribution for a sphere is not immediately derivable from Eq. (4.1), we can obtain one or more expressions for comparison with (4. 1) by going back to the basic analysis for a sphere given by Senior (1965). If we retain only the dominant term for high frequencies in the expression for S given in Eq. (90) of the above reference, we have 5 (ka)4/3 i7r(ka+l/3) 1 e cw 2 e 2 e (4.4) exp i7r (ka/2) 1/3 e3 (4.4) and if we now introduce the effect of all higher order (s > 1) creeping waves, L then S S, where CW CW SL = 2 (k/2)43 eir (ka - 1/2) (4.5) with = (ka/2)1/3. (4.6) Alternatively, the exponential in Eq. (4.4) can be written as the product of two exponentials, each with half the exponent shown in (4.4). To introduce the effects of additional creeping waves into each exponential separately now entails the physical assumption of the waves being launched and re-born at some point, and is mathematically justifiable only at such high frequencies that the leading term in each sum above suffices, but if we do sum in this _________ 159 ___ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN — 8525-1-F manner over each exponential separately, we can arrive at an approximate expression for S in the form S -- S, where cw cw cw pw i (ka)S 4 4/3 eir(ka- 1/3)~l {Ai(-,1)} [(2] (4 7) with e = 7T/2 (ka/2)1/3. (4.8) This is analogous to Eq. (4.1) with the series deleted, and it was using Eq. (4. 7) that the sphere curve given in Fig. 4-2 was computed. We note in passing that for all ka > 0. 5, S and SP differ by less than 6 percent. cw cw This reinforces somewhat our acceptance of the reasoning leading up to Eq. (4. 7) and certainly demonstrates the insignificance of all higher order creeping wave contributions. (U) In line with the analysis leading up to Eq. (4.1), we can seek to 'improve' the estimate SP of the sphere creeping wave by incorporating cw the series expansion contained in (4. 1), and in this way we obtain s -4 ()4/3 ei (ka- 1/3)3 Ai(3) (2 Cw 2 1.7r e6 {21/3 1 + (1 + 3O (ka)2/3 Ai(- 2s + 3/3s ) exp (-P2 Is e 6) (4.9) where 52 is as given in Eq. (4.8). This is, of course, a function of ka only, and its modulus is shown as the broken line in Fig. 4-2. It bears about the same relationship to the results for a = 1 as does the sphere curve based on Eq. (4.7) to the computations of Eq. (4.1) with the series deleted. 160 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) Having transfered our attention to the sphere creeping wave, we are in a position to estimate the accuracy of numerical data obtained from Eqs. (4.5), (4. 7) or (4.9) by using our knowledge of the true creeping wave contribution for this body (Senior, 1965), and in Fig. 4-4 the moduli computed from Eqs. (4. 7) and (4.9) are compared with the exact values. As suspected from our studies of Figs 4-1 through 4-3, the incorporation of the 'correction' provided by the infinite series actually worsens the accuracy of the numerical approximation, and Eq. (4.7) provides a somewhat better approximation than does (4.9). Even (4.7), however, does not yield a good approximation at those values of ka of most practical interest. At ka = 5, for example, the estimated modulus is in error by almost 60 percent, and the error is still of order 10 percent for ka approaching 100. This in spite of the fact that (4.8) furnishes the correct leading term in the asymptotic expansion for large ka, and is numerically equivalent to Eq. (4.4). This is, in turn, the approximation furnished by the geometrical theory of diffraction, and its inadequacy for ka < 10 has previously been commented upon (Senior, 1965). Moreover, the primary source of the discrepancy is known to be the neglect of higher order terms in the expansion for the 'true' creeping wave decay factor. (U) Inasmuchas the exact expression for the sphere creeping wave contribution is available, it is a trivial matter to find the numerical factor which, when applied to Eq. (4.7) (or, indeed, Eq. (4.9)), will yield the correct estimate for the creeping wave. Confining ourselves henceforth to the situation in which the infinite series is omitted, we show in Figs. 4-5 and 4-6 the modulus and phase respectively of the ratio of the true creeping wave contribution for a sphere to that computed using Eq. (4. 7), i. e. r(ka) = |(ka) ei(ka) = scw Sp I cw _______________________ 161 ______I_____ UNCLASSIFIED

UNCLASIFIE THE UNIVERSITY OF MICHIGAN 8525-1-F / 0 /P /z //d 0 /L /> 0 m,) 162 UNCL~ASSFE

(1) W Ir Ul I I SCWCW (2), 4 d 'A;d (m P - 004 (P, (A I-An zzC _IC --- 50 5 ka 10 MpUTED o. 5 VR~IG WAVEl AMPLI'rTJDE FO SHR TA FI.4-5-. RATIOO XC FIG. u SING EQ - (4 -7)* INn am

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 60 50 -40 - w a) (1) k ao (2) ro 30 -ft %4 : Now 0 710 &-O% c <~1 boJ Q rj2 20 - 10 0-..... I I I Il I I I I I I I F I I I 0.5 1 ka 5 10 FIG. 4-6: PHASE DIFFERENCE BETWEEN EXACT AND APPROXMATE (EQ. 4.7) CREEPING WAVE CONTRIBUTION FOR A SPHERE. 164 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F Clearly, such a factor is of no practical utility if it relates only to a sphere, and we must next ask ourselves whether it can be used to correct the estimates for the creeping wave contributions of non-spherical terminations given in Fig. 4-2. To apply the factor ['x) with the argument x equal to ka implies that the correction is primarily determined by the transverse radius at the shadow boundary, and it is found that this does not lead to a change in the modulus of the type satisfying the intuitive guidelines discussed in Section 3.5. On the other hand, it would seem most natural that the longitudinal, rather than the transverse, radius should be the relevent parameter, and if we use the factor P(x) with x = kb, the resulting moduli of the creeping wave contribution for bodies having various a = b/a are as shown in Fig. 4-7, along with the bounding curves provided by a flat-backed cone and a conesphere. The curves now have much more the character expected of them. Their levels increase uniformly with decreasing a and, providing the curves are duly cut off at their intercepts with the curve for the flat-backed cone return, all are contained within the bounds set by the cone-sphere and flatbacked cone. It is therefore suggested that for a cone with non-spherical termination a reasonable estimate for the modulus of the creeping wave return should be obtained by following the flat-backed curve out to its intercept with the curve for the appropriate (non-zero) value of a, and thereafter following the latter curve. The estimate is, of course, only an approximation whose effectiveness for cross section purposes has yet to be determined. In particular, the abrupt change of slope at the intercept is obviously non-physical, and a better approximation should reveal a fairing-in of each higherfrequency curve as its intercept point is approached from above. We note in passing that with the four curves shown in Fig. 4-7 for a / 0, 1, the point of intercept increases from ka - 1.8 to ka - 20 as a decreases from - -..165 UNCLASSIFIED

UNCLASSIFIED THE UNIVER SITY OF 8525-1-F MICHIGAN 10 1 Isf 0. 1 0.01 0. 1. 1 ka 10 100 FIG. 4-7: AMPLITUDES OF THE CREEPING WAVE CONTRIBUTIONS FOR VARIOUS FB MODELS, DEDUCED FROM EQ. (4.7) AND USING THE CORRECTION FACTOR r(kb). 166 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN - 8525-1-F 0.5 to 0. 025, corresponding to kb decreasing from 0.9 to 0. 5. Values of ka < 1 are just those where we would expect a fairly rapid change in the nature of the scattering, with changes in the nature of the birth and launch weights of the creeping waves and, perhaps, some direct backscatter. Such transitional behavior is presently being studied using the parabolic cylinder as a model. In spite of the round-about manner in which the curves in Fig. 4-7 have been derived, their theoretical basis is relatively simple. Bearing in mind that what we have really done is to 'correct' the estimate in Eq. (4.7) or, equivalently, Eq. (4.9) using the ttrue' values for the creeping wave contribution of a sphere, it follows that for an FB model a 2 2ik(a-b) + i7r/12 S - a (kb/2)1/3 e -- cw 2b 2k(a-b) X S(1)(kb) (4.10) (Ai -) where S (kb) is the true far field amplitude of the creeping wave for a sphere of radius b. In effect we have used the function S (kb) to account for the influence of the curved portion of the path, and modified the result by the space factor and differing birth and launch weights demanded by the FB model. The function S (kb) has a relatively simple (asymptotic) expression which is numerically accurate over a very wide range of kb, and a selection of the available computed values was listed in Table IV-1 of Section 4.3. (U) The amplitude of the function S defined in Eq. (4. 10) is certainl3 very close to that required for reproducing the measured backscatter for the very close to that required for reproducing the measured backscatter for the ________________________ 167 ___________ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F FB models. Moreover, since arg S(1)(kb) is almost a linear function of kb over a substantial range of kb, it is trivial to estimate arg S. In parCW ticular, for 0.3 < kb < 3.0 arg S = (114.592 + 73.42 a) ka + 186.312 cwy (degrees) with a maximum error of 0.6 degrees, and this phase is in excellent agreement with that demanded by the positions of the maxima and minima in the FB data when ka is large. In Section 3.4 we shall attempt to predict the nose-on scattering behavior of these models using the above expression for the creeping wave contribution. 4. 3 Various Approximations for the Creeping Wave Contribution of a Sphere. (U) In the course of our attempts to provide a numerically effective expression for the creeping wave contribution of the rear of a non-spherically terminated cone (e.g. one of the FB models), it has proved desirable to investigate various approximations for the creeping wave contribution of a sphere (U) For this purpose, consider a sphere of radius a illuminated with a plane wave, One of the major contributors to the far field amplitude is the creeping wave, and if we confine ourselves to the backscattering direction, then, as shown in Senior (1965), a highly accurate approximation to the creeping wave return is S(ka) = S (ka) where* \(1) 4 i 7r/3 7/3 S()(ka) = T ei /3 {1 +02 2 (32 1 + 1) 2Ai 1 -ir/6 i 7r/6 3 _ 3 I~ 6 Ai(-41) * exp i7r ka — e 60 e- ( - 9) (4. 11) 1/60/31 with T=(ka/2)1/3 1 = 1.01879297..,.1 Note the omission of a factor ir from the third terms in the exponents of Eqs. (90) and (91) of Senior (1965). 168 UNCLASSIFIED I

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN -- 8525-1-F and Ai(-j3) = 0.53565666.... The phase origin has been taken at the center of the sphere (or, equivalently, at the shadow boundary) and a time factor e has been suppressed. (U) The asymptotic approximations leading up to Eq. (4. 11) are fully detailed in the above-referenced Report and computed values of 2/ka S (ka) are given in Table 10. As a result of the checks that have been carried out, it is concluded that for ka> 0.7, and certainly for ka> 1.0, the estimate of the creeping wave contributed provided by Eq. (4.11) is accurate enough for all practical purposes. Indeed, it is so accurate that throughout the rest of this section we shall regard Eq. (4. 11) as giving an exact expression for the creeping wave amplitude. (U) A selection of the computed values listed in Table 10 of Senior (1965) are reproduced in Table IV-1 along with the results of some later computations of the same expression for both larger and smaller values of ka. Although there can be no question of the meaningfulness of the results when ka is large, it is by no means certain that the same is true when ka < 0. 7. (U) An alternative, but inferior, approximation to the creeping wave contribution is that provided by the geometrical theory of diffraction. For numerical purposes, the associated expression can be obtained from Eq. (4.11) by omitting all higher order terms in the amplitude factor and in the exponent. If we denote the resulting expression by S (ka), where the affix k is short for Keller, we have S(ka) ' Sk(ka) where Sk(ka) = r4 eilr/3 1 exp <ifr ka-e-i7Tr TWi. (4.12) Computed values based on this expression were also given in Senior (1965). _______________________ 169 _______]____ UNCLASSIFIED

UNCLASSIFIED - THE UNIVERSITY OF 8525-1-F MICHIGAN I TABLE IV-1 ka 0.3 0.4 0.5 0.8 1.0 1.5 2.0 2.5 3.0 4.0 5.0 8.0 10.0 15.0 20.0 25.0 50.0 S (ka) 0.372297 0.384624 0.394588 0.415880 0.425575 0.440747 0.447937 0.450308 0.449529 0.442182 0.430490 0. 386793 0.357137 0.291536 0.239536 0. 198751 0.088872 arg S (ka), degrees 228.296 246. 592 265.128 322.146 358.963 453.178 547.295 641.232 734.981 921. 945 1108.302 1664.667 2033.961 2953.574 3869.785 4783.771 9335.938 170 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F A selection of this, together with some later ones, are shown in Table IV-2. The last two columns give the amplitude and phase of the factor rk(ka) S (ka L S (ka) J by which S (ka) must be multiplied to yield the exact creeping wave values. Observe the considerable error inherent in Eq. (4.12) for small values of ka, and the fact that the discrepancy (particularly in modulus) is still quantitively significant for ka as large as 10. (U) We might seek to remove the part of this discrepancy by incorporating the effect of creeping waves other than the first, but still neglecting higher order terms in both the amplitude and exponential factors. In this way we obtain S (ka) - S (ka) where L 4 ihr(ka - 1/2) S (ka) = 2 4 e -1/ () (4.13) with 71 = Tr T. The function q q(O)( -- exp i e + 5 i7r/6} 1 2f Cr Ai(-/]l has been tabulated by Logan (1959) for g1 = 0.5(0.1)8.0 (see Table T of this reference) and the affix L used above is short for Logan. Computed values based on Eq. (4.13) are given in Table IV-3, and these are seen to be almost indistinguishable from the corresponding values of Sk(ka) even for small ka. It therefore follows that pL and rk are almost identical, implying that the higher order creeping waves are not responsible for the discrepancy between the exact and approximate values for the creeping wave., This is, of course, otherwise obvious from the fact that the so-called exact expression (4.11) involves only the lowest order creeping wave. 171 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN TABLE IV-2 ka 0.3 0.4 0. 5 0.8 1.0 1.5 2.0 2.5 3.0 4.0 5.0 8.0 10.0 15.0 20.0 25.0 50.0 Sk(ka) 0. 062515 0. 079104 0.093985 0.130797 0.150426 0.187865 0.213971 0.232604 0.246002 0.262324 0.269735 0.266684 0.255656 0.221080 0.187926 0. 159565 0.075535 arg sk(ka) 162. 718 185.621 207.762 271. 559 312. 775 413. 307 511. 691 608.771 704.960 895.524 1084.444 1645.551 2016.790 2939.480 3857.543 4772.797 9328.107 [rk(ka) 5. 9553 4.8623 4.1984 3. 1796 2.8291 2.3461 2. 0934 1. 9359 1. 8273 1. 6856 1. 5960 1.4504 1. 3969 1.3187 1.2746 1.2456 1. 1765 arg[-(ka) 65.578 60.971 57.366 50.587 46. 188 39.871 35.604 32.461 30.021 26.421 23. 858 19. 116 17.171 14. 094 12.242 10.974 7.831 I 172 UNCLASSIFIED

UNCLASSIFIED -- THE UNIVERSITY OF 8525-1-F TABLE IV-3 MICHIGAN ka 0.3 0.4 0.5 0.8 1.0 1.5 2.0 2.5 3.0 4.0 5.0 8.0 10.0 15.0 20.0 25.0 50.0 SL (ka) 0.062094 0.078555 0.093357 0.130187 0. 149859 0. 187596 0.213874 0.232548 0.245921 0.262442 0.269890 0.266701 0.255784 0.223762 0. 187782 0.159842 Beyond range of arg SL(ka) 163.735 186.301 208.221 271.698 312.838 413.237 511. 607 608.710 704.926 895.452 1084.382 1645.518 2016.737 2939.402 3857.461 4772.684 Logan's table. IL(ka) 5.9957 4.8962 4.2267 3.1945 2. 8398 2.3494 2.0944 1.9364 1. 8279 1. 6849 1.5951 1.4503 1.3962 1.3029 1.2756 1.2434 arg DL(ka) 64.560 60.291 56.907 50.448 46.125 39.941 35.688 32.522 30.055 26.493 23.920 19.149 17.224 14. 172 12.324 11.087 m 173 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) Still another approximation to the sphere creeping wave follows from Eq. (4.12) on rewriting the right hand side as 1 i -ir/ exp -i1/6 e {xp <2e / -1 l and if we now insert the effect of higher order creeping waves using Logan's function q(), we have S(ka) m SP(ka) where 4 iwr (ka - 1/3)2A 2 sP(ka) = -47r T e - 1 IAi(-3)2 [q( 2 (4. 14) with 2 = 7r T/2. The affix p denotes a product form, and Eq. (4. 14) can be visualized as resulting from the instantaneous launch and re-birth of all creeping waves after they have traversed only half their path. (U) In view of our earlier finding about the numerical insignificance of the higher order creeping waves, it is to be expected that the approximation k L (4. 14) will do little to remove the discrepancies inherent in S and S, and rather is it our aim to see if the incorporation of the product has worsened the approximation to any extent. This is important because of the role played by an expression analogous to (4. 14) in our estimation of the creeping wave return for a non-spherical body. (U) Computed values based on Eq. (4.14) are shown in Table IV-4. Observe that the differences between S and S or S are quite small except when ka is small; but what differences there are are such as to de _______174 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN ka Sp(ka) 0. 3 0. 070167 0.4 0. 085564 0.5 0. 099293 0. 8 0. 133481 1.0 0. 151664 1. 5 0. 186599 2.0 0.211430 2.5 0.229226 3.0 0.242272 4.0 0.258837 5.0 0.266692 8.0 0.264679 10.0 0.254356 15.0 0.220739 20.0 0.187841 25.0 0. 159657 50.0 0.075584 TABLE IV-4 arg SP(ka) 175.424 196.468 217.236 278.307 318.380 417.112 514.416 610. 847 706. 567 896.442 1084.974 1645.614 2016.774 2939.339 3857.420 4772.631 9328.035 I rka) 5.3059 4.4952 3. 9740 3.1156 2. 8060 2.3620 2.1186 1.9645 1.8555 1. 7083 1.6142 1.4614 1.4041 1. 3207 1.2752 1.2449 1.1758 arg rP(ka) 52. 872 50. 124 47.892 43.839 40. 583 36. 066 32. 879 30. 385 28.414 25.503 23.328 19.053 17. 187 14.235 12.365 11. 140 7.903 175 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F crease somewhat the discrepancy between the approximate and exact values for the creeping wave. This is presumed to be fortuitous. (U) The final approximation that we wish to consider is one in which we attempt to account for some of the effect of the transverse curvature of the sphere (all previous approximations take into account only the curvature along the creeping wave path), thereby incorporating to some extent the contributions of the higher order terms in the exponential factor that were neglected in Eqs. (4.12) through (4.14). The result is analogous to SP but differs in thei.presence of a series correction to Logan's function, namely S(ka) - S (ka) where i r(ka 3) ( 2) 1 Ss(ka) = -47 e 3i 3 jAi(-13)j [^(2) ei /6 11/3 3 +* _I 2 n exp ( 2 n e (4.15 The affix s indicates the incorporation of the series, which series has been hand-computed for the required range of ka. (U) Computations based on Eq. (4.15) are shown in Table IV-5, and the fact that immediately stands out is that the series does not carry out its intended function and does, indeed, increase the discrepancy between the approximate and exact values for the creeping wave contribution. The increase in the discrepancy is particularly noticeable for small ka, but is still evident for ka as large as 50. It must therefore to concluded that the approximation S (ka) is numerically (if not mathematically) inviable. 176 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F TABLE IV-5 MICHIGAN ka 0.3 0.4 0.5 0.8 1.0 1.5 2.0 2.5 3.0 4.0 5.0 8.0 10.0 15.0 20.0 25.0 I SS(ka)j 0.056553 0. 065129 0.074435 0.101880 0.118096 0.151450 0. 176681 0. 195578 0.207634 0.228700 0.240164 0.243499 0.234364 0.207960 0.179458 0. 153791 arg S Ska) 86.944 123.058 155.131 234.316 281.049 390.642 491.883 591.645 688.957 882.474 1073.220 1636.958 2008.934 2933.564 3852.819 4768.751 PS(ka) 6. 5832 5.9056 5.3011 4.0821 3.6036 2. 9102 2. 5353 2. 3024 2. 1650 1.9335 1. 7925 1. 5885 1. 5239 1.4019 1. 3348 1.2923 arg p (ka) 141.352 123.534 109.997 87.830 77.914 63.536 55.412 49.587 46.024 39.471 35. 082 27. 709 25.027 20. 010 16.966 15. 020 177 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) There is one overwhelming conclusion that can be reached as a result of our study, namely, that the higher order terms in the exponential factor and, perhaps, the amplitude factor of the creeping wave expression are numerically important for all but the very highest values of ka. Failure to include them, or to provide an adequate simulation of them, leads to estimates which are unsatisfactory for detailed computations of scattering behavior, and compared with these terms, higher order creeping wave contributions are comparatively insignificant. (U) If the surface over which the creeping waves travel is not spherical, and cannot be decomposed into portions which are either planar or spherical, we have no alternative but to use one or other of the approximations (4.12) through (4.15). The approximation S (ka) given in Eq. (4.15) is distinctly inferior to the other three, but of the others it would seem to matter little k which we use. Based on computational convenience, S (ka) given in Eq. (4.12) is the more attractive, and we would then have no alternative but to accept the error that this approximation entails. On the other hand, if we can incorporate the Texact' expressions S (ka), it is clearly to our advantage to do so, and one such 'non-spherical' shape for which this is possible is the rear of an FB model. Such an application is discussed in Section 3.4. (U) There is one final point that can be made in regard to S (ka) that is of some convenience in cross section estimation. Over substantial ranges of ka, arg S (ka) is almost a linear function of ka. Using the method of least squares, the following fits have been obtained: 0.3 < 3.0 (16 values): arg S() (ka) = 187.916 ka + 171.312 maximum underestimate is 0.610 degrees maximum overestimate is 0. 316 degrees - 178 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 1.0 < ka < 10.0 (23 values) arg 1) (ka) = 186. 033 ka + 176.327 maximum underestimate is 1.810 degrees maximum overestimate is 3.397 degrees. 4.4 Surface Current in the Illuminated Region on a Parabolic Cylinder. (Focal length is comparable to the incident wavelength). 4.4.1 Introduction (U) The surface current in the shadow region had been studied earlier and the results of the study were reported in Section 3.3.2 of Goodrich et al, 1967c. In the illuminated region, the surface current may be represented by the summation of a geometrical optics term and a residue series which may be defined as the reflected creeping waves. In the penumbra region, the surface current may be obtained by the series expansion of the integral representation about a point on the shadow boundary. 4.4.2 Integral Representation for the Surface Current. (U) In Section 3.3.2 of Goodrich et al, 1967c, the surface currents were shown to be.k -ikr (i tan ))n U (Z) ik e sec n n dn (4.16) D 2r r 2 2 sin r n W (z') 1 n o C2 and N 1-ikr I (itan 2)n U (z) J sec I 2 () dn (4.17) N 2 2 s n W' (z' n o C 2 for Dirichlet and Neumann problems respectively. If we consider the region x < 0 where e is negative, we have the following relations 179 UNCLASSIFIED

UNCLASSIFIED r THE UNIVERSITY OF 8525-1-F MICHIGAN 2n -2n U (-z) = - i V (z) - i W (z) n n n (4.18) (4.19) U (z) + V (z) + W (z) = 0 n n n z = iik ' where U (z) and W (z) are defined by Eqs. (4.18) and (4.19). n n (U) Following Rice's (1954) derivation, the leading terms in the asymptotic expansion for U (-z) along the contour C2 are obtained as follows: n 2 -2n 2n -2n U (-z) = (i2 - i )A' + i- A' n o 1 f(t' ) A T e At 2i (ik2 - 2m)1/4 (4.20) (4.21) f(t'.) 2 e(ik - 2m 2V7r(ik 2 - 2m(l/4 A! 1 (4.22) f(t') = zt' + m/2 - m n t' t'o 2 i? + ik - 2m t 1- k F - 2 m 1 = 2 ik - ik ~ - 2m (4.23) (4.24) (4.25) m = n + 1 z = ik, g >o. In fact, the asymptotic expressions of the function U (-z) in the various n 180 UNCLASSIFIED -

I UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F regions of the m-plane are listed in the following table when z = 1i /2, fkF > 0. The contour C passes through regions I and II in which U (-z) has different asymptotic forms (Fig. 4-8). Because the stationary phase point is found within the region I, i.e. a 2 ____< a < 2 o 2 2 we may use the asymptotic form in I for U (-z) along the entire contour C2. When (4.20) is substituted back into (4.16) and (4.17) we obtain ______________ 181 UNCLASSIFIED m

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F m. 1 III C2 m = i H III n - m r Im [ft - Mf(t )] = 0 2 = -i 2 C2 FIG. 4-8: REGION IN THE COMPLEX m-PLANE CORRESPONDING TO DIFFERENT ASYMPTOTIC EXPRESSIONS WHEN 182 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F JD = - 2i M * A' (io)n o d + M W (z') n n o 1 A' dn sin r nW (z' ) n o (4.26) and JN =- 2i N N o (i )n C2 A' o 'W (z' ) n o dn + + N wo I C2 where w = tan 2 -2 A' dn (/i)n sin 7rn 'W (z' ) n o (4.27) -ikr -ikr M = r e sec and N = e o 2irr 2 2 2 sec 2 In the following sections, we will see that the first term may be recognized as the geometrical optics term. The second term may be expressed by a residue series which represents creeping waves launched from the shadow boundary and traveling along the surface of the parabolic cylinder into the illuminated region. It may be called the reflected creeping waves (Fig. 4-9). 4.4.3 The Method of Geometrical Optics. (U) The first term of (4.26) and (4.27) may be calculated by the stationary phase method when k - co. The asymptotic forms of the functions W (z' ) and 'W (z' ) along the contour C are given by l n o n o 2 183 UNCLASSIFIED -

Y I I I I Reflected I,/// // I - I H> Incident.. / / ^ Shadow Plane Wave,", ---- ------ Z Reflected Transmitted Creeping Creeping Waves Waves rn /^', /' ^ j^~~~ (\^- ^ "^ f.// Reflec ed I 20 FIG. 4-9: GEOMETRICAL OPTICS AND CREEPING WAVES.

UNCLASSIFIED THE UNIVERS ITY OF 8S25-1-F MICHIGAN and W (z' ) = A n o o 2 ' 'W (z' ) = (z' ) - 2m A n o 0 0 (4.28) (4.29) where f(t ) 1oe ~ -2i (-i p2 - 2m)1/4 A o (4.30) o o o 2 no z = if p =1 = -2i k h Therefore, the first term (4.26) and (4.27) becomes (4.31) (4.32) JD = -2i M D o o J = - 2i N N o IC2 C2 is 2^ Al (in), ~ dn (i) )n - )2.m (4.33) At o A o dn (4.34) where A' A o V L ik g2 -2 m 1/4 f(tt ) - f(t ) e (4.35) 185 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F o - 2i(m/k) f(t ) -f(t ) = ikr - ik + 2i(m/k) + 2 {o2 p + 2i(m/k) - - i (m/k) n (-i) I (4.36) w - tan Q/2. Introducing the new variable of integration a = i (m/k), we obtain 2F- 1/2 = -ik _ o +k2 + 2a D si2 I2r I 2 rno-2a] L2 +2a +2 a 1/4 e-ik ( (a) da ecl (4.37) and - 1 NZ. o sin2 ik C2 1/2 -ik _ (a) e dca (4.38) [,2 + 2a) ( 2 0 - 2a)]1/4 where 4 (a) = 2+ 2~ + 2 2 2 r)- 2a -? 2 _ 2a /2a ~ (+~ n + 2a 186 UNCLASSIFIED (4.39)

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F The stationary point of the phase ( (a) is obtained from r + r) - 2CY '( = In W 0- =0 as W (1 + -2 o2a )=+ 2 +2. (4.40) The Eq. (4.40) has a real root if rioW < 2 +. On solving this inequality, we find > - rl cot ~. The point e = - ro cot t is the boundary of the shadow and the points 9 > - rj cot, are located in the illuminated region (Fig. 4-10). Thus the stationary point of the phase 4 (a) exists only if the point of observation is situated in the illuminated region. Solving (2.40), the stationary point is found as )2 2 a sin / - sn cos. (4.41) Substituting a int the phase function ~ (a) we have 0 2 2 () = g o sing - o2 cos. (4.42) = x sin/ - ycos b and also 2 +2a = rI sin - cos l r - 2 = 7r cos + + g sin ~ ______ 187 UNCLASSIFIED 0

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN y x <O < o Reflected Wave x>0 e > o Incident Plane Wave ' = ri cot b 0 x r = rn FIG. 4-10: GEOMETRICAL OPTICS. 188 UN CLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F = -1,, (to) sin (rb cos 0 + ~ sin ~) (r]~ sin c - e cos ~) r 1 2 +2 ) Now the asymptotic expressions of the surface current in the illuminated region are obtained as follows: rY cos i + e sin DJ 2 -2ik 2 exp -ik (xsin - y cos/) (4.43) D 2 2 o + oI JN = 2 exp -ik (x sin / - y cos ~ ). (4.44) It can be shown that the quantity (r0 cos qi + sin i)/ + r in (4.43) is the cosine of the angle of incidence 0 (Fig. 4-10), and the exponential factor is the incident plane wave U (Ivanov, 1963). Thus, J - 2ik cos U (4.45) D o o J =2U (4.46) N o o i.e. the distribution of current in the illuminated region is described asymptotically by geometrical optics. 4.4.4 Reflected Creeping Waves. (U) The second term of (4.26) and (4.27) may be calculated by the sum of the residues at poles given by W (z' ) = 0 and 'W (z' ) = 0. In Section 3n n o 3.3.2 of Goodrich et al, 1967c, it has been shown that all zeros of both functions W (z' ) and 'W (z' ) are located in the third quadrant of the nn o n o __189 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F plane. The contour C2 may be deformed into C3 containing all poles (as in the above reference). Thus the asymptotic behavior of the integral is defined by the poles of the integrand. Therefore, we obtain and ao JD = -27ri M c s=l OD JN =- 27r i N c S =0 (V/i)n A'1 sin n a W (z' ) an n o n=n s (4.47) (W/i)n A'. sin 7r n aW (Z ) n n 0 n (4.48) =n' s where n and n' are zeros of the functions W (z' ) and s s n o tively, and they are obtained as follows: m = n + - (p+ +2.8) +i (p + p+ 1.4) P s 2 L m' =n' +1 = - [(2 p+ ) +i (p2+ p] P s 2 I 5J 'W (z' ) respecn o (4.49) (4.50) and also we have - W(z') an n o n=n s a 'W (z' ) an n o n s = - n (to/t )] A - - ip2 - 2m' [n (t /t1) A. p o o (4.51) (4.52) Let us assume a plane wave impinges upon the parabolic cylinder at an angle __190 UNCLASSIFIED - -

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN = 2, then we have w = tan = 1. If we consider the leading terms of the 2 2 residue series, the reflected creeping waves are given by A' /A ____ 1 o ___ D c = 2r i M o n (i) sin i ns n (t /t ) s o 1 (4.53) Al 1/A 0 J = 21r i N N c I o n' (i) sin gr n' s V-ip -2mt n (t /tl) p o 1 (4.54) here A'1 A o t ' 92 1 -p -2m I L ik2 _ 2m j f(t' ) - f(t ) 1 e m = m p p P for Dirichlet problem for Neumann problem. Simplifying the expressions in (4.53) and (4.54), we obtain D C = 2kr kr -ip -2m 1 -p 4 2 i ik2 - 2m j P exp [d (g)] ei -ep(t1 - et) 0 1 (4.55) 2,f? i 3/2 e x ) I N c (4.56) [ik 2 - 2m' )(-ip P - 2m' )1 p i rnm i27rm' e P(1 - e P)n (to/t) where the phase (Q) is expressed by 191 UNCLASSIFIED - -

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 1 ( ) = (m -I) In 2 -2 2 im L f-2 ir'] t 2i 2 lk~- +2 im + +i 2p - 2im 2 1 + (m - -) In 2 (4.57) (U) Let S be length of the arc of the parabola between the point ~ = 0 and ~ = e, and D = (2h)1/3 in r+ V2 + 2h L then (4. 55) and (4. 56) reduce to [,k 1/3 2D J eCp-ikS - c (p +3.8)+i(p+ 1.4) D 2/3 P A (4.58) (4.59) {( k13 r 2 3 A)exp-ikS- (_p + 1.2) + ip C D 2/3. P } where A () and A' (~) are amplitude functions, and they are given by A( ) = 1/4 1 In (t /t1) -ir m eL i27' rm 1 -e pj 22 ' i3/2 2 72 2 1,/4 ik 2m )( — 2m' 2 ) (t /tl) p pj o 1 192 UNCLASSIFIED (4.60)

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 6525-1-F (U) The reflected creeping waves are exactly the same as the transmitted creeping waves in the shadow region except for a constant factor -mr / i2m7r) e (1 -e ). Both are launched from the shadow boundary. One propagates into the illuminated region and the other into the shadow region. Let us define this constant factor as the ratio of reflection to transmission, then we have -i mr C (m) e (4.61) i2m~r 1 - e where m = m for Dirichlet problem, m = m' for Neumann problem. In p p general, I C(m) is negligibly small for p > 1 (Fig. 4-11). Therefore, reflected creeping waves may be neglected in the case of large parabolic cylinder 4.4.5 The Surface Current in the Region of Penumbra. (U) The function U (z) in (4. 16) and (4. 17) may be expanded into a series about a point on the shadow boundary. If we expand exp (2zt) in the following integral U (z) = 2i | exp {-t2 + 2zt - (n + l)Ln tdt (4.62) n 27r i and integrate termwise, then the function U (z) becomes n U(z)= - sin 2n '. (-2z1 Pt1-n/2)/' (4.63) Therefore, we obtain the surface current in the following forms,-,-,-, 193 __ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-i-F 2 3 p -2, u 0 -1 z 3%009 -x-x-x Neumann Dirichlet FIG. 4-11: In Ic(m) VS p. 194 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F M oo =- ~ 27r 0 0=0 c2 (iw)n r( - n/2) W (z' ) n o dn (4. 64) N oo N ' 0 It N 2ir I (-2z) / Q! 2 c2 (to)n. r - n/2) 'W (z' ) dn. n o (4.65) Now (4.64) and (4.65) may be evaluated by the residue series. terms are considered, we obtain If only leading J = - M D o 2 -ff (-ip2 - 2m )1/4 (io) s f(t ) (t / t) e ~ 2 1/4 s P., (-2z)I I!.I-n F( 2r ) ' iM o In(t/t1) r (-n /2)(2m )1/4 s P p -4 t mp I = exp (m 1 - -) Sn - 2, (-2z) V! I-n (4.66) n' - 2 7,f- (ia) s e -f(t ) N = - No (-2z) Q! i- n! r2 - in(t /tl)Jo(-ip2 - 2m' )1/4 o t p n' 27r ~2 (w) s i iN % 0 In(t /t )(2m' ) 1/4(- n' /2) 01 p s (-ip2 - 2m' )1/4 p continued 195 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F t -n' exp,(m - n2 ),(4. 67) Yp-2 - -t (=0 where z = Fik e, n and n' are zeros of W (z' ) and 'W (z' ), respec' s s n o n o tively, and m = n + 1. (U) When w = 1, we have ~ > 0 for the shadow region, < <0 for the illuminated region. Equations (4. 65) and (4. 66) converge absolutely for all values of R. They are especially useful in calculating the surface current near the shadow boundary where: = 0. At e = 0, (4. 65) and (4. 66) are identical with (3. 72) and (3. 73) in Goodrich et al, 1967c. 4. 5 Zeros of Parabolic Cylinder Functions (U) In the study of creeping waves, surface currents in the shadow region are represented by the residue series which are evaluated from poles (see Section 4.4 of this report and Section 3. 3.2 of Goodrich et al, 1967c). It is desirable to investigate the location of poles in order to see how far the high frequency approximation can be extended into the low frequency region. (U) For this purpose, let us consider a perfectly conducting parabolic 2 cylinder x = 4h (h - y) of focal length h illuminated with a plane wave e (x sin - y cos) with the time factor e. Then surface current densities are = 2rri 2 c2 [ sin W Fr dn (4.68) | 4n o 1 -ikr, (i tan )" U (f () J sec r- - dn (4.69) N I 2 2 n sin ( n n W -ik(4 68 ) -c ______196 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F where JD and JN indicate respectively the Dirichlet problem and the Neumann problem. The contour c encloses all positive zeros of sin 7r n. The parabolic coordinates x = e rY, y = 1/2 (r2 - 2 ) in which the surface of given cylinder is defined by ro = 42. When h = 0 the cylinder reduces to the half plane x = 0, y < 0. (U) In the shadow region we obtain creeping waves by deforming the contour c and taking to account all zeros of the function W (1-i 2kh') or the n function 'W (i-2ikh). It has been shown that all zeros are located in third n quadrant of the n-plane (Rice, 1954). These functions are related to the parabolic cylinder function D (z) (Whittaker and Watson, 1927) through the equations n (z) 2n/2 z 2/2 U (z) = 2n e /2 D (/ z) / (n + 1) ( in/2 z /2 W n(z) = - i 2 e/2 Dni (i 42 z)/ if (4. 70) (4.71) (4. 72) in2n/2 'w (z) - -- n 4271 z2/2 e az [D (i z] az-_n- 1 Now zeros may be obtained asymptotically from the following three cases: Case 1: If kh -* 0, then the expansions of the parabolic cylinder function D (z) are given by the formula (Erdelyi, et al 1954) n 1 -D-n-1 1 2 1 1 2 2 2 2 - z) (4.73) D (z) = e (-1)s ( z)s (4.73) n - (-n) s where z = i'2j-i2kh ei r/4 V4kh.: Therefore the condition W (V -i2kh) = 0 implies that n 1 2eir /4 Y2kh r - /2)- 0 r(1 - a/2) [(-a/2) (4. 74)..II..... 197 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F where a = -n -1 = -m. When kh -e 0, the values of a for which the above expression vanishes are clearly near the poles of r(l - a/2). Therefore a may be given by the formula 1 a = - (s + e) (4. 75) where s = 0, 1, 2,... and | is a small value. (U) Substituting (4. 75) into (4. 74) e may be obtained approximately from the equations 1 / =sin 7r (s + (-1) sE! l (1 -.a/2) - s (s) inrsc) 2 2eI"/4 2kh =2 J. (4. 76) 1 r(-s - 1 2 Now the asymptotic behavior of the zeros of W (-2ikh ) in the complex nplane are given by (4s + 1)t4 4e 4 +2kh m = n + 1 - (2s + 1) + (4. 77) s s s! (-s- 1 When kh = 0, we have -i.n w (0) = n 2 P(i + ) Then zeros are given by poles of r(l + n/2) that is m = n + 1 = - (2s + 1) (s = 0, 1 2, 3,... ) (4.78) S S 198 UNCLASSIFIED

I UNCLASSIFIED -— THE UNIVERSITY OF MICHIGAN 8525-1-F This is the limit case of (4. 77). Similarly the equation which determines the asymptotic behavior of the zeros of 'W (n-2ikh ) is n 1 + F(-a/2) 1 eir/4 (a+ ) e 2kh r(i - a/2) = 0 (4. 79) For kh -- 0 the zeros are very near the poles of P (-a/2). Let us assume the solution as - /2 = - (s + c) (4.80) where s = 0, 1, 2, 3,... and e is a small value. Then we have (4s + 1) e e - e (2s+ 2kh s! (-s + 2 ) (4.81) From (4.80) and (4.81), the asymptotic behavior of the zeros of 'W ( -21i ) in the complex n-plane are obtained by (4s + 1) m' = n' + 1= -2S-e s s (4s + 1) f2khj s! r(-s +2 ) (4.82) When kh = 0, we have.n-1 -1 'w (o) = n(0) = (n + 1/2) Then zeros are given by m' =n' +1 = -2S (s = 0, 1, 2, 3,... ) s s The result is the limit case of (4. 82). 199 UNCLASSIFIED (4.83) (4.84) l I

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN — 8525-1-F Case 2: If kh - aoo, the asymptotic expression of the function W (z) and n 'W (z) may be obtained by the saddle-point method (Rice, 1954; Ivanov, 1960). The function W (z) is expressed by the integral. Ivanov, 1960). The function W (z) is expressed by the integral. n 1 w (z) - - n 2iri W exp - u + 2uz - (n + 1) nu du (4.85) where W is the contour which runs from + oo to - oo with - 7r < arg u < 0 in the complex u - plane. (U) It is well known that for large kh the zeros n are also large (Rice, 1954) n - -ikh +0 {(kh)1/3. Thus, to obtain the zeros of smallest moduli of W (V-i2kh?), the asymptotic n expression will be obtained in the case ns - ikh. It has been shown that 5 the asymptote of this function is expressed by the Airy function (Rice, 1954; Ivanov, 1960). From Ivanov's result we obtain W( ) i f+tn 2 i IV)[(t) n 2 47r 4 TI - t w" (t) +0(B-2) - -w 2 where B = (-ikh)1/3 f = B3/2(3 - In T The Airy function uo(t) is defined by (4.86) 3 B 3 1 [3 -) and t = lB -(n+ 1). 2 B 200 __ UNCLASSIFIED...

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 3 w(t) = 1/ 7 exp ( t - -) d (4.87) where the contour of integration F runs from o e (/) to 0 and further along the real axis to infinity. (U) The zeros of W ( -2ikh) are clearly near the zeros of o (t) due to the large parameter B. Denoting the sth zero of the Airy function by t and assuming the zero of W ( -2ikh) by t = t + + 0 (B ), we determine 1 n s B C = - - from (4.85). Transforming the zero from the t-plane to the complex n-plane, we obtain n =-ikh - (-ikh) /3 t +0 (B1) (4.88) s s 2 (U) From the definition of the function 'W (z) we obtain n 'W (z) e /2e W(z) n / [e 2 w(z)n r 2 - (2u - z)e- +uz - (n + 1)n d (4.89) J W The asymptote of this function is i B FV) ) 2 2 w'(t) + 1[ 2( )(t) n 2 o B 4 t -2 "' (t) + 0(B ) (4.90) C 2 Similary we assume the zero of 'W (J-2iklh) by t = t' + C + (B ) where n s B t' is the sth zero of the function w'(t). From (4.90) we determine C' = - s 2 ' ______ 201 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F In the complex n-plane zeros are located at nt - ikh - (-ikh) /3t' - 1 + (B 1) s s 2 (4.91) I (U) We note that one may use Rice's results (Eqs. 13.21 and 13.24) to calculate the zeros in the complex t-plane asymptotically. If zeros in the complex n-plane are requested, the Rice's asymptotic expansions should be modified to include the term of order (1/B). The reason is that it involves a large parameter B multiplication in mapping the zeros from the complex t-plane to complex n-plane. Therefore one should not neglect the term of order (1/B) in evaluating the zeros. Finally instead of the -1 in Rice's results we obtain - 2 in (4.88) and (4.91) (Keller, 1956; Ivanov, 1960). 2 (U) The zeros of the Airy function w(t ) andits derivative '(t' ) occur s s when t and t' are negative. The first five zeros are tabulated below s s s t t' s s 1 -2.3381 -1.0188 2 -4.0879 -3.2482 3 -5.5206 -4.8201 4 -6. 7867 -6. 1633 5 -7.9441 -7.3722 Equations (4.88) and (4.91) can be rewritten as follows m =n + 1 =0.866 t (p2/2)1/3+ 1 i - p2 ( 2/2)1/3] S S 2S 2 p I- ts p 1 /3 (4.92) 202___ UNCLASSIFIED.

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F m = n' + 1 = 0. 866t' (p2/2)1/3 + 1 i p2 s s s 2 2 - where p = 2-. (U) For the case of the zeros of largest moduli 'W ( -2ih), i.e., kh/ I|n << 1, Rice obtained t'(p2/2)1/3] of W ( -2ikh) n (4.93) and (4.94) W (ip) A(in 01 - i ) = -2iA sin (+ ipim) 'W (1p) - -i p -2m A (i 1 0 + i ) n o1 =2 V-i p - 2m A cos (- + ip '2i ) ~1 (4.95) where A = 2-3/2 -1/2 exp [m/2 (1 -n - m ) - ip/2] 01 = exp [p 2imr] = 1/0 m = n +1. Thus the location of the zeros of largest moduli are obtained from the trigo2 nometrical function. If s is a large positive integer such that 2 S>> p zeros of W (i Fp) are n m = - [(2s- 1) +4 (p/7r) jTS/2 -i4 (p/7r) ( s/2 + p/7r) (4.96) and zeros of 'W ( f'p) are n. 203 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F m' = - 2s+2 (p/7) 2s + 1' -2i (p/7r) (2s + ) + 2 (p/7r) (4. 97) 2 From the condition 2s >> p, we know that (4.96) and (4.97) will be useful in locating the zeros of large moduli when p =2kh is small. case 3: If kh - 1, that is the case of the incident wavelength comparable to the focal length, zeros may be obtained graphically from Rice's results. The asymptotic expression of W (4' -p) in the third quadrant of the complex n-plane is W (/p) = A -A (4.98) n o 1 where f(t ) A -2ii(-ip2 - 2m)1/4 (4.99) f(t ) A- 2 1/ (4.99100) 1 2i;' (-i p - 2m)/4 t = 1/2[<i'p p -ip 2m (4. 101) o m 2 t = 1/2 9+ - i - 2m (4.102) fp(t m/2 (1 - In 2m - In 0 ) P t (4.103) ______204 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN t f(t ) = m/2 (1 -1In M- 1-) 0 + i'p t1 (4.104) m = n +1 -37r/2 < argm < 7r/2 -37r/2 < arg (-ip2- 2m) <7r/2 -3 r/4 < arg t < 7r/4 -57r/4 < argtl < 37/4 therefore zeros of W (i~ip) are located at W ("-i'p) =A - A = 0 n o 1 or (4. 105) J exp [f(to) - f(tl)] = 1 iyo/l (4. 106) Using the following transformation w = n (t /t1) =u + iv 2 z m = w 1 cosh w + 1 (4. 107) (4. 108) (4. 109) f(t ) - f(t ) =m (sinhw - w) = o 1 2 z (sinh w - w) cosh w + 1 the complex m-plane is mapped into the w-plane in the region u > 0; v < Tr. Thus in w-plane (4.106) becomes 205 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY 0 8525-1-F F MICHIGAN 2 sinh w - w _ w cosh w + 1 - 2 + i 2 (1 - 4s)] 21 (4.110) where s = 1, 2, 3,.... By separating the real part and the imaginary part of (4. 110), we obtain two simultaneous equations p (cos v + cosh u + u sinh u) sin v - (cosh u cos v + 1) v = - 2 u (cosh u cos v + 1) + (sinh u sin v)2] (4. 111) p (cos v + cosh u - v sin v) sinh u - (cosh u cos v + 1) u 1 r 1r] 2 2+ = [v + (1 - 4s)r] (cosh u cos v + 1) + (sinh u sin v) (4. 112) Let (4. 111) be divided by (4. 112), we obtain (cos v + cosh u + u sinh v)sinv - (coshu cosv +l)v (cos v + cosh u - v sin v) sinh u - (cosh u cos v + 1)u -u v - (1 - 4s)7r (4. 113) This equation is independent of the parameter p. Settings s = 1, we calculate the first zero as the following: (U) First, Eq. (4.112) may be approximated by a circle in the w-plane as 2 Lu - (r - a) 2 + v -] 2 (4.114) where a + (r 12) r = 2a - 2a (4.115) 206 UNCLASSIFIED

UNCLASSIFIED -- THE UNIVERSITY OF MICHIGAN 8525-1-F a = u /2 = 0.575, s = 1 0 < u < a <1. Second, for u < 1, (4.112) can be evaluated approximately by 2 1 - -p uv sin v = (v - 3ir) L(cos v+1)2 (u sin v) (4. 116) The location of zeros may be obtained by graphical means. If we plot (4. 114) and (4. 116) on the w-plane, the points of intersection between the two curves determine the zeros. A typical plot is given in Fig. 4-12. Mapping the zeros 2 from the auxilliary w-plane with the help of m = -i p /(cosh w + 1) gives the location of zeros on the m-plane. If we consider p as the variable parameter, the locus of the first zero in the m-plane is expressed approximately by ms = - [(p + 2.8) +i (p2 + p + 1.4) (4. 117) where we limit the range of p as 0 < p < 10. Re m and Im m are plotted s s in Fig. 4-13. Similarly, loci for s = 2, 3, 4,... may be obtained by the graphical method. (U) Similarly the asymptotic expression of the function 'Wn(-FTp) in the third quadrant of the n-plane is 'WN ( p) = -i p - 2m n A0 + Al] (4.118) Therefore, the zeros are located at A +A = 0 o 1 (4.119) or 207 UNCLASSIFIED

UN CLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN v Q w-plane w = u + iv /- Im [f(to) - (t)] = 0 r P [-(r - a), 2 2 t --- a + (r/2) a I I r 2a 0 1 2 a = 0.575 u FIG. 4-12: GRAPHICAL SOLUTION FOR ZEROS OF W (z) WHEN z =7 p. n 208 UNCLASSIFIED

U N LASSFIE THE UNIVERSITY 0 8525-1-F F MICHIGAN +. 0 H4 I I I 01 —. I ll I (X C.4 CL \ V-4 1 cq h 0. \ 0 \ A \ I '-4 co +4 G) 1-4 Q. C4 LJ x w 1-4 0 0 --N PC4I 0N.1 k C Q. fr C) 209 UNCL~ASSFE

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F exp [f(t) - (tt) = i/ tt; (4. 120) Using the transformation w = In (to/t ) = u + iv, we obtain 2 sinh w -w 7Ir w -l coshw+ i i (1+4s) - (4.121) cosh w + 1 2 2 where s = 0, 1, 2, 3,.... By separating the real part and the imaginary part of (4. 121) we obtain two simultaneous equations p (cos v + cosh u + u sinh u) sin v - (cosh u cos v + 1) v 1 2 2 = u (cosh u cos v 1) + (sinh u sin v)J (4. 122) p [(cosh v + cosh u - v sin v) sinh u - (cosh u cos v + 1) u =- [v- T (1+4s) [(coshucosv+ 1) +(sinhusinv)2] (4.123) Dividing (4. 122) by (4. 123) we have (cos v + coshu + u sinhu)sinv - (coshu cos v + l)v -u (cos v +coshu - v sin v) sinhu - (coshu cos v + l)u v -r (1+4s) '. Setting s = 0, (4.124) may be approximated by a circle in the w-plane [u - (r - a) + [v - r /22 = r2 (4.125) where r +(7r/2) 2a _______210 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F a=u = 0.48 v = 7r/2 0 < u < a < 1 For u < 1, (4. 123) may be approximated by 2 1 2 2 - p uv sin v = 2 (v - it) (cos v + 1) + (u sin v). (4. 126) The location of zeros are determined by the graphical method from (4.125) and (4. 126). A typical plot is shown in Fig. 4-14. Mapping the zeros from the w-plane to m-plane gives approximately the locus of the first zero as m s - P( + 1) -i P + 1) (4.127) where p is limited in the range 0 < p < 10. Re m' and Im m' are plotted s s in Fig. 4-15. Similarly, loci for s = 1, 2, 3,..., may be obtained by the graphical method. (U) From the results (4. 77), (4. 82), (4.92), (4. 93), (4. 95), (4.96), (4. 11t and (4.127), we plot zeros as the function of p in Fig. 4-16 to Fig. 4-19. They show that the high frequency approximation may be pushed down to very low frequency region for the imaginary part of zeros (Fig. 4-17 and 4-19). For the real part of zeros the high frequency approximation may be extended down to p = 3 for the first zero (Fig. 4-16 and 4-18). 2 (U) For zeros near the boundary 2s = p, they may be determined by the graphical means. In the region 2s < p, the high frequency approximation may be applied.... 211 UNCLASSIFIED

UNCLASSIFIED THE UNIVER SITY OF 8525-1-F MICHIGAN V w-plane Im Eite ) - f(t = 0 0 r p4 Pa Er-a), a = 0.48 l- ---—, a + (r /2) 0 1 2 2a u FIG. 4-14: GRAPHICAL SOLUTION FOR ZEROS OF 'W (z) WHEN z = V p. n 212 UNCLASSIFIED

UNCLASIFIE THE UNIVERSITY OF 8525-1-F MICHIGAN 0) I I I 10 I I I cm I 9-4 1 1 1 a I I lo I I I ICL I CL %...O 4 1 CJ la. as s I Q-I. %-Th +-1c QJ.4 W-10 + -I- w Kq PI 1 PI I I z z. 0 --- H -N 0 0 N HoEr4 0 ( Q~ lt cvn clI V 213 UNCL~ASSFE

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F - Re m 2 s. s 2s = p 4th zero 3rd zero 2nd zero 22 I I I i I 20 18 16 14 12 10 8 6 4 2 I I I 1st zero High Frequency Eq. (4.92) 000 Graphical Method n --- Low Frequency Eqs. (4. 77) and (4. 96) 18 20 22 24 26 28 30 p 0 -.5 FIG: 4-16: REAL PART OF ZEROS OF W ( iIp), p =1k. (Dirichlet's Problem). 214 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN 1000 100 10 -Imm s 1.0 0.1 0 FIG. 4-17: 10 20 30 p IMAGINARY PART OF ZEROS OF W ( 1Fp), p 4S (Dirichlet's Problem). 215 UNCLASSIFIED

UNCLASSIFIED -R s THE UNIVERSITY OF 8525-1-F 2 [e m' 2s = p emI 4th zero I I I I I I I I / i/ zero MICHIGAN 14 2nd zero High Frequency Eq. (4.93) o 0 0 Graphical Method Low Frequency Eqs. (4.82) and (4. 97). 201st zero 20 30 p 2 -.5' FIG. 4-18: REAL PART OF ZEROS OF 'W (A p), p = 2. Problem). (Neumann's 216 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 1000 FIG. 4-19: IMAGINARY PART OF (Neumann's Problem). ZEROS OF 'W ( -i p), p = 2'. n 217 UNCLASSIFIED

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F V RADAR CROSS SECTION IN THE PLASMA RE-ENTRY ENVIRONMENT 5. 1 Introduction (S) The emphasis in the theoretical and experimental investigations during the last two quarters has been shifted towards the determination of the backscattered return from a plasma sheathed conical vehicle for nose-on incidence. Apart from the return from the tip which becomes blunted due to ablation effects, the key problem is the determination of the return from the base of the vehicle which for the exo-atmosphere case is the dominant return for most sharp nose re-entry bodies. The plasma will partially or completely shield the base thus reducing its cross-section. (U) The treatment of the electromagnetic scattering problem depends upon proper knowledge of the electrical properties of the sheath, which in turn depends upon knowledge of the flow fields. Because of lack of such information for a rounded base, the study was limited to the flat-backed base vehicle, although it was anticipated that with the advent of additional information that the rounded base could be properly treated in an extended program. (U) For a flat-backed base the flow fields have been computed by Weiss and Weinbaum (1966), who point out there is a rapid expansion and separation of the hypersonic boundary layer at the rear shoulder of a blunt based re-entry body. In the outer portion of this expansion region (the free shear layer) the electron density will rapidly decrease beyond the shoulder, whereas, in the inner portion and the recirculation region the electron density may be quite large (see Fig. 5-1). Due to the complexity of the problem, some simplications of the electromagnetic problem have been made. For the direct backscattered return from the rear edge, the effect of the electron density in the recirculation region and the free inviscid layer adjacent to it, has been assumed to play a minor role. Such regions are important when multiple scattering across the 218 SECRET

UNCLASSIFIED - THE UNIVERSITY OF MICHIGAN 8525-1-F base are to be taken into account. With the above in mind, attention is focussed upon the spatial behavior of the electron density in the vicinity of the rear shoulder of the vehicle, and the outer portion of the expansion region beyond it. Again here, further approximations are made in the model, which lead to three separate cases, which are described below. Boundary Layer Fre S r /0 Free Shear / Layer Recirculation Region FIG. 5-1: HYPERSONIC BOUNDARY LAYER SEPARATION FOR FLAT-BACKED CONES. (U) In the first case it is assumed that the expansion is sufficiently rapid in the outer portion of the free shear layer, such that the electron density becomes insignificant a short distance (compared to wavelength) back of the rear shoulder or edge. If N is the mean relative index of refraction of the bouns dary layer at the shoulder, and 6 is a boundary layer thickness at the s shoulder, and O is the angle typical of the divergence of the flow in expansion region, then on assuming frozen flow the mean relative index of refraction N at a distance D beyond the shoulder is given by (N - 1) = (N - 1) 6/(De) = (N - 1)k 6 /[D/X 27r S S S S ' _ _______219 UNCLASSIFIED

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F Hence it is seen that if (Ns - 1) k 6 < < 1, 0 - 7r /4, then (N - 1) < < 1, at a distance D = X/4 from the shoulder. This implies that for thin penetrable plasmas, the sheath in the neighborhood of the shoulder can be modeled by a terminated slab. Such a model will be useful in practice at lower operating frequencies, and higher altitudes where the sheath is less dense. (S) For the cases where the sheath is too thick or overdense, such that the electron density in the expansion region remains significant well beyond the shoulder, alternative models must be employed since the sheath can not be abruptly terminated at the shoulder. There are two separate cases, an impenetrable sheath at the shoulder and a penetrable sheath at the shoulder. In the former, the sheath effectively hides the inner conducting wedge. In this case the backscattered return will arise from the discontinuity in electrical properties located in the vic inity of the shoulder. There the rapid expansion of the flow field will cause an abrupt charge in the electrical properties. In the later case there will still be some return arising from the inner conducting edge, but it will be reduced due to absorption. (U) Only the first case (thin penetrable sheath) has been treated both theoretically and experimentally. It had been expected that the remaining cases would be treated in the follow-on program. A detailed discussion on a theoretical approach based upon the concept of an anisotropic impedance sheet along with comparison to the physical optics approximation is given in the next section. The associated experimental model is treated in Section 5. 3. 5. 2 Anisotropic Impedance Boundary Condition Approach to the Base Return for a Thin Plasma Sheath. (U) The backscattered return for nose-on incidence to the plasma coated flat-backed cone is comprised of the tip and base returns. The base returns (neglecting multiple diffraction effects) is calculated using well-known techniques of diffraction theory, wherein the base is approximated by wedge segments. 220

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F Wedge Segments \I FIG. 5-2: ILLUSTRATION OF WEDGE SEGMENT APPROXIMATION. The total field is the sum of the individual returns from the edges of the wedge segments. It can be shown that the total backscattered field for nose-on incidence for a base of radius a, is given by 2 2 a = r a RI - R, (5.1) where R 1 and R1 are local reflection coefficients associated with the twodimensional wedge return. To be explicit, if E is the plane wave incident upon the wedge, the backscattered return from the edge has the form i(kr + r /4) R Es= e E R E R (5.2) where r is the distance from the edge, and Ell and E are the components of the incident field parallel and perpendicular to the edge, respectively. Thus the problem reduces to solving the two-dimensional wedge problem. (U) For a special class of coatings (magnetic absorbers, good conductors) the electrical properties of the coating may be expressed in terms of a boundary condition on the outer surface, i. e., the effect of the coating upon incident radiation can be represented in terms of the impedance boundary condition invol ving the tangential components of the total field on the surface E - (E. n) n = 221 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F V E rj n x H. In this expression, n is the unit outward normal to the surface, and r- is a function of the electrical properties of the coatings as well as its thickness, being zero for a perfect conductor. For this case of coatings, r] is independent of polarization and angle of incidence of the incident radiation. E| Direction of Incident Radiation Plasma Coating FIG. 5-3: LOCAL WEDGE GEOMETRY. (U) To obtain approximate results for thin plasma sheaths, the concept of impedance will be modified, with rj taken to be a function of polarization. Except for very overdense plasmas, the appropriate choice of rY can not be made independent of the angle of incidence. (U) For a uniform plasma sheath of thickness d and index of refraction N, an appropriate choice for r, based upon the flat plane results, is given by. tan Lkd (N2 - cos2 6)1/2 (.3 2 2 1/2 11 (N -cos 6)/ 2- -2 (N - cos 6)1/2 tan [kd (N2 - cos2 6)1/2] (5.4) N where 6 is the angle of incidence measured from the locally flat surface and will turn out to be equal to the (small) interior half-angle of the cone. The index of refraction will be based upon the cold plasma result 222 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN - 8525-1-F 2 2 N2 =1- (5 5) 1 + i (f /f). c (U) Several observations can be made here. In the very overdense case, such that IN >> 1 and Im(Nkd) >> 1, the impedances become rll -r-' 1/N ~ 0 and the sheath appears effectively as a good conductor. On the other hand, for thin sheaths, such that kd < < 1 and INkd|< < 1, the impedances are approximately given by ri] - - i kd -0, 2 2 N - cos 6 r 1 - kd 2 N Again, the sheath appears as a conductor for the r11 impedance; however, the r1_ impedance can profoundly change the characteristics of the scattering process. For operating frequencies well above the plasma frequency, such that N - 1, the impedance rp behaves as rY] -i kd sin 6 - 0, and the sheath appears again as a conductor. Contrarywise, for operating frequencies close to the plasma frequency, such that N 0, the impedance rl_ behaves as Y1r ikd cos 6/N - o, and the sheath now appears as a magnetic conductor. The result is that near the plasma frequency, the two reflection coefficients R1 and (-RI ) appearing in Eq. (5. 1) will cancel each other and greatly diminish the backscattering cross section. This effect will be made more quantitative later. (U) For non-uniform plasma sheaths, the appropriate choice for the impedances is given by i1+R1 1+R2 ___223 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F where R1 and R2 are voltage reflection coefficients for a plane wave incident upon an infinite non-uniform plasma slab. Such reflection coefficients were calculated last year on the basis of the plasma profiles supplied by Aerospace. In the present notation, R and R2 correspond to electric polarization perpendicular to, and parallel to, the plane of incidence, respectively. Some quantitative results for the laminar flow case at 150 K ft, will be presented. (U) The above representation of the sheath in terms of an anisotropic impedance boundary condition greatly simplifies the scattering problem. The two-dimensional scalar scattering problem can be expressed as follows; to s i s i find the scattered field u such that the total field u = u + u, where u is the incident field, satisfies the Helmhotz equation V u + k u = 0 and the boundary condition on the outer surface of the coating au. ik 1 an The appropriate boundary condition on the rear face of the wedge may be taken to be that which corresponds to a perfect conductor i. e., u = 0 and au a = 0, for polarization parallel to the edge and perpendicular to the edge, respectively. (U) Fortunately, this problem has been solved by Milinzhinets (1958), and his solution appropriate to our geometry is, for backscattering: i(kr + ) u e ikr R where 224 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 2r 6 sm 23in7 26 + 3r, (-2 -4) 27r (6 - r) 2C r 6 26 + 37r - cos26 + 3r 7 2,) (- 6 + 3T 2- 4j 6 77r 2 4 2r (6 - Xr) os 26 + 3r 2r 6 26 + 37r (5.8) with 6 representing the angle of incidence. Direction of Incidence 6 of Incidence 6 The function L is given in terms of a special meromorphic function, by (a) = ((a + ~ + 01)(a - - 0l)(a - + (02)/(a - ~ - 02) (5. 9) with = 6 + 3r and 2 4 c = va' F21 = cos exp 2r COS j -O log (1 - i tan 'a = ~, (-a). X tanh co ) dsh 4$ cosh v (5. 10) This solution contains both polarization cases at the same time and they are to be distinguished by the relationships between the (complex) angles 01,2 and 225 UNCLASSIFIED m I

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F the impedances; in particular, on the illuminated face: - 1 (5.11) cos 9 = cos 01, (5.12) and for the shadowed (or rear) face, later taken to be perfectly conducting: = (5.13) cos 92 = cos. (5. 14) Thus, at present, the solution contains four complex impedances for two polarization cases in the single formal expression (5. 8). (U) It turns out that for the particular geometry at hand, Eq. (5. 8) can be vastly simplified, at least to the extent that the definite integral in Eq. (5. 10) does not enter into the final results. We begin with the b functions in Eq. (5.8): Co ( 6 - = 7r + 0) (7r + 0 ) (2 + 02)q(2 - r 0 6 3r' 3' r3f' 3-' 37_ =2 4 = 2 1)( -01)O(2- -'+02)0)(2( - -2 )) (5. 15) 2- + 4) = + 2 01)(2 01) 2 02) 2 2 Now, from Malinzhinets (1958), we have 226 _ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F ( /2 + )( /2 - a) = (/2) cos 0 (7r/2 +a)o 0 (7r/2 - a) =0 (v /2) cos i r /40 (5.16) and by means of this formula, we find (3 +a) (3 - a) = (1r /2) 01(- 2 +a)( 2 - a) = r/2) r(a+r ) it(a+Tr) cos 4C coss - - COS x~ 4a 7r (a + 27r) 0 r (a - 27 ) cos4p cos 4~- ~ 46) 7r (a + ) 7r(a r) 4& cos 4 (5.17) (5.18) With these we may simplify the various ratios of, functions involving 01 in Eq. (5. 15) and the quantity 2,(7r /2) gets cancelled out. Similar considerations apply to the lp( functions involving 02, only first the relation (Malinzhinets, 1958) O (ba + 2) 00(a - 2q) = ot(+ 24 (5.19) is employed to bring the 0( functions involving 02 to the appropriate form applicable to Eqs. (5. 16) through (5. 18). (U) Without entering into any more algebraic details, we found that the quantity R in Eq. (5. 7) can be written in closed form as: = 1 sin (3r /2v) 2v sin (r/2v) A I sin (ir /v) + B(2 sin (27T /V) (5.20) where 227 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 1 + cos (1/v) 1 - cos (02/v) 1+cos 02 cos (7r/v) + cos (01 /) cos (7r/v) - cos(02/v/) 1-cos (5 cos (2r/lv) + cos (0l/v) cos (27r/v) - cos (02/v) 1 A (5.22) COS (7T/V) + COS (j1V) COS (27T /) - COS (62/V) BA cos (r/v)+ cos (0(/) cos(/r v) - cos (02/v) ' (5.22) and v = (6/7T + 3/2). The nose-on backscattering cross section for the flat back impedance cone is then 2 |2 a = r a 2 R - R (5.23) where a is the base radius, RlI is obtained from Eq. (5.20) upon employing the definitions in Eqs. (5.11) and (5.13), and R_ is obtained from Eq. (5.20) upon employing Eqs. (5.12) and (5.14). For a perfectly conducting finite cone, = ioa, = e =7r/2, and 0 82 i "1 1 27r 7r a 2 1 R cscRL = - c = csc (2:r/v), 5.24) V in agreement with Siegel et al, (1959a), (1959b). On the other hand, for a perfectly absorbing finite cone (all impedances equal to unity) 01 t = 0, 01 = I 11 2 0 = 0 and R1I = R_, o = 0, in agreement with Weston's (1963) general theorem concerning absorbers. (U) We now specialize to the case where the rear face is perfectly conducting, 0 = i o, 02 = r /2, and the illuminated surface is governed by the impedances rn1, rL. The following explicit results are obtained: A = -/ 2 /(5.25) l +u /v cos-+u -1/ 1 + u cos - + u UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F l -1/v [ 2r + -l/v ( B-A 1 + u*1/ 2 cos 11i (5.26) 1 + ull/v 2 cos - + u /(5 BA 1+ L2 co- +llu]- sin ( 5/44i) (5.28) L /11 + 1/r - -1, 1 + -1 L2 cos y + -I s ULf l+ vi. 1we a ao coe +n ud ( aB t s e rAos a e-V non- n e in ini ve (5.28) 1 -+ 2 c +COS + ca = 1/ 4R +R2 +J, (5.29) u = A L +u cos- 1 and the square roots have non-negative real parts. where R and R are the voltage reflection coefficients defined in Eq. (5.6). where Ri and R are obtained from (5.20) with (5.25) through (5.29), has been programmed for the computer. The output of the program gives the cross section of the plasma sheathed finite cone normalized to the cross of the per fectly conducting finte cone; this ratio is denoted by ', where Z = a /a. (5.31) coated p. c. 229 -- UNCLASSIFIED

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F (S) The first results obtained are for the laminar flow case at 150Kft, for a sharp 11 cone of length 104". The plasma profiles were supplied by Aerospace (profile No. 3) and the appropriate reflection coefficients, amplitude and phase, were calculated in last year's effort. These are tabulated in Table V-1. The resultant cross sections are also included. It will be noted that the reduction in cross section is greatest for those frequencies close to the plasma frequency f = 6.10. Unfortunately, experimental data for this case are not p available. Some data are contained in de Ridder and White, (1966), for a flat-back cone of angle 80, length 110", although the cone tip is slightly blunted, and this can alter the flow field characteristics significantly. It is difficult to obtain an accurate measure of the cross section reduction from their report; nevertheless, some data of interest are reproduced in Table V-2. TABLE V-l: Amplitude and Phase of Reflection Coefficients for Profile No. 3, and Nose-on Backscattering Cross Section 6 = 110, I= 104??. f R1 01 R2 02 10.999 3.183:877 -2.140 - 3.3 db 5 ~ 108.992 3.338.638 -.303 -23.2 db 9 10.990 3. 529.682.852 -19.4 db TABLE V-2: Measured Cross Section Reduction (de Ridder and White, 1966) Altitude Frequency E 45 -50 km UHF -10 db 50 55 km L-band -20 db,,,, I - -230 SECRET.,

SECRET THE UNIVERSITY OF MICHIGAN 8525-1-F (U) The cross section reduction in either the wedge approximation or the physical optics approximation is quite sensitive to changes in the various parameters, especially when these changes occur for frequencies close to the plasma frequency. In order to assess these effect, computations were conducted for a uniform plasma sheath, where the impedances are given by Eqs. (5.3) through (5. 5). A thickness d = 1/4" was chosen, although other thicknesses can significantly alter the results. The cross section reduction, both in the wedge approximation ( ) and the physical optics approximation ( E ), was calculated for a sequence of values 0.4 < (w /w) < 2.5, 10 < f < 5.10; 00 for several values of the collision frequency f; and for 6 = 11, 8. Some of the results are presented in Figs. 5-3 through 5-10. (S) Figures 5-4 through 5-7 present results for the wedge approximation. Generally the reduction for 6= 8 is greater than for 6 = 11, as expected. 8 9 Further, the reduction for f = 10 is greater than that for f = 10, and, in c c addition, the effects for the lower collision frequency are much more severe. This is expected since one approaches a sharper resonance effect as the losses are reduced. It will be noticed that at the higher frequencies, the curves change shape, and in some cases display a kind of double resonance effect. At present, this phenomenon is not completely understood, the equations being too complicated to pinpoint the cause; however, at the higher frequencies the plasma layer is thick and one may question the validity of the wedge model. (U) Figures 5-8 through 5-11 present results for the physical optics approximation. The physical optics cross sections are consistently larger than the corresponding cross sections based on the wedge approximation - - this is also true in the case of a perfectly conducting finite cone. 231 SECRET

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (db) 0l f= 109 f = 5- 108 f = 2. 109 f = 3 10 f = 5 109 -10 -20o -30 - -40 - -50 I I I I I I I i. I. I.4.6 4- W /w p.8 1.0.8.6.4 "IG. 5-4: NOSE-ON CROSS SECTION (6= 11O, f C - 108). 232 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (db) f = 108 Ok -10 f = 5 - f= 109 f= 2. f = 3 f = 5 108 109 109 109 -20 - -30 k -40 - -50 - I I I I I I I 'I...4.:6.8 1.0.8.6.4 OWP/ p/ p FIG. 5-5: NOSE-ON CROSS SECTION (6 = 8, f = 108). C 233 UNCLASSIFIED

UNCLASSIFIED THE UNVE..~? "8525-1-F" S (db) f = 5* 108Hz ol l,%*" = -010 f= 109 f = 2 - 109 109 — J f = 5 -20 -40 -50.4.6.8 1.0.8 p — = 0).6 WP V.8 / 9 CROSS SECTION (6 = 11 f C = 109). FIG. 5-6: NOSE-ON 234 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (db) ok0 -10 f = 5. 108Hz f = 109 f =2 109 f = 5 10 -20 -30 - -40k -50t I I I I I I 0 m- (0.4.6.8 1.0.8.6.4 *- S /u S I p FIG. 5-7: NOSE-ON CROSS SECTION W1S p (6 = 80 f C = 109). 235 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F zp.o. (db) 0 -10 f = 10 Hz f = 5. 108 f= 109 f = 5 * 109 -20 - -30 -40 -50 - I I I 1 I I I.4.6.8 4 —u 1.0.8 P.6.4 FIG. 5-8: PHYSICAL f = 108). OPTICS NOSE-ON CROSS SECTION (6 = 11~, 236 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F po(db) P.o. o0 - f f f 108Hz 5 108 109 f = 5 109 -10 -20 -30 -40 -50r 1 I I I I I I.4.6.8 1.0.8.6.4 4 "-I P FIG. 5-9: PHYSICAL OPTICS f = 10 ). c W Wu/u p NOSE-ON CROSS SECTION (6 = 8, 237 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F p.o.(db) 0 - ommum. _, — ---= f = 5 108Hz f = 109 f = 5 109 -10 - -20 r -30 t -40 -50 I I I I I I I I *. *. I.4.6.8 1. 0.8.6.4 4- Op/ FIG. 5-10: PHYSICAL f = 109). C p OPTIC NOSE-ON CROSS SECTION (6 = 11, 238 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F. (db) p. o, OF f = 5 10 Hz f = 109 = 5- 109 -10 r -20 - -30 -40 -50 1 - I I I I I I I - I I I.4.6.8 1.0.8.6.4 - w P1 p p FIG. 5-11: PHYSICAL OPTICS NOSE-ON CROSS SECTION (6 = 8~ f = 109). o 239 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) It is clear from the results that the nose-on backscattering cross sections of plasma sheathed finite cones are very sensitive to variations in the plasma sheath properties. No attempt has been made to account for secondary diffraction effects, due to the lack of information concerning the flow field in the base region on the cone. For those cases where the primary diffraction cross sections are very small, secondary effects can be of utmost importance. 5. 3 Re-entry Plasma Experiment (Task 2. 1. 5) 5. 3. 1 Introduction (U) Experimental re-entry plasma studies were undertaken during the third year (1967) of the SURF program in order to examine the effect of thin plasma sheaths on the backscattering (radar cross section) from simple geometries, i. e. flat plates, flat back cones, and cylinders. Wire grid structures with wire spacings less than a half wavelength, were used to simulate the thin plasma sheaths. Previously wire grid structures were successfully employed to simulate plasmas in propagation and antenna studies (Smith and Golden, 1965a, b) but our present investigation for backscattering has uncovered limitations to this technique which were not noticed in earlier work. For example surface field measurements are difficult to interpret because the discrete rather than the continuous properties of the grid structure are more dominant in the near field. Also at oblique angles of incidence the transmission and reflection coefficients for thin sheaths of infinite extent do not appear to be as accurate as at and near normal incidence due to the edge of effects of the finite structure. (U) The plasma sheath study has met with only limited success, especially so far as theoretical models to support the experiments are concerned. One of the primary goals of this part of the program was to determine the validity of a physical optics model for describing the behavior of backscattering 240 - U NCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F from plasma clad bodies. Experimental results indicate that a theoretical model based on the physical optics approximations is valid only near normal incidence on coated bodies which have large dimensions and radii of curvature com pared to the incident wavelength. As the angle of incidence becomes more oblique complex surface waves are excited and cause the scattered fields to depart noticably from the ordinary physical optics model. It also should be noted that the physical optics model becomes less accurate even for perfectly conducting, flat bodies as the angle of incidence becomes more oblique (Ross, 1966). (U) Attempts have been made to explain the effects of complex surface waves on the monostatic radar cross section of a thin current sheet (a grid of wires) in free space without any conducting backing. For this case the complex waves reduce to ordinary surface waves (the type which propagate along the surface and exponentially decay away from the surface) which are easier to work with analytically than the more general complex waves. This model is discussed in more detail later. (U) Before describing last quarters work on the current sheet, a brief review is presented for the work done on the flat plate and flat back cone covered with thin sheaths. For these two coated bodies, the mathematical representations yielded only a limited amount of information, particularly for the coated cone case. Experimental results from both the plate and cone geometries indicate that the composite reflection coefficients for the wire grid backed by a perfect conductor are not accurate for large oblique angles of incidence and it is believed that this is the cause of our poor match between theory and experiment, particularly for the coated cone at nose-on incidence. 241 UNCLASSIFIED

UNCLASSIFIED - THE UNIVERSITY OF MICHIGAN 8525-1-F 5.3.2 The Plasma Covered Flat Plate (U) In Goodrich et al, 1967c experimental and limited theoretical results were presented for the radar cross section behavior of a grid of wires backed by the conducting flat plate. This model was designed to simulate a thin plasma sheath separated from a plate by an air gap. When the wire grid is fabricated from copper wires the sheath is purely inductive and can be described by a surface impedance Z = j X. For the wire grid the normalized inductance is s s approximately Xs/Zo.n (c/2r r) (5.32) s o Xn where X is the wavelength and r and c are the wire radius and spacing. For the plasma sheath the approximate inductance of the sheath is (Smith, 1963) X /Z /)2 1- (5.33) Xs/Zo (W/Wp) kd (5.33) where k is the free space wave number, d the sheath thickness, w and w the plasma and incident frequencies, and Z = 377 2 for free space. (U) Experimental measurements for the flat geometry and VV and HH polarizations have two noticable features: 1) In the vicinity of normal incidence the backscattering for the plate coated with a lossless sheath has the same return as a conducting plate and theoretically follows the physical optics model. If losses (collisions) are introducted the composite refelction coefficient will be reduced and in turn the scattering cross section is also reduced as indicated by the physical optics formulation described in Section 2. 5.2 of Goodrich et al, 1967c. 242 ___ UNCLASSIFIED

U NCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 2) For large oblique angles of incidence usually large lobes, which are dependent on the size of the air gap and the sheath impedance, appear in the scattering patterns. These lobes are due to complex waves generated by edge diffraction and are not part of the usual physical optics model. (U) The approximate location of the large lobes in these patterns can be determined from the poles of the composite reflection coefficients. Unfortunately as the angle of incidence increases these reflection coefficients become more inaccurate and therefore the precise location of the pattern peaks becomes less predictable. For an infinte flat plate with a thin sheath placed a distance I in front of it, the composite reflection coefficients Ri and RI are j 2k x cos 0 Z (e '-1) + 2 Z cos 0e R =- (5. 34) 1 - -j2k Icos - Z (1 -e ) +2 Z cos 0 0 s 1 for WVV polarization and j2kl cos 0 Z cos 0 (e -1) + 2 Z R - 0 II (5.35) jj = - ~ --- -j 2k icos 0 Z cos 0 (1 - e )+2 Z for HH polarization. Z and Z are the free space and surface impedance, 0 5 0 and 06 are the angles of incidence for the two polarizations, and k is the free space wavenumber. These coefficients become inaccurate for finite plates as 0 or 0i increase (depart from normal incidence). (U) For the sake of completeness let the surface impedance Z = R + jX 5 and allow the angle 0 to take on complex values a + j r-. Then if the denominators in (5. 34) and (5. 35) are set equal to zero and one solves for X and X, the sheath inductances for the WVV and HH polarizations, it is -- 243 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F found that RV [S cos (2k C) - C sin (2k C)] - sin (2k C) X (5. 36) V S sin (2k I C) + C cos (2k C) (5.36) and RHS cos (2keC)+ C cos (2kC)]+ 2(S2 + C )sin (2kIC) X = 2 5. 37) H S sin(2k C) - C cos (2ki C) (5. where S = sin a sinh rl and C = cos a cosh rf. Solutions to (5. 36) and (5. 37) indicate the presence of pattern peaks located at angles of incidence a for corresponding values of X, R, ki and Yr. These solutions are poles of the reflection coefficients given in (5.34) and (5. 35). Physically these poles result in complex surfacewaves whichare excited by the edges of the sheath and propagate along it giving rise to large extraordinary pattern lobes at oblique angles. As r] becomes small the amplitude of the extraordinary lobe increases. (U) For a purely inductive sheath R = RH = 0 and if we look for poles when rY 0, then(5.36) and (5.37) reduce to -1 XV 2 cos 0 tan(2klcos 0) (5.38) and = cos tan (2k cos 0) (5.39) H 2 where a has been replaced by 0 since rj = 0. Roots do exist for these equations for various values of 0, XV and XH so long as kl is large enough to cause the expressions to become positive. Negative roots for these equations indicate that a capacitive sheath is necessary and physically the wire grid or current sheath is never capacitive. _ 244 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) Lack of time has prevented this part of our study to be carried any further. Theoretical techniques such as those found in Chapter 11 of Collin (1960), Chapter 4 of Brekhovskikh (1961), and Chapter 3 of Clemmow (1966) regarding treatment of poles near saddle points and complex surface wave propagation would be useful in seeking a more complete answer to the behavior of the problem. Some of these techniques will be demonstrated in the section on the thin current sheet. 5.3. 3 Plasma Coated Flat Back Cone (U) Radar cross section measurements were made on a flat back cone with and without a wire grid sheath covering the cone, the results of which were described in Goodrich et al, (1967b). Measurements for the perfectly conducting cone (without the grid sheath) agreed well with theory for nose-on, broadside, and end-on incident angles. When the wire grid is covering the cone the broadside and end-on cross sections are the same as the perfectly conducting case as expected, but the nose-on return is reduced by 5 to 13 db depending on the incident wavelength. (U) So far two theoretical models have been examined in an attempt to explain the nose-on cross section with the wire grid present and both models only predict reduction of approximately 1 db compared to the uncoated cone case. The two theoretical models (the physical optics approximation and the coated wedge approximation) are discussed in Section 5. 2 of this report. The weak point in both of these models for this particular experiment is that they depend upon the composite flat plate reflection coefficients given in (5. 34) and (5. 35). Near nose-on these coefficients are being applied at steep angles of incidence where their validity is questionable. In addition traveling wave modei can be supported by the grid coated cone and they are not considered in thes e models, in fact it may be these modes which are causing the reflection coefficients to appear erroneous. 245 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) The flat back cone measurements with and without the wire grid produced consistent experimental results even though we cannot explain what is happening at and near nose-on incidence. One way of gaining more insight into this problem is to study the scattering from a cylinder coated by a sheath at oblique angles of incidence, a problem which should be easier to examine theoretically. A few experimental measurements were made on a rough model of a cylinder with a wire grid sheath surrounding it. No time remained to examine or extend this work on the cylinder model, although our limited cross section tests indicated that surface wave modes were having noticable effects on the patterns at oblique angles of incidence. 5.3.4 Backscattering by a Current Sheet. 5. 3.4. 1 Introduction (U) During the final quarter of the 1967 program efforts were made to gain a better understanding of the radar cross section characteristics of a thin current sheet (flat wire grid) alone in free space. This is the primary component of our physical system and a clear understanding of its behavior is essential in order to explain the effects of a wire grid in the presence of conducting bodies such as cones and plates. A distinction is made here between the words "sheet" and "sheath". Both words are used to describe a thin layer of material such as a wire grid or a thin plasma, but "sheet" means a flat layer alone and "sheath" refers to the same material surrounding a body. Both theoretical and experimental results are discussed for the sheet problem. In the experiment a uniform wire grid is used to represent the current sheet whilE in the theory the current sheet is any thin layer of material which can be described electrically by its surface impedance (Smith and Golden, 1965a, b) Z = 1/ad (5.40) s 246 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F where g is the complex conductivity and d the thickness of the sheet. Results are given for the two linear polarizations, vertical (VV) and horizontal (HH). (U) The main difference in the monostatic radar cross sections for the two polarizations is that the HH case supports surface waves and the VV case does not. The effects of the surface waves become evident for aspect angles beyond 300 where the lobes of HH patterns increase in amplitude and beam width somewhat like the pattern of a traveling wave antenna (Wolff, 1966). Otherwise the backscattering for aspect angles between 00 and 300 for both polarizations has the same characteristics as that for a perfectly conducting flat plate of the same dimensions with the amplitude reduced by an amount equal to the power reflection coefficients. (U) The feature which distinguishes the sheet from a perfect conductor is that part of the incident field is transmitted through and part is reflected by the sheet whereas all of the incident field is reflected by the conductor. Because the sheet is thin its analytic solution yields to a simplified form of the Stratton-Chu Equations which is no more difficult to handle than the conducting plate problem with the exception of accounting for the surface wave in the HH case. 5. 3.4.2 Formulation of the Problem (U) Consider the geometry given in Fig. 5-12 where a plane wave of either vertical or horizontal polarization is incident obliquely on a square current sheet which is very thin compared to the wavelength in the sheet material. It is assumed that the length andwidth, a, of the sheet are large compared to the incident wavelength (a > > X) to such a degree, that locally the transmissiol and reflection coefficients Tll, T, Rl, and R are those for sheet of infinite extent along the x and y axes; this is essentially the physical optics 1 247 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F x H E y (-L ELL Current Sheet - Region 2 r z I I legion 1 R I 1 '.L (a) Vertical Polarization / Trransmitter r and Receiver in the Far Field x a 2 T - II) E. z I R11 2 2 I H p (Surface wave) (b) Parallel Polarization FIG. 5-12: SCATTERING GEOMETRY FOR THE CURRENT SHEET. 248 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F assumption. At the sheet surface, z = 0, the Poeverlein-Wait jump conditions are used to describe the sheet behavior (Wait, 1960; and Smith, 1965a) z x ( - E ) 0 (5.41) z x (H1 - H2)- J (5.42) where z is the unit vector normal to the surface, J is the surface current s (amps per meter) and the subscripts 1 and 2 refer to the electric (E) and magnetic (H) fields in regions 1 and 2. The surface current is related to the electric fields and the surface impedance in (5.40) by - J (z (5.43) s Z 2 2 s on considering (5.41) s Z1s s = 1 E1l -_(. 1) (5.44) (U) For two-dimensional, flat geometries with homogeneous and isotropic materials the W and HH polarizations may be examined separately and the entire E and H fields can be expressed as A - jkz cos 0 kzcos 0 jkxsin E1 = y E (e os +R ek ) ekxsin (5.45) E2 E T ek (z cos + x sin ) (5.46) 2 y 0 Jk for VV polarization and H1 = y H (ejkz cosO+ Rj e-jkz cos 0) ejkxsin 0 (547) Hi yH e+ 1 O(.7 ______249 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F H2 y H T elk(z cos 0 + x sin 0) (5.48) H 2 y H T ek( OsOxsif9) (5.48) for HH polarization where E and H are the intensities of the incident o o fields for the two cases. The remaining E and H fields can be derived from Maxwell's two curl equations. No surface wave fields are included in the above, but they will be considered later. (U) An application of the Poeverlein-Wait jump conditions (5.41) and (5.42) to the total fields in regions 1 and 2 leads to the following transmission and reflection coefficients (Poeverlein, 1958) i 1 1 2 Z cos (5.49) +- 1 - s1 1 2 Z cos 1+ Z s 0 for VV, and T 1 R = 15.50) TII Z C —os ' i (5.50) 1+ - 1 + s 2 Z Z cos 0 s o for HH polarization, where Z = / and Z = 1/od according to (5.40) or more generally Z = R + j X. (U) In order to determine the monostatic radar cross section, where the transmitter and receiver are located together at the same far field point, it is necessary to find the scattered fields. The Stratton-Chu equations (Stratton, 1941), which are a vector form of the Kirchhoff-Huggens principle, are evaluated in the physical optics approximation to obtain the scattered far fields. Assuming a time harmonic form e, the scattered E and H fields can be expressed by a surface integration over the total fields E(_L, I I) and H(J1, I ) _______ 250 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F at the surface of the scatterer (Johnson, 1965), = - (z x E x VG + (z ) VG - Si -j k Z G ( xH dS (5.51) 00 1 + |(z xoH~ x VG+(,-c HsV Hk where d Si = dx dy is the differential surface element on the region 1 side of the sheet, and G is the spherical Green's function -jkr G - e ~ jk (z cos 0 + x sin e) (5 4 a r e(5. 53) 0 4n ro and in the far field VG z - (jk) (z cos 0 + x sin 0) G (5.54) where r is shown in Fig. 5-12. The fields at the surface in (5.51) and o (5. 52) are related to those in (5. 45) through (5.48) by E - - \ 1 n1 E 2 and (5. 55) - H11:H1 - 2 2 251 - UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 5.3.4.3 Radar Cross Section of Current Sheet-Excluding Surface Wave (U) Since the surface integrations in (5.51) and (5. 52) take place at the sheet, the Poeverlein-Wait jump conditions are applied to the surface fields and only the difference in tangential magnetic field components, which are equal to the surface currents J remain; thus -Es 5 j- j k GzO G V x jw x 1 Edx dy (5. 56) 1 and 1 x (H1 - H2)] x VG dxdy (5.57) where (5.55) and stitution of (5.45) sheet, it is found H = l/jw4 V x E have been substituted. Upon further subthrough (5.48), (5. 53) and (5. 54) and integrating over the that / 2 \ YEo J ] jka cos sin (ka sin e j) k I2 rr L 1 _ —Lka sin0 O (5.58) and Hs = yH II 0 sin (ka sin 0) 1 -jkro 'II ka sin 0 (5.59) As the current sheet approaches a perfect conductor (Z -- 0) both R and s 1 RII approach unity and the scattered fields for the two polarizations have the same form 252 UNCLASSIFIED m -

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F j ka cos 0 27r r 0 sin (ka sin 0) -jkr ka sin 0 which is the usual physical optics result showing polarization independence. (U) Radar cross section is defined by 2 2 E H 2 1E 2 ]l ~ —S 0 E ~ H E H 0 0 and for the case of a perfectly conducting, large, square plate, both the E and H- formulationsyield and H formulations yield 0) 2f 1 a (kacos )2 sin (ka sin 0) 2. = (ka cos (5.61) pc r ka sin 0 If the two polarization cross sections, cl and l are normalized to pc and the Z is expressed in terms of a surface resistance R and inductance X s s s then =R R- = PC 1 (5.62) e and c1 = R R pc I 2 cos 8 (5.63) 2 cos 0 + 4(R /Z ) cos 0+4 S 0 where the * quantities are the complex conjugates of the reflection coefficients 253 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F in (5.49) and (5.50). In the case of VV polarization (5.62) is as good an approximation for a sheet of arbitrary Z as o is for a large conducting s pc plate. Beyond an aspect angle of 0 = 30 all physical optics models increase in error with increasing 9 as indicated in an article by Ross (1965). (U) For small aspect angles 0 through the first few sidelobes off broadside o'a /c' in (5. 63) is also accurate, but as 0 increases and depending on the value of Z, the shape and amplitude of the scattering pattern depart noticeably from ordinary physical optics behavior. This is due to the excitation of surface waves by the edges and the support of these waves by the sheet for HH polarization. A rough model is presented to account for the surface wave, but it is only a qualitative model which shows that the lobe amplitude and beam width increase, but the agreement with experimental data is poor. This is to be expected because the physical optics model is poor in this aspect region independent of whether or not surface waves exist. The larger the inductance X is compared to R s, the more noticeable the surface wave contribution is to the pattern. 5.3.4.4 Surface Wave Contribution (U) Actually two surface waves are excited for HH incidence one originating from the edge at X = a/2 and the other at X = -a/2. If these surface waves are treated like traveling wave antennas, ignoring any multiple reflections, then the edge at X = -a/2 will contribute considerably more to the monostatic scattering than the a/2 edge; it is the only surface wave analyzed at this time (Fig. 5-12). Our model is based on the work of Tamir and Oliner (1963), although J.R. Wait(1960) also has looked at this problem. In order to formulate an expression for the surface wave, assume the sheet in Fig. 5-12b is extended to + co along the X axis and that a magnetic line source _________________________ 254 _____________ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F is located at y = -a/2 and z = 0 which is just on the region 1 side of the sheet. According to Tamir and Oliner (1963), the surface wave contribution from such a source is of the form RH (- L ) -jk(x + a/2) cos ( -/2) p dR (0p) df where L is the launch factor representing the strength of the line source generated by the edge at x -a/2 and p is the complex pole of the reflection coefficient, R cos (5.65) Z cos + 2 - 0 o This last equation is actually (5. 50) after 0 = a + jr] has been substituted for 0, the aspect angle. To find the pole location 0p, set the denominator of (5.65) equal to zero and expand 0 and Z in their complex forms to 2 cos (a + jr1) = - Z (Rs +j X) (5. 66) 2i S S 0 which leads to two expressions upon setting real and imaginary parts equal to one another. 2 cos a cosh -- R Z s 0 sin o sinh r = X (5.67) Z S 0 Returning once again to (5. 64), the term 255 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 0 p dR,(0) 2 Z sin (5.68),IIp ) s p do is the residue for the pole contribution. (U) In the experiment the current sheet is purely inductive with R = 0; s thus Eq. (5.67) indicates ao= ir/2 or 7 = ir/2 + jr7 which leads to sinh ri = 2 X /Z cosh rt = 1 + (2 X /Z ) (5. 69) and after further substitution R i (p) -Z sinh2 r j 2 X R (0) 2(j X ) cosh -2 2 (5.0 p s Z +4X 0o o Therefore the surface wave in (5. 64) is approximately H 2yLL s -jk (x + a/2) b (5.71) H p y L Z e (5. 71) where b = 1+ 4 (X /Z ] 1/2 and the approximation sign is used to indicate that the launch factor L is unknown. This is to say the amount of the incident energy which is converted into line source energy is unknown. - -- (U) When H1 - H in (5. 56) is replaced by H in (5. 71), the inte2p -s gration over the current sheet gives the scattered field HP due to the surface p wave jk a2 cos0 \ ~2X L sink a (b- sin 0) -jk(roH=y 2 1 Zbe (5.72) p 27r r Z b a - o o ' -~ k (b - sin 0) 1 256 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F S S Now H and Hj can be combined as phasors and after using the definition in (5.60) the total HH cross section c[i can be expressed as -s s 2 T +H 47r r H (5.73) o H 0 where HI and Hs are given in (5. 59) and (5. 72). This expression which II p atempts to account for surface wave contributions in the HH scattering patterns only gives a rough indication of this behavior as will be shown in Fig. 5-13. (U) Before going on a word is in order to explain why the VV polarization case does not support any surface waves. If the denominator RE in (5.49) is equated to zero in the same fashion as in (5. 66), then - Z cos (c + j) = 2R + 2 (Rs sj XS) and after equating real and imaginary parts -Z X O S (real) cos c cosh rl 2 2 2 (R + X ) s s -Z X O s (imaginary) sin c sinh ry - 2 (5.75) 2 (R + X) s s The latet equation indicates that for VV polarization the sheet will support surface waves only when it is capacitive reactive. Under most circumstances thin over dense plasmas, especially cold ones, are inductive reactive. More discussion on this topic is found in Wait (1960) and in Collin (1961). 257 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F K 4 i.. Ii,C i i I. 1. I... I i i llr CD CD r1i I i I --- -.. -. -... --. I j i i - i — t = - I ---- llIll9 ----*i —..A-.-.l i +;r4 I I.4 I p. -4 — o — I I i:, I I - L i.! I I. I. e4 -7 7' T I I. i I I, I C14 I I I I I i -C qp1 83M~d 3AI1V136 1 i I I i I I. i I i AI p I - -r —v~ ~h17 -t7 co cm-o0 iqi _ _ _ _ _ __3 A_ _V_3_ r... I i........ H N 0 "0 0 0 C, C) p 0 r z H-: H4 C,) Q t: 0 N z N 0.1 I _ _ _I.: ItT _ _ a20 I. 'IA I7 T 7'-= I I! - 1 1 1!II la.I I-1.:I:.I - - I - -.I 1 I r —f P4 4 I 8 aI -r4 FI I li r —4 i < CY) r —i I Lo00 2 4 - I -. I. I. I - - i. I. I I 1 1 -t I- I.. I I! qi= I I 14., - i.4 - i I, i 41 —l i I. I.,-. I:.,!. - i 6 I t I I. I i. i -,. I r!- - q l I i I. I. I.... -.. -. —. I I. I -f — - -. -.4. - - I — T --- I 258 UNCLASIFIE

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 5. 3.4.5 The Current Sheet Experiment (U) In the experiment, wire grids with two mesh sizes were measured at four X-band frequencies. Table V-1 displays the frequencies, wavelengths, sheet sizes normalized to a wavelength, and surface inductances for the tests. TABLE V-1 Xs/Zo x. f (GHz) X (cm) a/X C = 1/2 cm C =1 cm 8.5 3.53 6.48.31.82 9.5 3.16 7.25.35.92 10.0 3.0 7.62.37.96 10.5 2.86 8.0.39 1.0 The sheets are 9" x 9" and the mesh spacings c are 1/2 and 1 cm. Equation (5. 32) is used to calculate the wire grid inductance and is valid for c/X < 1/2 (Golden and Smith, 1964). The grids were made by gluing No. 33 gauge wire (dia =.007") to 1/4" sheets of styrafoam. These models were measured on an indoor scattering range 25' from the transmitter-receiver location. (U) Four experimental patterns are displayed in Figs. 5-13 through 5-16 for tests made at 10 GHz, VV and HH polarizations, and grid surface inductances Xs/Z equal to 0.37 and 0.96. In each of these patterns the right hand portion, e between 00 and 900, is the wire grid results and the left hand portion, e between 90~ and 1800, is the calibration reference, an aluminum flat plate. Both the grid and the plate are 9" x 9". The broadside peak for the plate has an absolute cross section in db compared to one square meter of 259 UNCLASSIFIED

II - 't:T 0 0 0 1 —d ) —a.5 0 r5 'IC 1 — C-t-p CD CD 0 rl) I- -4 - NV9IHOUN 10 ALISRIIAIt~fl a111 GJIJI&SV19Nf

U N LASSFIE THE UNIVERSITY OF MICHIGAN 8525-1-F c 1. t 0 2 0 V!; I AIliV1d. - I.-M. smpcm!=! -' -'. r0 0 F1 0 H z. S 0 0z OQ!v4 'I -1I i1 261 U N CLASSFE

THE UN UNCLASSIFIED IVERSITY OF MICHIGAN 6525-1-F I ' J".... - I. CD I.. 3., E.1 --- ' N p 3M 3.........Al13 ---- -. -....:.!,,... qp 83~d JAl V1 ---~:: 0................: d r4 262 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 2 a2 2 C = (ka) =+ 15.8 db. (5.76) pC 7T This recipe is obtained by setting 0 = 0 in (5. 60). (U) These measurements show clearly that the HH grid patterns produce large, broad lobes beyond 30 aspect angles compared to VV grid patterns or either polarization results for the aluminum plates. These broader lobes resemble the pattern structure of a traveling wave antenna (Wolff, 1966) especially in Fig. 5-14. Surface wave behave in the same was as traveling wave antennas and it is reasonable to attribute these scattering pattern characteristics to surface waves excited by edge diffraction and supported by the grid sheet particularly since theory predicts this type of action. For the most part the VV grid patterns tend to follow the aluminum plate tendencies with one exception; the nulls are sharper for the grid at larger aspect angles. (U) Figures 5-17 and 5-18 are comparisons between theory and experiment for the 1/2 cm mesh grid and aluminum plate. In all these patterns the solid line is theory and the XX marks are the experimental lobe peaks taken from Figs. 5-13 and 5-15. Part (a) of Fig. 5-17 is the HH pattern for the grid with X /Z = 0.37. The theory for this pattern is obtained by renormals 0 izing (5. 75) to the form T 2XL C11 2 sin (ka sin 0) s 2 2 = cos0 RII kasin 0 + Z b a (ka) o sin k (b - sin 0) i 2 (5.77) 22 a - e k a (b - sin 0) 22 such that for 0 = 0, (5.77) is R2 or - 2 db compared to the plates. For _________________________ 263 ________________ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F 3X X -10 xx -20 x a2(ka) -30 -40 - I. — I -- 1 M ICHIGAN -Theory X Experimental Lobe Peaks X X 00 60~ (a) Wire Grid 0: -10 --20 - x x x 7tro PC2 a2 (ka)2 i -30 - -40 - x x x 5 9 N 00 10U 20~ 300 0 40~ I 50~ 60~ (b) Aluminum Plate FIG. 5-17: COMPARISON BETWEEN THEORY AND EXPERIMENT FOR HH POLARIZATION AT 10 GHz. 264 UNCLASSIFIED

UNCLASSIFIED THE UNIVER SITY OF MICHIGAN 8525-1-F Theory X X X Experimental Lobe Peaks X \A A X X T I I a2 (ka)2 (db) -20 -30 -40 0 10 20 0 30 40 50 (a) Wire Grid 60 ) -10 7T pc a (ka)2 (db) I -20 - -30 - A I - x x x x Xx X I -qtu - - - I i I I I I I~I 00 10~ 20~ 0 30~ I4 40~ 500 60~ (b) Aluminum Plate FIG. 5-18: COMPARISON BETWEEN THEORY AND EXPERIMENT FOR VV POLARIZATION AT 10 GHz. 265 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F the case in Fig. 5-17 L, the unknown launch factor was arbitrarily set equal to b in (5. 77) to produce the best fit near 30. The lower pattern in part (b) is for the aluminum plate and the theory is taken from (5. 61) and put into the form 2 pc C 2 sin (ka sin )(5.78) a2, 2ka2 ka sin 0 a (ka) and for 0 = 0, (5. 78) is unity or 0 db. (U) The theory for part (a) of Fig. 5-18 which is the VV polarization pattern for the grid is derived from (5. 62) and is 2 _L OS2 sin (ka sin) (5.79) a2( 2 ka sin 0 which also reduced to R for 0 = 00 or - 2 db compared to a perfectly conducting plate. Equation (5. 78) holds for the VV aluminum plate case in Fig. 5-18b. More agreement is found in the VV than the HH pattern for the grid. This is mostly due to the qualitative rather than quantitative model used to describe the surface waves. More effort is needed to develop a more accurate formulation for the surface wave behavior. (U) Figure 5-19 is a comparison between theory and experiment for all the test cases in Table V-1 at normal incidence as a function of the grid inductance. At 0 = 0 the theory for both polarization reduces to 7Tr = 2 1 2 R (5.80) a (ka) 1 + 4 (X /Z ) s O where the second term in (5. 77) is taken to be zero compared to the first. Good agreement is found between theory and experiment in Fig. 5-19 where all errors are less than 1 db. 266 UNCLASSIFIED

2 1 R = 1 + 4 (X /Z ) 3 0 0 HH Polarization Experiments X X X VV Polarization Experiments 1S3 m3 la 0s II C5 CI> P4 00 bo Cn l I I Pt ~-c Z c D z Cl) 0 - I'11 m 10. z 0-4 O: 1.4 Xs/Z (Surface Inductance) s'O FIG. 5-19: COMPARISON BETWEEN THEORY AND EXPERIMENT FOR THE RCS OF GRID AS A FUNCTION OF Xs/Z AT NORMAL INCIDENCE: sO THE WIRE

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 5.3.5 Conclusion (U) The work done on scattering by current sheets in the final quarter of 1967 shows that theoretically and experimentally physical optics assumptions do not hold for HH polarization at aspect angles greater than 30. Beyond 30 surface wave, which are excited by edge diffraction, propagate well along the sheet and add a significant contribution to the scattering pattern. Vertical polarization follows the physical optics model more closely and no surface waves are supported by the sheet in this case. (U) For coated body scattering problems with curvature such as cylinders and cones, both VV and HH polarizations must be considered together no matter what the incident polarization is. Therefore the surface wave, which will be more generally complex than not, must be accounted for at oblique angles of incidence if a reasonably accurate model is desired. This is demonstrated by the poor agreement between theory and experiment for the wire grid coated at nose-on incidence where surface waves have been neglected. (U) If and when time and money permit, experimental and theoretical research should be done on the finite cylinder surrounded by a wire grid. This analysis will lead to a better understanding of how VV and HH polarizations couple and what types of surface propagation they produce. Once the coated cylinder is better understood, the coated cone difficulties should be reduced. _____ 268 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F VI SHORT PULSE INVESTIGATION-TASK 4. 0 6. 1 Introduction (U) All aspects of the problem discussed in the Third Quarterly Report (Goodrich et al, 1967b) were continued, and the new work performed in the final quarter is described under the same section headings used in that report. (U) The work to date on this task represents completed significant researc on preliminary, relatively simple problems and on finding productive formulations of more physically interesting but difficult problems. In the course of studying the physically simple problems various interesting and unexpected limitations on the methods to handle pulses were uncovered. Some more specific descriptions and evaluations of aspects of the work are as follows. (U) The work on this task which we feel has the greatest potential for providing knowledge of short pulse returns from relatively complex targets is the integral equation numerical approach formulated in Goodrich et al, 1967b Section 4.3. One part of these equations have been developed in the present report's Section 6. 3 to a form very nearly ready to be programmed for a computer. These equations give the time dependant surface fields. The remaining equations give the fields off the surface and can be reduced in a directly analogous manner for numerical computation. This method is fundamentally different than the CW method used in Section 4. 4; it is much better adapted to handling arbitrary pulse excitations, since no Fourier super-position of CW results is needed and there is no limitation to symmetrical geometries. We forsee no special numerical analysis difficulties in the method. Although written up in our work only for perfectly conducting bodies, the method is applicable to bodies with general impedance boundary conditions; however, the details are more complex. _______269 UNCLASSIFIED

U NCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) The formulas and numerical results of Sections 4. 4 and 4. 6 of Goodrich et al, 1967b and of 6. 4 and 6. 6 of the present report are quite specialized though they do illustrate the possibilities of pulse distortion. Some of them, primarily the perfectly conducting sphere results, are not basically different from results in the literature (Rheinstein, 1966) and were meant as trial problems to get us started prior to doing similar work on coated bodies. However a simple side investigation stimulated by these sphere results brought to light some pronounced effects of bandwidth on pulse reception which were not anticipated. These results are described in Section 6. 7, below and are not limited to particular shapes or materials for the radar scatterers. In addition the simplicity of the formulas in the flat-backed cone section (6. 5) brought to light certain unexpected theoretical difficulties in the general approach of synthesizing pulse returns from approximate CW responses by superposition. These are discussed in a separate section, Section 6. 4 since they are in fact of concern for the treatment of any shape of scatterer which supports creeping waves. (U) The investigation of pulse diffraction by means of the pulse phase fronts, their normal trajectories (rays) and transport equations for the discontinuities of fields and their derivatives has proven far more difficult than anticipated. (U) Various approaches to the problems which arose have been tried (as reported in the previous quarterly reports under the heading Ray Optical Techniques, and in the present Section (6.2). While progress has been made we are far from having applicable results. To a large extent it turns out that we cannot avoid, by these methods, the difficulties which occur also in the CW highfrequency asymptotic diffraction theory. These difficulties are tied up with 270 U NCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F divergences of the electromagnetic fields on the surfaces of the diffracting bodies. These divergences are in fact unphysical, but more refined theory than we have previously presented is needed to remove them from the formulas. (U) A somewhat different and promising approach to the determination of the time dependent surface fields has therefore been tried in the last quarter. In particular an extension was made of the Fock method (1965) to treat the surface CW fields. This extension is reported in Section 6.2. As carried out it is limited to a fairly restricted class of pulse shapes incident on a smooth perfectly conducting parabolic cylinder. It is felt that further work could result in suitable generalizations. 6.2 Transient Surface Fields (U) In this section an attempt is made to extend Fock's (1965) analysis of the launching and propagation of surface fields at and into the shadow side of an irradiated smooth obstacle to include an incident plane pulse with arbitrary time dependence. It was found that for a certain class of pulse shapes the method of Fock could be used to obtain the propagation of the pulse along the surface of a smooth perfectly conducting parabolic cylinder. (U) For the two dimensional problem we desire to the solution of 2 2 2 au u 1 a u 2 2 2 2 ax ay c at in the neighborhood of the origin or coordinates as shown in Fig. 6-1. The assumed solution will be of the form u (x, y, t) = E gn(x, y) f (x - ct) which we choose as satisfying the wave equation term by term. Substituting 271 UNCLASSIFIED

UNCLASSIFIED THE UNIVER SITY OF MICHIGAN 8525-1-F y! | F L Incident Plane Pulse FIG. 6-1: COORDINATE SYSTEM. g(x, y) f (0); 0 = x - ct into the wave equation we obtain 2 gx fr + (gxx + gyy) f - 0. Using Fock's order argument that y >>gx ' gyy gxx we obtain the equation 2f~ gyy f gx = Making a change of variables 5 = ax r = 3 F (x,y) and letting x (x, Ys 272 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN 2 F (x, y) = y + 2R 0 where R is the radius of curvature of the cylinder at the origin and a, B o are constants we obtain 2 G + - rlq f a f? 1 a 0 R O G1 =. li_ Letting 2f and 2 a e 1 + 32 4 a RO I 2 ( Q) d5 G (g, r) = A I we obtain for v ( rY) the equation v +2 -P+ 2aR ir7 - a~ 2p fr~ v + n + q a 2=aR] a 2 a3R0j - &v where p = (d e To obtain Fock's equation we set 2 a + 2aR 2O 0 0 which becomes 273 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN - 8525-1-F 1 2a 3R ^-,~ 2 - which is recognized as Bernoulli's equation. The solution for ~ (5) is given by 1 [c2 1 r - 2 0 2 1/22 <^ L c1-2 0R^ Noting that - = a x and ( = of functions: 2 f' (x - ct) f (x - ct) we solve for f and obtain a class I R 2e 2 a e f () = C1 for which the equation v nfl + rv = 2 v 3 defines v ( rl). (U) The solution of this equation is 132 x v(g rn) = (i) ( w (X - r) where w(i) (X - r) is an Airy function which satisfies (i) w (X - r) = (X - n) w(X - rl) and X is an eigenvalue of this equation. 274 UNCLASSIFIED - -~

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) To satisfy the Neumann Boundary Condition for u (x, y, t) requires that av/r =0 o. rl= 0 Satisfying both the boundary and initial conditions on v (Q rj) we obtain { e 2Xf wd ( wI(X) v(e n) = e ( w(X) w2( - r) - dX where c is a contour to be chosen. Thus v (Q r]) is known and hence G(Q rl) and g (x, y). Depending upon the values of a and 13 the function fc1(0) is also determined and thus u(x, y, t) is specified. Now we may write 3 2 3 1 I (9) d a 4a R G ( r ) =A e v (e, rl) which reduces on the surface of the cylinder, rY = 0, to the expression. 4a3 R2 G(, 0) = Ae 2 e dX w (X) c Thus, for the given f which is a member of the above determined class of function we have determined u (x, y, t) on the surface of the cylinder and in the neighborhood of the origin. The resulting integral over c is not explicitly the representation of the Fock functions but it is hoped that the integrand could be represented in such a way as to make G (Q, 0) given by a 275 UNCLASSIFIED II -

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F series of Fock functions. This is yet to be done. It is noted that the function f is real only for the interval 0 = x - ct < (c a )/(2 32R ) which means that this technique will only yield the result for a certain predetermined period of time after the passage of the pulse front. (U) By repeated application of this procedure at successive locations along the surface, the propagation of the pulse along the surface of the cylinder can be found and at the point of reradiation of the creeping wave it would appear that the radiated component of the creeping wave could be found. (U) Additional work must yet be done on the details of this procedure which when worked out will provide an analysis for the surface diffracted field, which would be combined with our previous ray-optical work. Finally it is noted that the case c1 = 0 represents Fock's assumed time dependence for the incident field. 6. 3 Integral Equation Formulation of Time Dependant Scattering Problem (U) In this section an integral equation is derived which governs the electromagnetic field when an electromagnetic pulse is incident on a smooth perfectly conducting object in three-space. A numerical solution of this equation for the unknown tangential components of the magnetic field on the surface; i. e., the surface current is outlined. The resulting expression is of the form -~ K n. x = - 47r n. x. + n. x B. x,m A j q m jk 3k1 k ( n x, m-) 1 < j < K where the surface is divided in K small subdivisions over which the field is _____ 276 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F assumed constant, th n. is the outward unit normal at the j subdivision, -th jm is the magnetic field on the j subdivision at the time m T after some initial time, T is the increment in time, m an integer, and A. and B are constant "influence" coefficients which are known J jk functions of the surface geometry and independent of the field and time. The summation over q always begins with q > 0 so that the field on subdivision j at time t + m T is always expressed in terms of the incident fields and the total field in other subdivisions at previous times. (U) The method outlined in this section provides a basis for the numerical solution of electromagnetic scattering of pulses by perfectly conducting objects. It is recommended that the following tasks be undertaken in a program concerned with scattering of electromagnetic pulses. (U) 1. Carry out the calculation of the surface current for a particular incident pulse and a particular object. This first example should be chosen to be as simple as possible to demonstrate the workability of the method, or more to the point, the absence of any fundamental errors in the derivation. (U) 2. Derive a numerical method for finding the field at any point in space and demonstrate its validity in a particular case. This appears to be directly analogous to the expression of the surface current but the details should be carefully considered. (U) 3. Investigate alternate methods of solving the pulse scattering problem numerically. The present method appears workable but is not neces _______277 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F sarily the best. The idea here is that it may be possible to carry out some of the iteration analytically and arrive at a simpler, i. e., more economical, numerical problem. (U) 4. Attempt to extend the above techniques to find 'the numerical solution of pulse scattering problems involving imperfectly conducting objects. (VJ) As derived in Goodrich et al, (1967b) Section 4. 3, the scattered pulse, S(r, t) due to an incident pulse I (r, t) in the presence of a smooth perfectly conducting surface B is A -r rt) = Vx x (a x a)f (t — - -- )da + x x (r t - ) da, r extB (6.1) BB A where the incident field propagating in the a direction is of the form \A \ ^I it a. r^ { (r, t) =Y * ( x a)L(t - ) a is a constant vector such that a x a = 0, r is a radius vector to a field point, rB is a radius vector to a point on the surface B, B R = | -B J, is the distance between them, Y =:c/t' is the intrinsic admittance of free space, and is an arbitrary function. An alternate and more complete form of this integral representation is 278 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F I 1 Vx 4r V B n R i.t — l_ R x I( t - -)da + (r, t) = (r,t), ext B, (r,t) B = 0, rintB (6.2) where ~ denotes the total magnetic field (incident plus scattered). (U) This expression results from Eq. (4.23) of Goodrich et al, which may be written, again more completely, as (1967b) 1 4V Vx 4ir ~B n -B, da - ) x, t- ) da = (,t) r extto B R B C = - (r, t) int to B or adding i to both sides, V x i x (r, t -R da+ (,t) =d(rt) r exttoB 4r fB c = B ntt = 0 r int to B (6.3) When r (U) integrand is on B we take the principal value and arrive at Eq. (6. 2). By bringing the curl operator under the integral sign in (6.2) the becomes V x R X (rB, t - where V operates on the variable r. Then 279 UNCLASSIFIED m

UNCL~ASSFE I THE UNIVERSITY 0 8525-1-F 'F MICHIGAN V xRxX rBI t - B) c 7 -I A — I.- R = V - x n xX(r t - -) R L B 't c I 1-ICU Rn r 1 t -B) C-i (6. 4) but, since n depends only on r B and is indep endent of r, 7 -— L A, t - V x n x (r B a cj I nx ax f( rB, Ri-B t-c y A x nx L. ay rB'. t cJi zLx az rB'I (6. 5) Furthermore, (x rl B' "t — ) at (r"B I __ %. R =ay~ ay B'p c ) at B' R 1)(IC aR R aR - B) (-1/c) pR I az rB'I t 15B = C atx B - -) (-i/c) C aR az Thus, Eq. (6. 5) may be written In addition = - I V R R R2 1 C L xn x (rB t - ] (6. 6) (6. 7).280 UNCLASIFIE

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F Substituting (6.6) and (6.7) in (6.4) we obtain V c x x B t - ) *- RJ (6.8) - v n x n rB t - )](6. 8)) Finally, we note that V R = - VBR where VB operates on "r and use 1 B a B this fact together with (6. 2) and (6.8) we obtain 7rB R L c (i(r B t)+. + 4 R B Ax {(rg, C +C at (rB t c ) da =A((U t)0 r extB =1 z(,t), r e B = 0, i int B (6.9) (U) It is this form of the integral equation for the magnetic field which is most analogous to the scalar equation treated in Soules and Mitzner (1966). They proposed a numerical solution which may be extended to include the vector problem. An outline of this procedure for the vector case is presented next. (U) The idea involves decomposing the surface into zones in which the field is taken as constant. The best way to effect this decomposition is not necessarily the simplest, according to Soules and Mitzner; however, this question will not be treated here. 281 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) The surface is subdivided into K zones. If we think of these as being formed by the intersection of the surface with a set of parallel planes, see Fig. 6-2, then in order to make sure that the field is relatively constant in each zone, there must be a further subdivision (see Fig. 6-3). FIG. 6-2: ZONING THE SURFACE. FIG. 6-3: ZONES OF CONSTANT FIELD. This was done in Soules and Mitzner and is probably convenient if B is a surface of revolution. On the other hand, it requires two subscripts to denote one zone and this is tiresome in formulae that are cumbersome at best. 282 -. UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F Furthermore it is not necessary if we think of Fig. 6-3 as representing the zoning of B. This is essentially a matter of notation and is in no way a change in method. (U) Thus we assume that B is subdivided into K zones, each one small enough so that 74 may be considered constant. Furthermore, the surface must be smooth enough so that n may also be taken as constant in each zone. Then, with equation (6. 9), it follows that 2 (r t) = (r., t) + 47 VBR x Bk R R. R. R. [n x{(B' t- — )+ a (r - da (6.10) c c at rB'tc and with the assumed constancy within zones 2 4r k k c tt R B Rj da (6.11) c Trk t where Rjk r. - rk and 1< j < K. (U) Furthermore the remaining integral may be evaluated and is a constant vector: i.e., let * Section 6. 3. 1 shows how to handle the term involving the singular integral (6. 12) in the final expression (6. 33). 283 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN —.b I jk Bk 1 R V R da R2 Bk jk jk (6. 12) then 2 i i (r,j t) + I 41r K k = 1%k kt- ) L x(rk, tR C IC A jk CA Rk + x (r tk c at k c (6.13) 1 < j < K. (U) In these formulae, r. denotes a radius vector to a point in B.. J J B is to be small hence r is essentially constant over B.. Since the pre3 3 cise zoning rule has not been formulated, neither can a precise rule be given for choosing rj, given B.. However, roughly speaking, r. should be taken J J 3 as the radius vector to some interior point of B., probably the "center" if one can be easily defined. The reason for this is that it is necessary to choose a time increment, T, and it is convenient to do so in a way which quarantees that Rjk > CT for j = k jk - (6.14) In general Rjk/c will not be an integral multiple of T but may be written as Rjk/C = (njk + jk T (6. 15) 284 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F where nk is an integer ( > 1 unless j = k) and 0 < k < 1. jk jk then nj = 0 and yk = 0. jk jk (hk' t - r = which is approximated by With this notation If j =k (6.16) (rk. t- n.jk7 - jk (rk t- ~-1(. k' C + jk Xf rk, - Yjk)(rk t - njkT) + t - (njk+ 1) (6. 17) (Note that the approximation is exact if yjk = 0 or Yjk = jk jk 1.) Similarly at k at r k' Rt C t - - k) (1 - 'jk) at (rk t - n. ) + jk + jkX Lrk t - (njk+1) ] Now assume that time progresses in integral multiples of T, i. e. t = t + m, m integer and rewrite Eq. (6. 13) using Eqs. (6. 17) (6. 18) and (6.19) obtaining (6.18) (6.19) a((i t +m7) = (, t+ mT) o 3 I K 4r k=ljk k =jk (1 - )k) n k x [rt, t Continued +(m- ) x( rk' t +(m - 1) i knjk) o jk. 285 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F +(nk+yjk)T(1 -jk)nkx - t E +(m -nJk] + jk -+ jk) Tjk nk x -(k', + (m - nj - 1) T (6.20) ik k k k t ko jk (U) The ence formula. at The constants approximation, next step is to replace the derivative with a backwards differ - Let [k' t +P Tj Cq [k o + ( - q) * (6.21) q=O c and the number of terms, Q, will depend on the e.g., if Q = 1 then c = 1 and c = -1 so that o 1 particular 'r k' to+P Tp- (rk' to +PT)-I k' t + (p- 1)Tj Soules and Mitzner give an example of a three term approximation. stants c are subject to the restriction q c = 0 q=0 q (6.22) The con(6.23) Introducing the general approximation, Eq. (6.21) into Eq. (6.20), leads to the equation J ] k==1 j Continued 286 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F -, jk)n x rk t + (m - njk) [k t +(m -n- 1) + (n k to+(m- nk -q) + (nJk rk t + (m - n 1 - q) To simplify notation somewhat, let =(p, t +PT) P i o MICHIGAN ] jk k x Q + jk,). Cqnk +k jk k qn x )k) ) q Oqnk x 0. (6.24) (6.25) that is, the field at subdivision B. after p increments of time after t. J o Then (6.24) becomes 2 2 0j,m Xj,m + 47r jk. x (1 - jk k x^ k, m + sjk nkx tk m - n. -1 jk + (njk + jk) (1 - jk= q =O - njk A c n. q xX k, m-njk-q jk + (njk + 'jk) qnk q=0 A c nk q (6.26) k, m - nk 1 -q jk K Further, denote by 1> the summation with the term j = k deleted, re 287 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN - calling that Rjk = 0 if j = k which with Eq. (6.15) implies that njk = 0 and yjk = 0 if j = k. Then Eq. (6.26) becomes j*^ 2 j,m 2,x j(m j, m K 1 IV I.k x (1 - 'jk)k x k, m-njk + k'k mnkL.+ k jn x;Y knk k k,m-njk- 1 + (njk + jk)( 1 - k) q=O0 A c nk q — A. k, m-n -q Jk + (njk + k)k Q q=0 x, m-nk- 1-q jkJ 4i r jj x(nj x j j,) m The awkward sum in Eq. (6.27) may he simplified by writing (6.27) (1 - Yjk)nk x km-n + (njk + k)(1 - yj k - - 1 m-njk q=0 q =O A c nk x qK Xk, m-nk-q + (njk + Yjk) 'jk q-= cqn x q K k, m-nk-1 -q q [(njk + jk)(1 - ^)c + (1- 'k 6] q i k kq nk X km + I(nk + yjk)7Yk c + jk q]1 = = j k jk q jk q g=0 k x k, m-n. -l-q ]k (continued) 288 UNCLASSIFIED

U N CLASSFE a THE UNIVERSITY OF MICHIGAN - 8525-1-F Q+n j q=n i X k, m-q Qni+nj +1i Qnjkl - -Y'jk ) Cq +(1Yj)6-n] [n jk + 'Yjk)'Yjk Cq-. -1 q =nk +1 0 A + -Y. 6q 1 nk jkQ -n +1 k x X k m-q jk BqknA (6. 28) where Bq = (n + jk jk )k)(1 - )k) Co + 1 - )k ' q = n jk = (nk +'Y 7 jk C Q q =n j+ 1+Q (6.29) = (n~k + [jk)(1 - Yjk) C 0 + jk 6q-n. n l nik < q < Q + nik + 1 and q = 0 q *0 = 1 q = 0. With this result Eq. (6.27) may be written as ________________________ 289 UNCLASIjiE

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 2 j~ x( x x A =/ ) K 2 j, m 47r j j j, m I 47 k=1 Q+n +l q = njk jk k x(nk x Xk, m-q (6.30) (U) Note that since j * k in the sum, njk f 0 which means q / 0 hence all terms in the sum are evaluated at a time less than t = t + m r. (U) In actuality, it is only the tangential components of ( which are unknown. An equation for these components is found by forming the vector product of Eq. (6.30) with n., the unit normal to the j zone on the surface, B, as follows - n. x + - r 2 j j,m 47r jj A A - n.n. x,. J j 3,m ^ <i, 1 ^ = n. x +- n. j,m 4r j K k=l Q+n +1 q=njk = k B x k x ) jk jk k k,m-q (6.31) where the identity n. x.. x J lJ i J,m i - i d 3, Imf I ji 'j r.,I, m~ J j33 j j, (6. 32) has been used. This may also be written 290 UNCLASSIFIED m

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F K Q+nk+l1 47r n. x +n. x Bk jkX(nk x m J k q= k m n.jx. m (6.33) m 7r + 2.. + n. JJ J 1 <j <K (U) This is the vector analogue of the scalar formulation of the pulse scattering problem as developed by Mitzner (1966). It expresses the tangential components of the total magnetic field in zone B. in the scattering surface at time t + mT in terms of the incident field in the same zone at o the same time and the total field in other zones at previous times. If the surface is zoned so that the incident pulse reaches zone B1 at m = 0 and has not yet reached any other zones. Then -A-, 47r nl x,1,0 n. x -- A j 1 x j,o 2ir +1 ' n =0 j + 1 (6.34) and the field at successive times may be computed using (6. 33). Since (6. 33) is a vector equation, it is perhaps more convenient for computational purposes to write it in scalar component form. In any particular case it is undoubtedly most efficient to employ a surface coordinate systan in which one coordinate varies along normals to the surface and the other two vary on the surface. Only two component equations will then result, though proper care must be exercised in distinguishing contravariant and covariant components should the coordinate system be non-orthogonal. In rectangular components, three scalar equations will be generated. Introducing the notation 291 ____ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F A X X 1 2 A 3 A n. x h. = h. + h x + h x h. (6 35) j j,m j,m 1 j,m 2, 3,m (6.35) 1 A. x + 32 x3 jk = jk 1 + jk 2 + jk 3 (6.36) these scalar equations are i Q+nk +1 ix + B n h x x -h,m jk- nj,m-q k j jk k, m-q X k= 1 qI 2ir + O.. n. p = 1, 2, 3 (U) To obtain the magnetic field off the surface, an expression comparable to the scalar equation derived in Mitzner may be derived in a manner following the method described above. The details of this analysis are not yet complete. 6. 3. 1 Addendum on the Evaluation of. (U) The numerical solution of Eq. (6. 33) is dependent upon the knowledge of the geometric quantities, Qjk defined in Eq. (6. 12) as Bnk jk This may be a serious problem since any numerical integration must cope with a highly singular integrand. A method of treating this is considered here. 292 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) The straight forward application of a vector identity yields 3k = - jk + Bk JBk V - da = -I Bk Rjk k ^ 1 ^ nk VBk R nk da k jk A xA 1 x ) k jk Since nk is assumed to be constant over Bk this is rewritten as A- -A 1A 0. = - % nda + nk x B k Bk 'k "k x ---- da nk xVBRk djk (U) The first integral is precisely that which occurred in the scalar problem. It is the solid angle subtended by zone k at r. and was evaluated J approximately by Soules and Mitzner. The second integral remains to be evaluated. However, with Stokes' theorem, this may be written A A 1. A n x nVB H da = nk x k k k Bk jk k Ck dr k R. Ok where Ck is the curve enclosing zone Bk. In this form, there is still a singularity in the integrand. However it is seen in Eq. (6. 33) that the quantity which occurs is not.Qjk but _J* nj x [Qjk x (nk x k,m - q] 293... UNCLASSIFIED -

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F Thus the possibly troublesome terms are A n. x J n( fk x ck drk R. jk x (nk x k,m q)] For convenience, call this expression I and let - k, m- q K =n k,m - q I = n x (k x Ck (Note that nk K = 0) since dk jk x K n. k) k =j - x" x n. k- ^k x ), A A.jnk)2 xn j) X[(nkX ck d r jk x K but x (n xj d k Rjk ) x K = O thus I - - kx( x () x j x Ck R — jk Since (nk x ck d rk - )xK jk = -K. Ck d rk Rjk A %k 294 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F - /I -. I- x k (ikxn.) = n x n. K 3 x nk K ck d r jk jk ck d r Rjk which may be rewritten as AA n xn. I = X,J In x n. I i K ' k nkxn. d rk Rjk or in dyadic form I = Ck n, xn. K J Rjk d rk In either case the integrand is non singular. Since nj x I = sin 0k where.k is the angle between n. and nk, it follows from the law of jk j sines (see Fig. 6-4) that n. x nk sin 0 k sin a R. R A jk jk (hence the integrand bounded) provided only that the surface is smooth. 295 UNCLASSIFIED m

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN \ I A I A\ jk V FIG. 6-4: ZONAL GEOMETRY. 6. 4 Causality and Reality in the Synthesis of Pulse Responses. (U) In the previous reports on this contract we have used the procedure of building up the response to a pulsed input by superposition of the CW responses. This method is valid in the sense that it yields (a) a causal response; i. e., zero response at a distance r from a point on the target until a time r /c after that point is first irradiated, and (b) a real response (zero imaginary part) to a real excitation. (U) Conditions (a) and (b) are satisfied automatically when the exact CW solutions are used. However, an exact CW solution is known only for the sphere. One may wish to approximate the complicated solution for the sphere and for all other shapes exact analytic solution are not available. One then uses approximate formulas such as those of geometric optics, physical optics, creeping waves and Rayleigh theory. A dilemma then appears since one can verify that both (a) and (b) cannot both be satisfied when the 296 _ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F creeping wave or Rayleigh CW formulas are used. In addition if an approximate CW response is constructed by patching together different solutions in different frequency ranges, there will of necessity be discontinuities in the function or its derivatives with respect to v. These will be translated as physically unreal contributions in the time response. To illustrate the latter point let represent the Fourier transformation operation. Then T f" (v) = (27r i ct)2 Of (v) Thus if f(v) has a discontinuity at v = v, f" (v) has the form f (v) + A6 (v ). 2 z a 5 6(va) = A exp (-27r iv t) and so a a -27r iv t A e a f(v) = 2Ae + tf2 (v). (27r i ct) (U) In Section 6. 5 we illustrate these remarks by the simple formulas for the flat-backed cone derived in Goodrich et al, (1967b). We first realized these difficulties when treating this case. However, Kennaugh, Moffat and Schafer (1967) have also observed an example of this phenomenon in the creeping wave expression for a sphere. They point out that if the sphere creeping wave expression is multiplied by - i = - IIj superposition then yields a causal response. They obtained the creeping wave expression from Levy and Keller (1960) who derived it be using Keller's generalized geometric optics approach and inserting a phase change of ei / along each geodesic (ray path) as it passed through the common intersection of the ray paths on the shadowed side of the sphere. Kennaugh et al, pointed out that a phase shift independant of frequency v, 0 < v < 0o, is physically unrealizable and hence questioned Keller's formula. However, we feel that this is not the root of the difficulty since, for the sphere, one can derive a result comparable to ________________________ 297 ___________ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F Keller's formula by purely mathematical high frequency asymptotic approximation applied to the exact (Mie) solution for the diffracted field. This is done in Senior (1965) where the creeping wave solution which is derived to 0 [a/X).4/3 coincides with Keller's result to 0 (a/X) 1/. Thus, an arbitrary multiplication of the creeping wave terms by a constant phase factor is not permissible. Another way to see that this is not possible is to note that the CW sphere (and also cone) formulas, which combine creeping wave and optics results taking into account their relative phases, agree very well with experimental results. We refer to Kleinman and Senior (1963) for the cone case. We have verified that the agreement is destroyed for the cone results if the relative phase is so changed as to make the flat-backed cone creeping wave term causal in the manner advocated by Kennaugh et al, (1967). (U) The problem lies deeper. Our CW results are for positive frequency (v > 0) only. To evaluate the transform F(m) = C f(v) e-2 iv dv v must be known over the entire real axis. One might make an analytic continuation of f(v) into the complex v plane. Then if T = t - r/c causalit requires F(T) = F(T) H(T) where H(T) is the unity step function i.e., the response at a distance r is zero when t < r/c. This requires that all poles of f(v) lie in the half plane Im v < 0 or, more precisely, below the point of integration. Then, for T < 0, when the integral along the real axis is replaced, using Cauchy's theorem, by an infinite arc in the upper half plane plus the sum of residues of the poles enclosed, the result is zero. For T > 0 one must use an infinite arc below the axis in order to have a convergent integral along the arc. The integral on the infinte arc is in fact zero so that F(T) is then - 27r i E residues. 298 UNCLASSIFIED

UNCLASSIFIED -THE UNIVERSITY OF MICHIGAN 8525-1-F (U) The continuation of f(v), v > 0 into the complex plane must, * however, satisfy f(-v) = f (v), v > 0 when F(T) is real. This is evident from the inverse transform formula o0 2\\ r ivt f(v) = F F(T) e2i d In Goodrich et al, 1967b we made use of this fact to write Do0 F(T) = 2 Re f(v) e27r iVT dv However, when the asymptotic creeping wave formulas derived for v positive, real are analytically continued to the negative real axis one finds that f(-v) - f *(v) so that one is led to an imaginary function F(T). (U) On the other hand if one requires f(-v) = f* (v), the function of complex v which accomplishes this is not an analytic function of v alone but depends on v and v. One can for example start with f(v) defined for v > 0 and replace v by Ivl, i by i sgn v. One is then led to a real F(T) which, however, is not causal. (U) We do not know how to get out of this dilemma; until it is resolved, it is necessary to use only exact results or purely numerical approximations to them. Geometrical and physical optics formulas, while not causing difficulties in satisfying (a) and (b) are too crude by themselves to be truly useful. This is evident from, say, Kennaugh and Moffat (1965). (U) An alternate approach to the problem would consist of using the moment method suggested in that paper. This is based on the fact that the existence of a convergent low frequency expansion of the CW result in powers of the frequency puts requirements on the time moments _ 299 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F OD t1 F(t) dt J0 (n an integer) of the time response F(t). These requirements can presumably be used to modify the high frequency physical optics CW response to take into account low frequency effects without an objectionable "patching" procedure. Kennaugh and Moffat did not pursue this approach in their paper. It has been investigated in this laboratory (not on this contract) without very favorable results). (U) As an illustration of the extent to which we satisfied causality in using a numerical approximation to the Mie sieries we reproduce here graphs of the field backscattered from a sphere when irradiated by an impulse and by a square pulse of length 4 carrier cycles. Both graphs are extensions to negative Tr of figures given in Goodrich et al (1967b), section 4.0. The small non-zero responses for T < 0 are due to truncation of the Mie series and using a numerical approximation to the contribution by the remainder in the series. 6. 5 Pulse Scattering from a Perfectly Conducting Flat Back Cone. (U) In this section we use the rather simple formulas derived in Goodrich et al, (1967b) to illustrate the points brought out in 6.4. The formulas T=A give the backscatter response to a 6 function excitation E = x E 6 (x + ct). 0 They are rewritten here in a normalized form. Cone length is included in the normalization of the independant parameter T = c/a (-z - t). 3 c ts( z - t) c c I T + I E/ = 4 6 (r) + 1 - I +1I cEz 4w 1 1 2 3 1 _______________ ~300 ___________ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F P0 a) rZ4 a) v C) 14 C,, a) a 0s W a) 4) I 7 8 T -1 1 2 3 4 Fig. 6-5: IMPULSE RETURN. 301 UNCLASSIFIED

14 T li E UNIVERSITY, 8525-1-F 0 F to (::) CV U-5 M'CIUIGAN cv -4 I 11 lahi If 0 e ff E-4 w m D u 44 co,11 Ul) m a 0 0 a (=) I I LO a CV)" 0 9 0 a I I V uwj. d d I I LO 0 V CV C; C; I I I 0 > zi co 0, 0 14 'a -04 Aq 0) " M P4 4?:1;q 4) Q) e k 1:34 H I co I to -0 k;-.4 A4 PlalLt' JaMOS OATIgl9j, 302 14,

U N CLAStFE THE UNIVERSITY OF 8525-1-F MICHIGANI 1= 2 I1 = 2 1=1 I2 = rj 1 (1 + sgn Tr2) sin 27ruCT (2 F2j) + sgn r2 s (2 u~ H '31 4 1=r~ I rcos (27wu r, + (4ir 'u 2 2 3 c 3 c 3 - 2) sin (2ir where = W= C08 ( 7 t2) dt sin ( ~ t2) dt I 27r cosec 7 sn7r V H I Cl1 2 7r = til 0=0 F 2 1 H, ir 37r1 Os8- -008S (Coi C 4-(-i7r cotG0 + 4 e 3 3 o cot 9 /4 0 It ) 1 3 303 UNCLASIFIE

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 'T2 T + 2 C Z T c - ( - t). 3 a c (U) In these formulas C1/47r 6 (T1) + I1, I2 and I3 are simply the normalized versions of the integrals I, I2 and I3 respectively in Goodrich et al, (1967b). The derivation assumed f(-v) = f * (v) and really yielded the + sign in the + factor in I and I2; the (-) sign is inserted arbitrarily in the discussion to follow. The physical source of the terms is as follows. By using a formula valid for frequencies v > v, v = 0(1) for the response integral (which is evaluated over all frequencies v > 0) one gets (C1/47r)6(tl) as the direct return from the cone base and I as the return from radiation which travels across the base and then back as sketched. I2 is the result of integrating the same formulas only over 0 < v < v. This was subtracted from I1 and replaced by adding the presumably more correct integral 13 which is obtained by integration over 0 < v < v of the scattering formula valid in that range, the Rayleigh scattering formula. z a 304 ____ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) Thus - I2 + I3 is intended to be a correction of the approximate result I + (C /4r) 6 (T). For a fixed observation distance z one would expect I1 = 0 for all time for which 2 = 1/a (z + 2a - ct) > 0. Yet this 1 is not true for the formula as derived, in fact it is zero for T2 < 0. If the arbitrary minus sign were used this violation of causality would be rectified, but one cannot of course simply do this arbitrarily. Examination of the CW equation from which I comes, via. EC i(kz- wt) 1 / 2 e iB i /] E x^ -- - [C e + C e e /4 27r z L with C and C2 real, shows that this change amounts to inserting a factor 1 2 i in the second term (which accounts for waves diffracting about the cone base the "creeping waves"t of this problem). While this term is an approximate result (due to Keller) its phase relative to the term representing the CW return directly from the edge, (the first term) is crucial in the CW response. This is the formula mentioned in 6.4 for which Kleinman and Senior (1963) have verified the accuracy by extensive comparison of numerical and experimental fixed frequency results. We have checked that insertion of a factor i in the second term gives results which do not agree well with experiment. As an alternate to having a non-causal creeping wave result, we could make the result causal (use - ), but imaginary, by the use of true analytic continuation into the complex v plane as discussed in 6.3. (U) The difficulties with the Rayleigh region, which result here appears as symmetric in r and hence non-causal, are best avoided by realizing that in practice one uses frequency bands about high frequency carriers which + Kennaugh and Moffat (1965) apparently treated the ramp response wave form from a flat-back cone using the same Keller formula that we have used. They do not mention the causality difficulty there, but discuss other physically unreal aspects of the results. ________________________ 305 ______-_____ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F exclude the low frequencies. This does not of course solve the basic theoretical problem. It does indicate that it is folly to try to use the impulse or ramp responses and the convolution method to solve problems of high frequency pulsed excitations. One merely brings in the difficulty which could be avoided, of a proper handling of the low frequencies. 6. 6 Pulse Scattering from a Perfectly Conducting Cone-Sphere. (U) This section is a revision and extension of the analysis in Goodrich et al, (1967b) Section 4. 5 of backscattering from a perfectly conducting conesphere irradiated nose-on by a pulsed CW wave. The present analysis differs from the former one in using more accurate formulas for the creeping wave contribution and for the creeping wave enhancement factor y. These are formulas derived recently by Senior (1967a,b). However, the result of this creeping wave part is not valid (it is not causal) as discussed in Section IV. On the other hand the terms in the following which represent the tip and join returns are indeed causal and real. They are manipulated into a form, Eq. (6. 52c) where this is clearly evident. This and the fact that we have written the formulas in a manner more closely resembling the earlier analyses of spheres are the main reasons for this section being included in spite of the difficulty with the creeping waves; to be closer to the sphere analyses. In addition to this, the presentation follows more closely that of the earlier analysis of spheres, the integrals have been rewritten in terms of a variable involving the base radius a, namely, y = wa/c instead of w. Also the variables = cT/2a and n= Tw /27r (6.38) are used instead of the pulse length T and center frequency are used instead of the pulse length T and center frequency w. 0 ______ 306 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) We are considering backscattering from a cone-sphere irradiated nose-on by a pulse f(r, t) = x cos w t tl < T/2 0 = 0 It > T/2 (6.39) which in the normalized frequency domain corresponds to F ) A T sin ( y ) + ni) sin (<y - nir) (6.40) X 47r y + n'r y - n7r (U) To compute the scattered field we will assume a linear time invariant system so that in the frequency domain, the response to (6.40) will be F(r, y) = x Fi(y) E (r, y) (6.41) where E (r, y) is the impulse response of the cone-sphere and is given by Senior (1965) as i-y s Ar a e a E (r, y) = x - S (6.42) r y and S is the scattering amplitude. (U) Thus, from (6.39b) and (6.42) we get that r s aT sin (y + r) + sin e - n6 e F (r,y) = x 4 + Ir S (6.43) 47 r ly + nt ly - mrw y Using the inverse Fourier transform, we can write -oo -i tc s( t a y F (r, t) = a Fs(r, y) e dy (6. 44) J -a) U307 UNCLASSIFIED

UNCL~ASSFE — THE UNIVERSITY 8525-1 - OF MICHIGAN Since f (t) is real., it follows that FS (r y) = (Y and 00 f (r, t) = Re 7ra 10 tc -s a F (r, y) e1 dy (6-..45) The scattering amplitude S is given by (Senior, 1965) i 2 i 2 - sec a exp - iy sna — tan a exp -i 4 4 csca$ + -y S (6. 46) where -yS is the creeping wave contribution. -s 1Fsin(Iy +nir) sirn f (r,t = R e tv+mr +I Normalizing f5 (r, tO lhy e -~iTYy S(Y fd (6. 47a) i 40 ~{sin (ey + nir) + ey + v sin (Qy - nir) R y- nR e {e -T SY y (6.4Thb) where frt) = 47r 2r cT f Ur, t) (6. 48a) and T - ct - r a (6. 48b) (U) By substituting (6.46) in (6.4Th) we get: 2 T(r,t) =Ax 4e n4 JOD 0 {sin (iy + nir) ey + r + Continued 308 UNCLASI)iE

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN sin (y - nr) + si(y - n) sin (2y sin a) cos T y + - tn n + cos (2y sin a) s + sin (gy - nr) ly - n; dy _ ^ in Ty - x Y 2 tan a 4 Ii sin (y + nr) + Iy + n1r Isin (2y csc a) cos T y + + cos (2y csc a) sin T y + x Re Y D sin (iy +n r) Iy + n7r sin (y - nr) ly - nr c y dy (6.49a) 2 & sec a =- 4 GO 0 sin (ly + nmr) ty + nr + sin (ly - nr) [(2 sin a + + ly - - r sin (2 sin a + + y - nTr L + T)y1ay Jy - x sin (Uy + nr) ay + nr + + sin (iy - n7r ) ly - ant csc a + Tr dy y 00x + xRe 0 sin (iy + n7r) Iy+ r sin (iy - nr) ()e-iTy + ty - nir 'Y ScYe (6.49b) Y S is given approximately be Senior (1967a) as C 4 i7r/3 S (y) = T e C 'I i7r/3 e 1 + 60 22 60 7.c 1 (323+ a) -r1 2a ei7r/6 0r _73 31 - a)} e 60T-c_,1 exp ir y -ir/6 -e r6TI - (6.50) 309 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F with T = (y/2)1/3, = 1. 01879297... and Ai(-j31) = 0.53565666 C Equation (6. 50) is a highly accurate approximation to the creeping wave return. (U) An asymptotic approximation for 7y (Hong and Weston, 1965) valid for large y, is given by F1 2/3221 e-ir/31 ~ 2(1/3 + Ai(-x) dx) 1 + 1 (y/2)2/3 2 e /3 + (y/2)2/3 2 /3 exp (y/2) (6. 51) (U) Equations (6.49b), (6.50) and (6.51) may be used to get numerical results, but instead an empirical relation for y (Goodrich, et al, 1967b) will be used. The integrations will then be carried out using an analytical curve which is fitted to the empirical one. Computations for n = 4, x = 1/4 (U) Computations were carried out asing n = 4, I = 1/4 in Eq. (6.49b) 2 {co f-(r,t)= xseca sin(y/4), 1 + 0 4 +4n s 1 1 + ~] d- ^ tan a co + sin -2 sin + T)y - x ' sin (y/4) Y- 4n/ y ( -— + -n) sin (2 csc a + T)y] dy +Re + 4n (6.52a) + x Re sin (y/4) + 7 S e (6. 52a) - ~+4n I-4n 310 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F fS(r,t) = 2 secsin (ya4) n 0 y2 - (167r)2 sin [(2 sin a + r)y] dy - x ta 0 O sin (y/4) 2 16)2 y -(167r) sin [(2 csc a + r) y] dy Resin(y/4) e-i d +x8Re 2 2 YSc e ct y - (16) ) 00 ( = x - sec a sin 67r (2sina +T) sin sin tan a sin + sin| - +)dy - x - tan a sin 6x ~(4 T Y y T (6.52b) ( 1 T )y (2 csc a + 7)T sin[( - T2)Y + sin [ + T2)Y + x 8Re J 0 sin (y/4) -i Ty 2 2 S e dy y -(167r) (6.52c) with T1 = 2 sin a + T, and T2 = 2 csc a + T. The first and second integrals clearly exhibit the causal property of the response from the tip and the join respectively. This is evident on using o 1 7r/2, m > sin my d: | - - dy = y J o - /2, m < 0 311 UNCLASSIFIED

U NCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F The third integral does not give a causal response as we have discussed in Section 6.4 and as numerical checks verify. Once we learn how to handle the creeping wave properly we can use the correct creeping wave expression together with the first two terms in (6.52c). 6. 7 Bandwidth Effects on Pulse Return 6.7.1 General (U) The purpose of this memo is to show how the effect of frequency pass band and the dispersive characteristics of radar echos particularly the creeping waves, combine to affect the shape of pulsed radar returns. It is shown that considerable care must be used in trying to infer information on the scatterer from comparison of incident and creeping wave pulse shapes. However, when sufficient bandwidth is allowed, the creeping wave pulse shape can be an indicator of the target structure which will not be easily duplicated by returns from other targets, regardless of whether the other structures yield the same time delay between geometric optics and creeping wave echoes. (U) Assume incident fields of the form - A E = f (t - z/c) x (6.53) Then the resulting backscattered field will be s(z/c -t) c- R i2B v(z/c t) S(v) F(v) di of 0 where Fo F(v) = f() e-i2fr VTd (6. 54) -OD I*_____________________ 312 __________________ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F and S is defined in terms of the CW backscattered field E A Es _ S (6.55) 27rz (U) If now the "desired" incident pulse shape (6. 53) is modified by a rectangular bandpass or if the return is so bandlimited in the receiver then (1+0) A ^ "wS. XC -i27rv (z/c - t -/l rs(z - t) = c Re| e S- (v) F(v) U(v) dv (6.56) c 7rZ v (l-0) where 2v Q is the bandwidth and U(v) = > Three ranges of 0 can o1, v < 0' be distinguished. (U) (i) 0 very small (approach to CW). In this case 0 is so small that F(v) S(v) -constant in the integration range. In this case the output is, s(t = x2.- S (v ) F(v ) cos 2Tr v T sin 27r v 0 T (6. 57) 7TZ 0 0 0 0 where t = z/c - t. This is a modulated cosine wave of carrier frequency v and modulating frequency Ov. Thus the pulse aspects of the response are completely lost. In the limit as 0 -4 0, the modulation frequency goes to zero. The field given in this way also goes to zero in the limit because the power per unit frequency interval is bounded; in the limit as 6v = 20v -> 0 there is zero flux. The idealized non-zero single frequency CW case requires a 6 function term 6 (v - v ) in the spectrum. (U) (ii) Intermediate 0: This case will be concerned with situation which may arise, in which S(v) = S(v ) e in the interval 20 v whereas F(v) has no such representation (except for the trivial case obtained by multiplying and dividing by common factors). 313 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F v (1+ ) xc - - irlv i27r-vt s(t) = - S(v )Re e e F(v) U(v) dv v (1-0 ) 0 0 A = S(v ) F(t- r7) (6.58) 27rz o0 The response is simply a demagnified time delayed replica of the original pulse. (U) This is the situation considered by Hong (1967), for convex bodies. He writes that S(v) = Sg(v) + S (v) where S (v) is the creeping wave response and Sg(v) the geometric optics part. Then he states that S (v) can be written in the form S (v ) ellV over the radar bandwidths considered. o Our computations however show that this assumption does not apply to the bandwidths used in present day short pulse radars. (U) (iii) Neither of the Above Two Cases for 0. In this case the dispersive character of the creeping wave due to varying creeping speed of each frequency component will be able to show up without excessive masking by the band pass characteristics of the receiver. Creeping wave pulse shape is then an indicator of the target characteristics. 6. 7.2 Examples: Sphere Computations (U) The analysis of Goodrich et al, (1967c) can be used to illustrate the above remarks. The target was a perfectly conducting sphere of radius c. The' functional form of f (t - z/c) was taken to be fi(A) = e (A) cos 27rv A| < T/2 =0 A > T/2 (6.59) 314 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN Then Eq. (6.54) becomes K (1+0) K (1-0) S) X R 0 (K;l, n) G*(K) U(K ) e-iK dK (6.60) where K = 27r v a/c, T= (ct - z)/a + 2 (K; 1,n) = sine (n + K /r) - sinc(n - K 1/ ) n = v T, I= cT/(2a) O, i2K G (K) = i - K OD j (K 1)i hl~) i K n~n (6.61) Here sine x = sin7rx/(7rx) is the so called filtering function while j (x) and h ((x) are the spherical Bessel and Hankel functions respectively. n (U) If Ov exceeds the frequency range from v to the first zero of sine (n - ci /r) the bandwidth is then determined by the zero. The zero occurs n - Ke = 1 i.e., T (v - v ) = 1 giving a fractional bandwidth (g = 1/n (6.62) In Goodrich et al, (1967c) the variable 0 was not introduced explicitly; (alternatively one could say that e was infinite). The computational results given there, Figs. (4-7) and (4-8) for I = 1/4, n = 4 thus correspond to a 25 percent bandwidth. This is a bandwidth which is considerably exceeded by 315 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F actual short-pulse radars''. One observes the considerable distortion of the pulse shape in the creeping wave. This is an example of case (iii) above. Such a pulse shape could be considered as an identifying indicator of the target since the natural pulse distortion is pretty well reproduced and would not be at all the same for other targets. (U) The opposite extreme (case i) is given by setting 0 = 0.5 in what is otherwise the same example. Numerical evaluation of Eq. (6.58) yields Fig. 6-7 which illustrates clearly the predicted modulated cosine response. Clearly the pulse distortion effects are submerged. See, for example, Milburn, J. (1967) "Short Pulse Model Measurement Studies,, Aerosystems Laboratory General Dynamics, RADC-TR-66-785. Vol. 1 which cites a 50 percent bandwidth as typical. __ 316 UNCLASSIFIED

-i C z 5; Cl) l)> n!> co 'TI m i-il *s <a * -l I(2 <D 0m z P0Z4 cin IND PTI C) z z C) C> rn 4.5 5.0 5.5 6.0 6.3 T FIG. 6-7: Time BAND LIMITED CREEPING WAVE RETURN CREEPING WAVE. n = 4, S = 1/4, 10 percent bandwidth.

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F VII HANDBOOK OF RADAR CROSS SECTION FORMULAS 7. 1 Introduction (U) For convenience, the radar cross section formulas for the various basic shapes studies under SURF have been listed in "handbook" form in the final reports at the end of each year's work. (See Goodrich etal, 1965 and 1967) This report is the third in this series. In this Section, are given the formulas for: a) Cone-sphere with indented base (revised), b) flat backed cone (radius of curvature at the join is a parameter), c) coated cone-sphere with indented base, d) coated cone-sphere with annular slots (representing slot antennas) near the tip of the cone and near the cone-sphere join, and e) coated cone-sphere with longitudinal slots. In Goodrich et al, (1967 ), formulas have been given for two other coated shapes. These were the coated cone-oblate spheroid and the coated cone-prolate spheriod. Formulas were also given for the cone-sphere with an indented base (such as is to be found in the Mark-12 re-entry vehicle). A revised formula for this latter shape is given in this Section reflecting improvements in accuracy obtained from further analysis in 1967. 7.2 Indented Cone Sphere (revised) a^ a-b -\ 9 — 1 / / FIG. 7-1: CONE-SPHERE WITH CONCAVE INDENTATION IN REAR CAP. 318 UNCLASSIFIED

UNCLASSIFIED I THE 2 = 1 /ir x UNIVERSITY OF 8525-1-F MICHIGAN i2 SI + S2 + S+ J3 (2 ka sinO1) I OK9K (7. 1) x2 1i 1 /7 2 spec <OK< 0 - - 2 (7.2) where s S _ 1 4 tana exp[ -2i ka cosec a cos O] i2 2 3/2 (1-sBin2 Osec a) (7. 3) s sec2 a aB (kb) J (2 ka cos a sin 9)e 2ikbsinacos 9 2 4 b o b>b -1 = *ka 2i 2 -ij-cosec-J (2 ka cos a sine6)e 2ikb sin a cos I (7.4) b K b n = (3/2) + (airr) (7. 5) kb1 3n sin 27r n 2 16 cos ar (7. 6) B (kb) = 1 -El 64 6n sin 2ir n 2 kb cos a )o.6 (7. 7) S3 = y (ka) B(kb) S 3 cw (7. 8) 319 UNC~LASSFE

UNCLASIFIE THE UNIVERSITY OF MICHIGAN 8525-1-F 'yka 3 + Ai (-x) dx)( +(ka/2)2/ a~ 0 i r/ 10 2/3 2 -iir/3Ai ), Lka)1/3 c43e1/6] (79 + (ka/2) a e Ai(p)ep ) e,79 ka > 3 -y(ka) =1, ka< 3 S =a'(kb/ 2) 5/3 cw 2bI e2ik(a -b) + i 5,r / 12 C 'I eir/3(320+ 9) 1 + 672 2 6 exp licrkb% - e -i7r/6 r~rf1 -ei- /6 T 6ff01 /60T (p3 7. = (kb/2) 1/3 (7. 10) (7. 11) (7. 12) (7. 13) =, 1. 018793.. Ai (-p31) = 0. 535656... (Ai is the Airy function) Ssec Iiffr/4 kaos tn+6 2ika cot ae cos (a + e) S - e ta (arsinO { 1- F j/kaotcos (a +6 } & K < r/2 - a e = ir/2 - a = ka cosec ayka seca' (7.14) 320 -UNCLASIFIE

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F I 2Ix F(x) = e 0 it e dt (7.15) The angle j3 at which transition smallest value of 0 where (7.1) first zero of J (2 ka sin 0) i.e. 7. 3 Flat acked Cone 7. 3 Flat Backed Cone between (7.1) and (7.2) is made is the and (7.2) intersect which is larger than the sin-1 2.405 2 ka I FIG. 7-2: BASIC FB SHAPE. = 1/7r S + 1 2 + S3 J(2 2S 3 0 2 ka sin 0), 0 < 0 < 3 (7.16) (7.17) ( = 2 1/7 -% AL spe spec, A3 < 0 < r/2 - a where S 1 S2 S3 S spec is is is given given given by by by (7.3) (7.4) (7.8) with b now the radius in Fig. 7-2 with b as above is given by (7. 14). 321 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F The angle 3 at which transition between (7. 16) and (7. 16) is made is the smallest value of 0 where (7. 16) and (7. 17) intersect which is larger than the > 1 2.405 first zero of J (2 ka sin 0) i.e. 2 > sin - k o 2 ka 7.4 Coated Cone Sphere with Indented Base. x2 S1 R(0)S2 + S3 2 J (2ka sin ), 0< 0 < 3 O (7.18) X2 2= 1/T R(0) 2 = 1/7r R(0) S x 2 spec B < 0 < 7r/2 - a (7.19) where S1 is given by (7. 3) S2 is given by (7.4) a (kb/21/3 ik(a-b) + i /12 (S(e)(b) + S(m)(b) - 3 2b 2 b2 k(a-b) 31 i ( 13) S()(x) =(kx/2)4/3 i7rkx-ier/6 1 S2 Ai( )2{3 i7r/3e2j s i(-B s-e qe 2 M) 4/3 i7rkx-i~r/6 6 m S (x) - (kx/2)4/3 e i. J /3-2 s Ai'(-a) {ei /3a -qm q - (kx/2)/3 r e q = - (kx/2)1/3 1/n 322 UNCLASSIFIED (7.20) (7.21) (7.22) (7.23) (7.24)

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F aC are roots of Ai' (-a ) - qmAi(-a )= 0 f3 are roots of Ai' (-/s) - q Ai (-3 ) = 0 S s e S R(0) =1 - r sin (0 + a) 1 + rl sin (0 + a) (7.25) S given by (7. 14) spec For lossless coatings, S3 is given by (7. 10) For lossy coatings, S3 = 0 The angle j at which transition between (7.18) and (7.19) is made is the smallest value of 0 where (7. 18) and (7. 19) intersect which is larger than the first zero of J (2 ka sin 0) i. e. 3 > sin 2 ka o ' 2 ka r = - i /7; ' tan ( 1/ k6) 6 = coating thickness, e, = coating permittivity and permeability lim er 7.5 Coated Cone Sphere SP with lotted Coating 7. 5 Coated Cone Sphere, LSP with Slotted Coating. Coating Nose antenna FIG. 7-3: COATED CONE-SPHERE WITH NOSE-TIP ANTENNA. 323 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 2 = IS + R(0) S + S3 J (2 ka sin 0) + S, 0 < 0< 2 ir 1 1 S- -3 x 2 a- = I/f R(0) S < 0 <7T/2 - a / () Sspec - - k where S1 is given by (7. 3) R(0) is given by (7.25) i 2 -2ika sin a cos 0 S= sec a J (2 ka cos a sin 0) e 2 4 o S3 S(e)a) + s(m)(a) (e) S )(a) given by (7.21) S (a) given by (7.22) 2ika cosec a cos a cos 0 47r ka e. S r-ln -r I _ (7.26) (7.27) (7.28) (7.29) (7.30) - % sp. Y,+Y tu) Jd \t Ka sin ut o s r YP Y r a s is slot load admittance is slot radiation admittance is spacer radius at its midpoint 324 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F iv ir/2 (2v +l) e n s, = C E r a 1 l (ka cosec a) - i nP (cos n m 1/~~,~.... im 7r /2 (2pm + 1) e m /, (ka cosec a) V' (ka cosec a) m Im I III. _ III (7.31) sin2 a a P (co a) a a sin aa p (oos a) /~r /Sm / (ka cosec a) m v are the roots of P (cos a) = 0 n v m are the roots of - P (cos a) = O m 3ac u %v(x) = 2l x v + 1/2 ( x- H(1) d; (x) = H (x), (x), x (x ) Alternatively Alternatively S 2- = ka rei J (2ka sin )e sp 9 s o s -2ik(a-a cosa)coseca cosO) s (7.32) re 0 is the complex excitation strength which may be evaluated from surface current measurements as discussed in Section 3.2.4. S is given by (7. 14) spec For lossless coatings, S3 is given by 325 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F iir/3 S = (ka/2)4/3 e~'3{1+ e 2 (32 3 + 9) (2/ka)2/3 3 60 (ka/2) /3 1 1 Ai(-_31)T ' expi ir ka - ei7/6 /31(ka/2) 1/3 P^At(^)2 - e i /60 (31 - 9) (2/ka) 1/3 (7. 33) 81 and Ai(-f31) given in (7.12) and (7.13) For lossy coatings, S3 = 0. The angle: at which transition between (7.26) and (7.27) is made is the smallest value of 0 where (7.26) and (7.27) intersect which is larger than the first zero of J (2 ka sin 0) i.e. B > sin 2405 o '0 2 ka 7. 6 Coated Cone Sphere, LSH with Slotted Coating. Annular Antenna Coating -7\ FIG. 7-4: COATED CONE-SPHERE WITH ANNULAR ANTENNA NEAR JOIN. 326 _ UNCLASSIFIED -

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN c2 2 1 = S + R(0) S +S3J (2 kasin0) + S 7r 1 2 3 0 sp 2 I, o0< < 3 (7. 34) (7.35) = 1/7 R (0) s spec 2, 3 < i r/2 -a where S1 is given by (7. 3) R(0) is given by (7.25) S2 is given by (7.28) S3 is given by (7.29) 3 S sp 2ika(cota +r /2 - 0o) ' e g rka/2) 13 (7 /2 - OD. '. T (ka)3 (Y, + Y ) sin 0 r o 2 J (ka sin sin 2 0) I o o (7.36) (7.37) J1 g({) = 1/7 1 -oD i t e e -dt ' (t) w(t) = - [Bi(t) + iAi (t)] (7. 38) 0 is the angular distance from slot to join Y is slot load admittance Y is slot radiation admittance r S is given by (7. 14) spec 327 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F For lossless coatings, S3 is given in (7.33) 3 For lossy coatings, S3 = 0 The angle 1 at which transition between (7. 34) and (7. 35) is made is the smallest value of 0 where (7. 34) and (7.35) intersect which is larger than the first zero of J (2 ka sin 0) i. e. > sin-1 2.405 0 2 ka 7. 7 Coated Cone Sphere with Longitudinal Slots (U) The effect of the slots is negligible and the cone-sphere without slots apply, i. e. formulae for the coated -2 = 1/ S + R(0) S + X2 1 2 2 - = 1/7r R(0) Spec S3 J (2 ka sin 0) 3o 2 < 0 < (7.39) (7. 40) < e T/2 -a where S1 is given by (7.3) R (e) is given by (7.25) S2 is given by (7.28) S is given by (7.29) S is given by (7. 14) spec For lossless coatings S3 is given by (7.33) For lossy coatings S3 = 0 The angle 3 at which transition between (7.39) and (7.40) is made is the smallest value of 0 where (7.39) and (7.40) intersect which is larger than the first zero of J (2 ka sin 0) i. e. 1 > sin 2.405 o 2 ka 328 UNCLASSIFIED

UN CLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F VIII COMPUTER PROGRAM FOR A ROTATIONALLY SYMMETRIC METALLIC BODY 8. 1 Introduction (U) Under other studies sponsored by Department of Defense agencies, attempts have been made to develop a computer program to calculate the radar cross section of rotationally symmetric metallic bodies. These attempts have met with some sucess but have serious drawbacks so far as application to the type of re-entry bodies investigated under SURF. They are characterized by the fact that either (a) the computer program is unavailable (b) the computer program is not sufficiently accurate when applied to pointed re-entry shapes and/or (c) the computer program is limited as regards electrical size of bodies to which it can be applied. (U) It was a goal of the SURF study, under Task 3. 1.3, to develop a computer program to calculate the surface currents and radar cross section of cone-sphere-like metallic shapes for all angles of incidence and extend the pro gramming to handle coated shapes as well. This goal has not been realized. A computer program has been developed for the metallic body but the analytical problems which arose during the development have been difficult and obtuse and the program is not yet at a stage for practical application. Although worthwhile advances which should lead to the effective solution of this problem have been made in the state-of-art and a method of matrix inversion proposed by earlier investigators (Schweitzer, 1965) has, with some modification, proven to be practical, further programming work would be required to bring it into operation. (U) Section 8. 2 summarizes the numerical analysis of the problem. Section 8. 3 describes the program at the stage of development which was reached at the conclusion of the SURF investigation. For a given metallic shape appropriately parameterized, the Fortran IV computer program calcu_329 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F lates the surface current for an arbitrary number, N, of uniformly spaced sampling points. The output of the calculation is in the form of 2N complex numbers representing the values of the orthogonal surface current components at those sampling points. There is no inherent limitation on the number of sampling points that can be handled. The choice of N as ten sampling points for this version of the program was dictated by the desire to test the system with sufficient accuracy without incurring undue cost. The computer program described in Section 8. 3 is capable of handling bodies larger than those treated in other available computer programs which attempt to solve a similar problem. Because of the column matrix inversion system which has been adopted, the number of sampling points and the size of the body which can be treated is limited only by the amount of computer storage available to the user and the necessity to obtain a computation of given accuracy at the least cost. (U) The program described in Section 8. 3 was written to test the programming development. It was devised to compute the surface currents on the smallest re-entry body to which the results of the SURF investigation might be applied. The example chosen is for a body for which ka is approximately 1. 5. For bodies with cone half-angles of interest in the SURF study, L (see Fig. 8-1) would be approximately 6a. By the simple criteria stated below, L- < x/6 = 3 - 1/K N 1 - 3K 0 0 N- 1 > K L 0- o a reasonable choice for the number of sampling points, N, is 10. (U) In this test program, 0 equals zero, i.e., the nose-on direction of incidence has been chosen and only one azimuthal mode is computed. The program is capable of extension to oblique incidence. __ 330 UNCLASSIFIED

U N CLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F L FIG. 8-1: REPRESENTATION FOR SAMPLING. 8.2 Description of the Numerical Analysis 8. 2. 1 Introduction (U) This Section describes the solution of numerical analysis problems not solved in Schweitzer (1965) which outlined a method of solution of the Maue integral equation for the surface current on the surface of a metallic rotationally symmetric shape. In Schweitzer (1965) and Castellanos (1966) will be found the fundamental description of the earlier analysis which led to development of the program given in Section 8. 3. This Section summarizes the continuation of the numerical analyses of problems which arose in devising a practical program. (U) The input parameter is f(z). The function f = t(z) is then defined by (4.10) and (4.11) of Schweitzer (1965). The Tij. 's are then computed by use of formulas on page 26 of the Schweitzer report and quadrature scheme described in the present report. (U) Next, the elements di are computed by use of their definition on page 42 of the Schweitzer report, and methods of Section 8,2.4 of the present report, and the b m are expressed in terms of the dm by use of (11.15) report, and the b. Is are expressed in terms of the d.i by use of (ii1. 15) and (11.21) of the Schweitzer report. 331 - UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) The T matrix is then inverted by use of the scheme described in m Section 34 of Schweitzer report, and the system solved for the c. as often as 1 is desirable. 8.2.2 The Evaluation of G (U) A. Definition of G: We begin with the definition of G given in Eq. (3.177) of Goodrich et al, 1967c *7r/2 G = 2 (-i)m m 0 -r 2 2 cos2m d. rR 1 - k sin 0 (8.1) Letting 0 = r/2 - 0 (8.2) we get 7T/2 G = 2(-i)m m 2 2\ ifR /1 - k cos 0 e cos 2m d0. 7R l -k cos 0 (8.3) Now let x = cos 0 in Eq. (8.3), to get (8.4) G = 2(i)m m Jo i7R 1 -k2 x e ', 2 ^ T (x) dx PR - x -kx (8.5) where Tm (x) is the Chebyshev polynomial of degree 2m. 332 UNCLASSIFIED m

UNCLASSIFIED I where -THE UNIVERSITY OF MICHIGAN 8525-1-F 1ie (U) Using e = cos 0 + i sin 0 we write G in the form G = 2 (i)m Rem + i m m L 22 T (x) 0 cos rRR 1 - k2x 2m( Rem dx 0R1 > si1 2'dx -R l -kx 1 -x 22sin T (x) m sin R 1-kx 2m dX 0 KR l-k x (U) B. The Evaluation of T2: With x = cos, T (x) = cos m 0 m Lve 2 2x -1 = cos 2 0 2(2x2 - 1) (2x2 - 2) = cos 4e cos (m + 2) 0 = 2 cos m 0 cos 2 0 - cos (m - 2) 0 (8.6) (8.7) (8.8) (8.9) (8. 10) (8.11) we ha or Thus, with 333 -- UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F T = 1 0 o2 T = 2x - 1 (8.12) 2 2r + 2 = 2 T 2r 2r- 2 and (8.12) may be used to evaluate T + 2 for all r. 2r + 2 (U) C. The Evaluation of em: We evaluate Sm in (8.8) by the use of Chebyshev quadrature: t;t 2N F(t.) + N; t. cos (8.13) Note that the integrand in (8.8) is entire. Applying a result of Stenger (1966) we obtain an error bound rR 1ke 2N - m l < 8 *22m eR e (8.14) rL 2(2N - rm) upon evaluating (8.8) by use of (8. 13). (U) D. The Evaluation of Rem for all Cells not Bordering the Diagonal: We also use Chebyshev quadrature of (8. 7) for all cells not touching the diagonal. Here the nearest singularity is a pole at the point x = 1/k. Upon again using (8.13) we get an error bounded by 16 e N 1/2 2m k 8N 8 e f C B t D (2/k)2m (8.15) (U) E. The Evaluation of G for Cells Bordering the Diagonal: For mIL the cells touching the diagonal it becomes important to appropriately treat the 334 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY 0 8525-1-F >F MICHIGAN nature of the singularity of G. We thus write (8.7) in the form m Rem = J o F (x) m dx (1 - ) (1 - k x2) F (x) (1 )( dx) J1 + x) (1 + kx) (1 - x)(1 - kx) IJ H (x) dx m (1 - x) (1 - kx) (8.16) where H (x) m = F (x) / (1 + x)(1 + k x) cos [R l- k2x T2m(x) rR (i + x)(1 + kx) (U) Substituting /1 t1 dt = a I (l - t) (1 - kt)' x y (8. 17) dt (8.18) where as in Goodrich and Stenger (1967) 1 dt a = -- J 1 - t) (1 - kt) 335 UNCLASSIFIED (8.19) I

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F in (8. 16), or x = (1 - k - y) (8.20) a 2 -1sinh k 2 ( 1+ Then 1 Rem = a H [x(y)] dy (8.21) 10 where x = x(y) is given by (8.16). This integral is now evaluated by Legendre-Gauss quadrature in the form Rem = a Wj H Ex(y) + EN(H (8.22) 256 N e+2m 2 -4N EN(H) |< ( ( 1 + ) (1 + k)(2-n) -1+2 2 sinh-1 -12k/(1 -k) =2 1 (2 ak (8.23) and where the yj and W. are the positive zeros and weights of 2N-point Gaussian quadrature. (U) In all of the above approximate sums the error is approximately equal to the difference between two evaluations using N and N + 1 points. 336 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 8.2.3 The Evaluation of T1. --— ij (U) We split these into two parts, those not bordering the diagonal and those that do. (U) A. Cells not Bordering the Diagonal: The Tij 's for cells not bordering the diagonal are done by repeated Legendre-Gauss quadrature. The integrals are all of the form a+h b+h a b Fl(s) F2(t) G (s, t) ds dt 1 2 m (8.24) They are more explicitly described in Schweitzer page 73ff; there is no need to give their explicit form here. (U) Putting 1 h 1 h s=b+-h+x, t = a +h + y 2 2 2 2 (8.25) the integral I becomes 2 1 1 I = h-1 1 - H (x, y) dx dy m (8.26) where 1 11 1 H (x,y) = F (b+ h + 2 hx) F (a + 2 h + 1 hy) m 1 2 2 2 2 2 1 1 1 1 G (b + h + hxa + h + h y) m 2 2 2 2 337 UNCLASSIFIED (8.27)

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F and h 2 (1) (2) 1 - ' W.2 H (. y>.) + m 8N8) 4 E E 1 m i + M,N (88) - j j=l " " The error M, N satisfies, approximately M, N EM+1, N+ 1 M,N - M+ 1;N - M,N 2LM + 1, N+ 1 M+1,N]i (8. 29) these results can be used to arrive at the correct choice of M and N. In (1) (2) (8.28) the xi, W(, yj, W. are the corresponding zeros and weights for M and N-point Legendre-Gauss quadrature. (U) B. The Cells Bordering the Diagonal Cells: We recall that G m is of the form G = 2(i)m + i m]. (8.30) m We now write Rem = log _ JG (8.31) where G is a bounded function, in order to explicitly display the singularity. Thus to evaluate the integrals (8.24) we, for either a = b or a = b +h, split Rem up into two parts. We achieve this using 1 338 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F lo = og s t + log 1 r * - j- h ~ + h (8.32) (8.33) Letting Remn = g +(2) where a +h j b+h Ja lb Rem Fl(s) F2(t) ds dt = I1 + 12 (8.34) and where a+h b+h I1 1= a b a+h b+h 12 = Ja;b (1) g( F () F (t) ds dt 1 2 (2) g F (s) F2(t) ds dt 1 2 (8.35) (8.36) Of these the integral I2 presents less difficulty; we can evaluate it directly using Gauss-Legendre quadrature, just as we evaluated (8.26) above. (U) In evaluating I1 we consider three cases: 1) a = b; 2) a = b +h; 3) a = b = 0, or 1 - h (U) a. The Case a = b * 0, 1 - h: The integrals are of the form aa+h a+h Ja Ja F(s, t) h ds dt (3.37). 339, UNCLASSIFIED

UNCLASIFIE THE UNIVERSITY OF 8525-1-F MICHIGANA We set s =a + -h + 2 t =a h+ 2 in (8. 37) to get*c u -v ufv (8. 38) ' =1 G (u, v) In JhF2-(v +h/$T2) h V du dv f C h/ f'W ( + I 0 J-(h/'r2-v) G (u,v) In Fq v du dv (8. 39) In the first term on the right we replace v by - v, so that [h/p' — hF2 -0 J -(h/I/?-v) v) +G (u, -] In - v du dv (8.40) Next we set h V = r2 x h U = IT(I - x) y in (8.40) to get, finally (8.41) (8.42) Compare Goodrich et al, (1967c), pp. 111-112. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 3 4 0 UNCL~ASt FE

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F I = h x=o y=-1 [F(s, t) + F(t, s)] (1 - x) in x dy dx (8.43) where s = h/2 (1 - x) (1 + y) + a t = h/2 1l - x) y + x + 1 + a } (8.44) (U) variable We evaluate (8.43) by use of Legendre-Gauss quadrature in the y and Gauss quadrature with weight in x in the variable x: I M N (2) I = Z W.l W. H(xi, yj) + MN i=1 j-= 1 J M (8.45) Where M, N again satisfies (8.29). (U) b. The Case a = b + h: In this case we want to evaluate the integrals /a+h ra I = I/s=a a=a-h F(s, t) In s - t dtds h (8.46) Let us make the transformation u - v s = a + -- V, t = a + v urv (8.47) in (8.46), to get 341 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F 1 I0 L v+2h I =;-(h/f2) -v G (u, v) in h- v du dv h fh/1 / -v+2h + G (u, v) In u -v u+ v where G(u, v) = F (a + - a +,- ). the first integral to get vI du dv h (8.48) We now replace v by -v in h/ v+f h;o J-v [G(u,v) + G (u,-v) in - v du dv h (8.49) Finally, putting h v = A P/ u = h/V'j [(1 + x) y + 1 (8.50) $ we arrive at the formula h2 1 1 I- ~ I = 2 F I (s, t) + F (t, s)(l+x) n |x dy dx 0 -1 (8.51) where s = 2 h 1 + x) y +(1 1 t = - h (1 + x) (1 + y) + 2 - x) +a a (8.52) We evaluate the integral (8.51) just as we evaluated the integral (8.43). 342 UNCLASSIFIED I

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) c. The Case When a = b - O or a = b = 1 - h: These are very important cases. They need special careful consideration, since of these cells T approaches zero only as O(h), whereas T approaches zero as O(h ) mn mn at all other cells. Thus T at these cells is O(1/h) times as large as mn T elsewhere. mn (U) We assume f'(0) / 0, f'(L) X 0. In this case G = ( 1 n Is- ) m s + t s + t (8.53) or s or t -a 0. As in the previous section, we therefore write (1) (2) G =g +g i In _C s + t) - = 1 s- - t G = In G s+t s+t G + -t in s +t 1 +ki 81 -k. sG (8. - t Byuef(854e a = b = 0. By use of (8.54) we 54) where G is well-behaved. (U) Let us first consider the case need to develop two quadrature formulas: jh Fh I(1) = | F1(S, t) + t 0 In -sl dt ds s + t (8.55) 'h h (2) If0h10h 1 F2 (s t) S+'t dt ds 2 's +t (8.56) - 343.. UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) Let us first consider (8.55). On setting U - V S = 2 u+v u + v t 2 v 2 (8.57) we get (1) h/ u G(u, v) *;u in Iv/ul dv du J 2' h /(2 hh+u) +h/ hu h/4- J -(4h-u) G(u,v) ~ -, n | dudv 1= I + 2 (8.58) where I1 is the first integral in (8.55) and I2 is the second. We next set u = I2 x, h y = uy = xy f2 (8.59) in I1. Then " I 1 1 II 2 I l(S, t) + Fl(t, s] In IY dy dx ly{ dyo (8.60) where h S = - xx(1 - y) t x (1 + y) F2 (8.61) Similarly, by setting u = h( l - 1/i x), v = h/42 xy (8.62) 2 Note the factor h/2, not h /4 344 UNCLASSIFIED

UNC CLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN in I2, we get 1 1 I2 = h/2 Fl(s, t) + Fl(t, s)T o O 1 2 - x [n y + In x + In ]- x x dy dx (8.63) (U) Before collecting terms, let us digress for a moment and evaluate I2 h h 12 - 1 F (s, t) ~ ds dt 2 S + t (8.64) (U) Upon setting u + v S -- '-'X ' t = u-v (8.65) we have | h/2 u f h "2h-u 1 (2) (v ) I (u, v) _L J-u Jh/F2 - (;f2 - uj ' dvdu (8.66) where G(u, v) = F 2(s, t) under (8.65). In the first integral on (8.66) we put the right of v = ux, u = h/2' y (8.67) and in the second 345 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY C 8525-1-F )F MICHIGAN 2 h - u = h/2y, We then arrive at the formula v = h /V xy (8.68) 2 h I = 2 (1 (1 h h G(,y xy) -1 ~~Y~ + G ((2 2-y (2 - dxdy - Y). xy) * dxdy h 2 F ( ( + x), y (1 - )) + 2- F2 (2 - y + xy), (2 - y - xy) dx dy We are now in a position to collect terms: i 1 1( I1) + I(2) = h/2 F(S, tl) + F (tl, sl) 1o 1o 1 (8.69) 1 + -x [Fls2 +2- x 1 t2) + Fl(t2, s2] + Y 2 -y F1(s3. t3) + F (t3 s3)J n y dy dx 1 3 (8. 70) 1 h/2 + h/2 0o Jo 2y In 1 - Fl t3 )F(t + 3)] 2 - y 2 - y L 3I 3 ( 3 + F2(s1' tl) + F2(t1 s1) + + F2 (t3, s3)] dx dy Y -y (s3 t3) 2 -y 2 3'0 (8.70) 346 __ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN - 8525-1-F where h s1= 2 x(1 -y) 2 h t x(l+y) 1 2 s2 h 2 - 52 2 x (1 - Y a t=: [2 - x (1 + y = 2 - (1 + t =h 2 - y (1 + x) (8.71) S3 = 2 -y(1 -X J a The first of (8. 70) is evaluated by use of Gaussian quadrature with weight function Iny in y, and weight function 2 (Legendre-Gauss) in x (compare 8.43). The second of (8.70) is evaluated by use of Gauss quadrature with weight 1 in both variables, x and y (compare 8.27). (U) Near s = t = 1, the singular portion of G is of the form G 1 s-t m m 2 - tn 2- -t If we set s' = 1 - s, t' = 1 - t we see that G satisfies (8.54) with s replaced by s' and t by t'. Thus we end up wi (8.70) with s. and t. replaced by s' and t'. The s' and t'. are the 1 1 i 1i 1 1 given by (8.71) together with the relation s' = 1 - s., t. = 1 - t.. m1 1 i 1 8.2.4 The Evaluation of d. 1 (U) The formula for these is given in Eq. (9.3) of Castellanos; these take the form r IT m ith:n e 1 I =Jo ik z(t) F (t) e J (k f(z)sin 0 )dt m o o (8. 72) (U) The Bessel function is evaluated by use of its truncated expansion M J (x) = E Im=0 u =0 (-1)p (x/2)2m+2 I! (m + ) + M M (8.73) 347 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F where r 2 2 M Il 2M EM < [x/2 e M(m +M) < x (8.74) (U) The evaluation of I in (8.72) is then done by use of the midordinate formula: f Z 2i- 1 f(x) dx h i f( 2 h) +;h = 1/N (8.75) i=l which is suited to this type of integral. The error TN satisfies the approximate relation.N - N N - 'N; this can be used as a test of when to stop the computations. 8.3 Description of Computer Programming (U) This Section gives a listing of the computer programming. The main program is shown schematically in Fig. 8-1. The inputs to the program are as given in Table VIII-1. It will be noted that as shown in this particular listing, the angle of incidence alpha is set at zero, representing nose-on incidence and the number of modes is entered as one. These representative entries to the program were chosen so that the first test programs would be as simple as possible and still provide data for comparison with theoretical calculations or experimental data. This is not a fundamental restriction. TABLE VIII-1: Main Listing Inputs Read: M N. CIR XKO A ALPHA 2 O 1 10 7 1 7T No. of No. of 1/2 cir- Angle of modes sampling cumference incidence _____points____ 348 UNCLASSIFIED

UN CLASSIFIED THE UNIVERSITY OF 8525-1-F MAIN MICHIGAN FIG. 8-2: SCHEMATIC REPRESENTATION OF COMPUTER PROGRAM. 349 UNCLASSIFIOED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F MA I N IMPLICIT REAL*8(A-HIO-Z ) COlIMMnN FMM, IM,CIR,XKO,A,ALPHA,T DIMENSIOFN T( 100),MAT(2,210),IMAT(2,210),TM(2,420) nIMENSION RRI (2,20),RI I (2,20),CRI (2,20),CRII (2,20) RFAL*R MAT, IMAT, READ (1,101,END=200) M,N,CIRXKO,A,THETA lnl FR MAT (2I3,4D16.7) + N1 =N-1 M —M+2 FACT=..DO/(3.DI*CIR) r( I ) =n. H=1.nn/Nl... n- -I-(_.._...._ _..._._. n — - - -':'2 --- 1 1 T(K)=T(K-1 )+H - - """rTT1rT 1 )- -'1-.l')-nn —_ _ -- FM = M nn 4 T=1,NNO ---- - -_FTn'-4- =T ----_ ---7 -4 MAT(K, I ) =0.DO) "T nzn TIT CN rr. MTST - - - ------------ W,,RITTF (6,,201) (MAT( 1,K),K=1,NNO) WRI I ( v,201) (MAI1?,.K),K=1,NN —) 201 FORMAT (1H1(8015.5)) C A — T LTTRS T AT -......AT. I Dn 2 K=3,NN - - - - - - = — A Irr -.....,Al.L I.t, V I IF ((K/5)*5.ED. K) CALL REFINE(MATIMATTMK,1.D-5,20) - - CnFlNTIN4EF CALL RRGEN(NTHETA,BRI,BRII) WRITE (6 102) RRI,BRI-ICALL CRMAKR(CRI CRI I,RRI BRI 1, IMAT,NN,FACT) 10? FWRITE (6,1 CRI,CRIA10? FnORMAT (,1 REAL'12X,'IMAGINARYl/2(D17.7)) I1 I I1 _) 200 CALL SYSTEM.~-r~ 350 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F The geometry of the shape to be studied is specified in the FMAKER listing. The parameter, T, ranges from zero to one and the parameterization in terms of this T for p, z and their derivatives is done in the FMAKER routine. FILL (U) All integrals are of the form ra+h b+h Tk, I ds dt F (s, t) G (s, t) a b The integral may be done three different ways depending on its relation to the singularity of the Greens function at s = t. If the range of integration is not near the singularity then the subroutine REGUL is used. If the integration goes up to the singularity the subroutine OFDIAG is used. If the integration goes across the singularity the subroutine DIAG is used. There is a subroutine FST which calculates the functions F(s, t). A subroutine GGEN calculates G (s, t). There are subroutines G1MST, G2MST, HMST which calculate g( (S, t), gf (s, t), and h (s, t) respectively which are used to m m m handle the singularity. +Oh b+h REGUL a b a+h b b h2 N N T f - - A.A. F (s., t) G (s., t.), 4 1 j 1 j m 1 j h s. = a + (1 + a 1 2 1 351 UNCLASSIFIED

UNCL~ASSFE THE UNIVERSITY OF MICHIGAN 8525-1-F F ILL S;1kRHI'JTINF F ILL N1,AT ) - _ IPLICTT REAL —R,( A-Hgfl-7) _ _ _ TN ITFP G F-P D __ _ __ _ _ _ _ 1< = (NI 1 / -3) * 2 f = N 1 -K C,_ _ _ _ I-) 0 2 I= 1,K C IF (nl-1) 11,1.2,13 r',n TO 2 12 TL AFDTA((4,JMAT) 4 K 0=f K n =n? NKDr=?**+KD CAL RFI (IJ,N\PIG- 9vAT) 2 TIRT~f-(-6 — flT- -__ 101 FAPOviAT(iX?5 p F- TI I F "\1 n 352 UNCL~ASSFE

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F t. b + (1 +.) J 2 j A. are the Legendre weights from Stroud and Secrest, Gaussian Quadrature Formulas, Table 1. a. are the Legendre nodes from Stroud and Secrest, 1 Table I. (U) It has been determined that this quadrature converges but it has not been tested under these circumstances to see exactly how rapid the convergence is. GGEN iPfR. G (s, t) (-1)m Cos m - m n j=j 2n = () -z(sJ 2 +p2(t)+ p2(s) +2p(t) p(s) cos (- 2n z(t) and p(t) are calculated by the subroutine FMAKER. The imaginary part of this quadrature is not affected by the singularity of G and has been m found to converge rapidly for all s and t with an n of 10 being adequate for 8 digits of accuracy. For the real part the degree of convergence is determined by the proximity of the singularity as s approaches t. When s < < t an n = 10 is adequate for 7 - 8 digit accuracy. For s - t = h it was determined that n = 30 was adequate for 7 - 8 digit agreement. (Discussion continued on page 360.) ______353 UNCLASSIFIED

UNCL~ASSFE THE UNIVERSITY OF MICHIGAN 8525-1-F RFGHiLI SIIRRniJTTNF REGUJL(Il,,JNPTGMAT)________ TMPLICIT RFAL*8(A-Hg('.-7) __ RFAL*8 MAT(100),(;M(10)_ _ _ __ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ - - TWENSTON F!I ( 5), F.1V-5h A,A ( 10 ),pX ( 10) COMMON FMMTMCIRXKOAALPHAT nTMENSION 1(100),F(9),LI(4)L,LJ(4),KK(494) O)TMENSION GLW(_10),GLN( 10) _______ _____ DATA X /.9324695142031529.6612093864662645,.02386191860831699,-93 __1.24695142031529-.66120938646626459-.2386191860831969/____ D(AT!WA J/.17132449237917039.360Y76i13573:04813R6,.4679139345726919,17l3 1?449237917039, 3607615730481386,9.4679139345726910/ __ H=( T(?)-T( 1 ) )*.5D0 CALL CFLL(IJgKKpK,~lLT,L) nn0? KJ=1;NPTTJl=T( J)+H*( l.+X( KJ) ).- A I a r-. A.1 r- - A IV. r,.. nnO _KTI=NPT __ T!=T( I!)+H*( 1.+X(KI )) ___ CALL FMAKFR (T T,9FI ) 2I ) - FJ~,(?~)0=0*On+FT(l1)*FT(l1)+FJ1(1I)*FJ1(1) CAIA GrFN(0,RNPT (;GC M)__ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ C,rA( MG) =,M (MG ) *AJ( K!)*AJ( KJ )*? W I 3 L = I,K _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ r, E l * K K ( T,L )-J __ __ __ __ _ __ __ __ __ _ KK?=?*KK (,19)-i ( K 3=2*K K (4 3,L IKK4=2*KK (4,L )-l nO 3 TO=1,? MAT(KKI)=MAT(KKI)+(F(1)+F(3))*G~m(mG,)+F(2)*SIJM M A T (K K M AT( K K 2 ) +F ( 4 )*I -+ F( 5)GM (MG fv.AT(KK3)=M4T(KK3)+F(6)*S~im+F(7)*GM(MG,) 0AT(K4) rAA( KK4) +F *D F+F (9 *GM (MG) K K 1 KK 1 +J I_ _ _ _ _ _ _ _ _ _ _ _ _ _ -KK?=KK?+l K K 3=K K3+1 KK4=KK4+1 3 PN r=M r+J P F TI I RN __ _ _ _ __ _ _ _ __ _ _ _ __ _ _ FwTP.YCHR(NLWG ) I' P T =\ N\1 A. I( L =L W(L) X I (9 =lro( L ) P F T I IPN F; N I) 354 UNCLASIFIE

UNCLASIFIE THE UNIVERSITY OF MICHIGAN 8525-1-F GGEN SI1IRRnoITINE GGEN(QRNPTGM) TMPL IC REAL*8(A-HO-Z) COnmmON FMM,!MCIRXKOAALPHAT nIMFNSION GM(2,3),T(1OO) nATA P /3.14159265389793/ TN=4*NPT ___ 00 10 K=1, IM GM( 1,K)=O. i0 rGm(2,K)=0. X=X/TN*PI _ _ _ _ _ _ _ _ _ _ CALL CHER (X,IM,TM) R-W=DSORT(O+B*TM(2) )*CIR GR=DcOs(R)/R G =D S I N (R) I R nn 2 KM=1,IM _ _ _ _ _ _ _ _ _ _ -rTI-MTT~vK T1=R* KM)+GM(19KM)o GM( 2,KM)=GI*TM(KM)+GM(2?,KM) X=PI/NPT DJO 3 K=1,IM GM( 1,K)=GM( 1,K)*X_________ GM (2,9K)=77MC?:TR-i 3 X=-X R E TUJRN END 355 UNCL~ASSFE

U N LASS)iE I ~~THE UNIVERSITY OF MICHIGAN 8525-1-F 1 C HER M~ROARITINE rHER(lX,lM,,To4) nJMENS~flN TM(3) ____ Tu?.*X*X-1. nnf i Mu3,JTM -K i~-~bTTW —~~ ___ LEE TgIRN 3 56 UNCLASIFIE

UNCLASIFIE THE UNIVERSITY OF 8525-1-F M ICH I GA N F ST SIIRROUITINE FST(ST, TJFSFTF) TMPL A-HFL*(_7 T.p-Z COMMON FM,M,N,C!R ___ r (w F i ) F~TTU5)p T9 CALL KMAKER (S,9T,9FK I,9FK ID) CALL KMAKER(TJFKJFKJD)) YT=(FK ID+FS (5) +FK I )/CIR CFS 1. )*F K 1* PFir (17*FFK J YVJ=( FKJD+FT( 5)+FKJ)/CIR__ Rm~~VFS47 FTV — _ F?)=C*FS( 3)*FT( 3) F (3) =-FS ( I) *YVI*FT ( ) *YJ __F(4)=C*FS( 3) TR =FkJ-*FSTT7 *YV1TF7 M-*TW F( 6)=C, F UT7T- FK J* FRF UIF MFIWIfC TR $ [WF F(R)=C*FT( 3) F(9)=FTI()*YJ*FKI/(CIR*FM) RETURN 357 UNCLASIFIE

U N CLASIFE ~~THE UNIVERSITY OF MICHIGAN 8525-1-F KMAKER SUBRROUJTINEKMAKER(TIFKFKD) ___TMPLIITVREAL*R( A-HO-Z) __ COMMON FMMNDTR, XKOAALPHATTOB XT-WT( ITT1T __ _ FK=1.-DARS (X )*i)_ _ _ _ _ _ _ _ _ _ — _-f(FK1,V P_ 92? 1 FKn=O.__ RFTUJRN H-_ (X)3,p4,p5 3 FKO)=D '5 FKD=-D 4 PRTNT lOlTX 101l FOIRMAT (Z2H ****K IS (IN THEI SPIKFt —J —)ogl RFTWRN__ ___ _ _ _ --- -- F'TNTV 358 w UNCL~ASSFE

UNCL~ASSFE THE UNIVERSITY OF 8525-1-F M I C HI GA N mmm -FMAKFR I IRRC1IIITTINKF FMAKF-R(TF) TMPI-TCTT RFAL*R(A-.HO-Z) f) T NFNSTO (N F-(9S)____ lA TA PT /3*141 59?(6S3'i9793/ IF (T-1.F-5) 1919? TF (*99999-T) 19193 _ _ _ _ F( I3)= n._ ___ _ _ _ F(4) =0. REFTUJRN Fl 1 )=S/PT ___ F(?=.-C)/PTI Fl 3) =C F 4 4) =S1. __Fl 5)=PT*C/S R FTIJIRN E N D _ _ _ _ _ _ _ 359 UNCLASIFIE

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F ah a+ha+h DIAG | || Ja la h2 N N (1)( (+ i, S.)] + E E AA F(s, tj)g )(s., t) i=1 j=1 Im(T = A.A. F(si, t )Im (Gm(s, t)) i= j= 1 1 j mi j = a + (1 + a.) 1 2 1 t = a + - (1 + a.) sij = h/'(1 -.)(1 + a.) + a tij = h/2' (1 - i). + 1 + Ii] + a A. and a. defined as above and L. are the log weights from Stroud and Secrest, Table 9. 1. are the log nodes from Stroud and Secrest, Table 9. (U) These quadratures have been demonstrated to converge but have not been tested in these circumstances to determine how great an N is needed to assure 7 digit accuracy. G1MST (1) 2(-1)m 1 gm (st) = CR 'r h(s., t) 360 UNCLASSIFIED

U NCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F )I AG SIIRRIlUTINE DTAG( I,J,MAT) TMPLICTT RFAL*R(A-HO-Z) RFAL*R MAT,I1AM CO MMOIN FM,M, INt,CIR,XKO,AALPHA,T DT F.'lNS I)N T(100),MAT(100),FSI(5),FT'J(5),FTIJ(5),FSIJ(5),FS(9),FT 1 (9),F(9),AJ( 1),AL( 10),ALF(10),LAM(10),G1M(10),G2M(10),GM(2,10) DI MEN.S I BN GLW( 1 0),GLN( 10 ),T9W( 10),T9N( 10)_ Dl MENSTON l '-4') 4 (4),KK"(4',4 1hATA NPTG/10O/,NI\PT/6/______ ___ _ WATA l AM/.?V^40 5P44179)-l, t.12958339l 1549508,.3140204499147655. ).. 53RA57! 7351 O8? 1,t 7569153373774029,.9226688513721202/ nATA AL/.?Rf76366?;S7H5476,..3:R2?65732739468,.2453174265632104,.142 1.noP7565664767,, 5554562324886290-1,. 10168958692932280-1/ U-. AA-L 7.9.3 64, 54'"'.3T5, 2093864662645. 2386191860831969,-.93 24695142?0152,-,6612093864662645,-.2386191860831969/ - f-~ATA- AT/17T 64-93T971 47. 66076 1 573048l3 86,.467913934572691,. 1713 1244923791703,. 3607615730481386,.4679139345726910/ H=T(?)-T( )" H2=H*..5D)n H —O4=H2,H? HSO=H2*H H.O2:=H?* 1.414213562373095 A=T( J) CALL CFLL( vI,JKK,K,LI,LJ)nn00 1 JK=1,NPT..TJ =H-*(-_. ( JK) + l. O ) +A - CALL FMAKFR(TJ, FTI ) nn — i I K =!-;-...... PST=H2*(ALF( IK)+1.00)+A SIJ=H2*(1.f)O-LAM(tK))*(l.DO+ALF(JK))+A TII=H?*( (1.nD-LAM(IK ))ALF(JK)+1.DO+LAM( IK))+A AV=nARS ( SI-TJ 7-/H CALL FMAKER ( S I,F FS I ) CALL FMA —KER(TIJ,FTI-J) - - CALL FMAKFR( STJ,FSIJ) CALL GMlMST( FSIj,TFTTIJ,,Gl) CALL G?MST(AV,FSI,FTJ,G2M) O= F. I 2'i- — 'F -.................... - - - - O=FSJ( )-FTJ(?)0=0*0+FS I( 1)*FSI (1 )+FTJ( 1 )*FTJ( 1) -:-= ' —(-r *-* Sf r(f- -) -*FTJ ( - _CALL GrGFIN( 0,R,NPTG,,GM) nn 2 I.=IiM (;IM(L ) =(U1fM(L ) *( IK ) *AJ ( JK )*( 1 00-LAM( IK ) )*HSO___________________ GM(?,L) =GM (, L)*^AI K)*AJ( JK)*HS"4 2_ G2M(L) =(M(L)*AAJ( IK)*AJJK)*NS04 ___ __________________ Si IM ( M M ( M)+,4GM ( M+?) )*5. )O ---- 1(M2UM(,M)+G2M(,M+?))*.5 r --- —----------------- )TF1= (G1M(M)-G1M(M+?) )*5DO ______ ')TF=( GM(7,M)-GM(M+?,) )*.5O)n nn 3- '- 1,K.- ----—.. — KK1=2*KK(1,L ) KK?=?*KK (?,L) KK3=2*KK( 3,L) K 4 = 2 * K K(4,L - L..... CALIL FST( SI, TJ, LI (L), LJI (L), FS I, FTJ, F) CALL F. T ( S I J, T I, LI ( L ).,LJ ( L ), F S I J, F T I J, F S ) _ 361 ___ UNCLASSIFIED

U N CLASSIFIED THE UNIVERSITY OF 8525-1-F M I CHI GA N flALL FST(TIJSTJLT (L),LJ(L),FTIJFSTJFT) I_ IFS(2)+FT(2))*SIJMl+(F(1)+F(3))*G72M(M+1)+F(2)*SIJM2 --— V KK WTlI=gTTKWRTIV +TFTI1TF(3))*(M(2,M.1)+F(2)*StJM __MAT(KK2-1)=MAT(KK2-1) ___ +(FS(4)+FT(4) )*OIFI+(FS(5)+FT(5) )*G~lM - (M + 1) + F ( ) P F F U i G M M ) - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ MAT(KK3-1)=MAT(KK3-1) +(FS(6)+FT(6))*S1IM1+(FS(7)+FT(7))*G1I, M4T(KK2)=MAT(KK2) +F(4)*D)TF+F( 5)*rM(2,M+1) MAT(KK3)=MAT(KK3) +F(6)*SIJM+F(7)*rym(2,M+l) _ _ TF (L T(L-) FEO. L J(L)) W I TO 3 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - Y(K- )=ATY(_KK4-1 +(FS(A)+FT(P) )*D!Fl+(FS(9)+FT(9) )*GlI 1ICM+1 )+F( 8)*DTF2+F( 9)*G72M(M+l) 3 cnfNTTIKIJF ____ R FTIJR N K PT =NN nn) 7 L=IVNN AJ L )=(LW( L) _ E I =rLJU~(V: AL-CL )=T9W( L) 7T- UI-A Mfj =FTq-r{C- LI P F TI JRIN hN I 362 UNCL~ASSt E

UNCLASSIFIED THE UNIVERSITY OF 8525-1-F MICHIGAN C is half the circumference of the body 22 2 R =Yz (s) - z(t) + p(s) + p(t) k - 2 'p(s) p(t) R G2MST (2) gm SI 2 (-1)m t) = CR -i n I s -tl (1 + ') -1 - k h j h (s, t) m note Is -t 21f- p(t) C, s-t t R, k, p same as above h (s, t) cones from m the subroutine HMST. HMST h (s, t) = N i=E i=l '1 d cos LCR1 - k x 7 dy, (1 + x) (1 + kxi) 1 T2m () Ai os R - j T2 (X.) +x.)C 2m 1 2 1- a. sinh (3 2 2 sinh2 f x 1 i 1 = In 1 inh -k sinh f= 1 - k ----— 363 UNCLASSIFIED

THE UNIVERSX8TY OF MICHIGAN 8525-1-F C-l MST __ S~k~ThTI\F lMST( FS,FT,GrlM) ___RFAL*R_"PKST____ - n DTMFNS~~cTF-iNt-IflM (3), FS (5),FTP- ) I(-I rvA vf (\ FMMv TMN',C! TR P=F S( 2F )-T (2 T I mFS ( 1)+F T(I R =nF;()S TRT*R+TlI*Tfl rCAl-1. He1S.lT(KSTRG1Mv) 1 1 =2.F Dl/(CT R*P*KST) nri 1 T=1T TM (1M =(Z1M( T ) *T1I P F TJR N F- l\i n _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 3 6 4 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

U N CLASSFE - -THE UNIVERSITY OF MICHIGAN -- 8525- 1-F G2MST SUMRRIJTINE G2MST(AVqFSFTG2M) — TM[P7IT TTV'EAL*8( A-H Ito-Z) RFAL*8 KST niMENSION- GM( 3),FS( 5) FT( 5) COMMON FM,M, IMCIRXKOAALPHAT DIMENSION 1(100) Rl=FS(2 )-FT( l2) _ _ _ _ _ _ _ _ _ R=OSORT( R*R+Tl*Tl) ____ IF (AV.GT. 1.0-8) GO TO 1 Il=2.82Z842711?44619*FJ(1)/(T(2)-T(l)l GO TO 2 _ _ _ _ _ T1T=AVTDSURrTTTT-3TF 2 SOK=DSORT(KST) _____ T h1 T*TI - UYOWSS 0 T1=2.DO/(CIR*R*SQK)*DLOG(Tl) nDO 3 1=1,ITM MHVTT=C7MTT)*TlT 3 Tl=-TI * P F7TJWF - _ E N F 365 UNCLASIFIE

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F (U) A subroutine CHEB is provided to calculate the terms (T2 ) of the Tchebychev polynomial A., a., C, R, k all as above. It has been determined that this quadrature does converge but the size of the N needed to assure 7 digit accuracy has not been determined. OFDIAG [a a+h Ja-h Ja Re (Tk) a 2 2 N N Z I. L.A. i=l j=l1 1 J N _ AkA (1 -ak) k=l k ij (1) ' i 1 k=l 2 F 1 In - Fc (s,,t ) (1)(sk ) + a + 3 kp kp m q JK% N +z Imh(T ) k,i 4 S.. = -. (a. 1j 2 1 j N A A A F (si, t.) j=1 N N i J A. A. F i=1 j= 1 3 (2) (si, t.) m j (s., t.) Im G (si, tj.) 1 j mL 1 3 - 1) + a h 1j 2. (1 + a.) + a 1 3 h [ Skp = - p (1 - k) - a -3] + a t h [p(1 tkp 4 P -ak) + ak + +a I 366 UNCLASSIFIED

U N LASSFIE THE UNIVERSITY OF 8525-1-F MICHIGAN HMS T SIJRROTIJTNE HMST(KSTRSTHM) T M PI L I r I T-IF[URL* TA-14i9ZT COMMON FM,M,TM,CIR _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ DIMENSIONI GLW(?O),GLN(20) n_ ATA N/1O/,SIG/I.DO,-1.1)0/ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - OVNS -rWT R-JTTq —(-T-,A (?0)-f,ALCF( 20 n__ ATAALF/.99312859918509499,.9639719272779138,.9122344282513259,.839 1167F721,431 XO101I3039761950601587?,.37370608A71541969.22778585114164519,7652652113349733D-l/ 1AAA/.l-fbl4OU(l11lZ1ZD-1,,4ObO14Z9F90(395b94)-l1,Zb7Z6Jz483341090 16D-i,.R3276741576704750-1,.10193011981724049,.1181945319615184,. 131 — 77 63 R4wj '7-W- — I -7 T9 61 l~~zlT9129g47607.15Z753387130725 RFTUIRN -5 — R=U-fr-T1TTT)n41.WaR1T-KSTT17 T.0 IT13rrO-K ST) DO 3 K=iIM DO 1 T s=l,? ~I I 1I I = I. P ___x=nSTNH(8*(1flnO-SIG( IS)*ALF(I) )*.fl1)O) __ ____ W=l fff-xi'-YXJSTW(1l.DOt.KS-TV7'' -"6CALL CHFR(XIMTM) REAR(I T* IsTCTRI R-SThITURT 1Tn' ST WTST*)YX 1T71'WPMT bT'rD IKST*X))? HM(K)=HM(K)+R*TM(K)__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ RFTIIRN N=NN * M, 4* I. =I pIY ___A(L)z(;LW(L) 4 -M- - CFfl ICcN(IL).-. ---- - --- R___ RTUJRN 367 UNCLASIFIE

UNCLASIFIE THE UNIVERSITY OF MICHIGAN 8525-1-FI S(IRRn ITTINIF nFDTA( ITJJp,MAT) ITMPLICIAT RFAL*P (A-HU(-L) COmmON FrAM, MlCIRSKZAAALPHATT( 100) REAL*8 MAT( 2,210),FSI (5),FTJ (5),FSTJ (5), FTIJ( 5),FSKP (5),FTKP( 5) RFAL*8 FSTI(9),FSTIJ-(-9),FSTKP(9),GlIJ(8),GLKP(8)g,G21(8) - -... - - -.. - -. -. - -. C- -% A -, %. - &. V. r%. I v. ^.. v. r. &. I. r. t.. RFAL*8 UM,Mj(?,8) T( 2,9) U( 2.,4),LRNI,LRWI,L(4NlLRNJLRWJ I_ NTEGFR KK(4,4)_,NN/6/,_TP(4),JP(4),NPTG7/20/ ______ R-~FAL*F4~- LRNI(20)/.9324695142O31520,.661209386466S26459.239619186O8319 F69,-.?3Re6191R608319699-.66120938646626459-.9324695142031520914*O./ RFAL*R LRW(0)/.17132449237917039.36076157304R1386,.46791393457269 9109.4679139345726910,.3607615730481386t* 1713244923791703,914*0./ RFA&L*8 LGN14(0)/.02163400584411695,.01295833911549508,.031402044991 _ F.476559.*053 865 7?173518021, *07569 1533737740299.*09226688 513721202/ ~ L~W{TT3R736625785;476,.3082865723729468,.2453174265632l FE04,.14200875656647679.O5545462232488629,.01016895869293228/ H = TT(?)-TT( 1) __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ HOF = H *.?500 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ HS? = H*H1nT - - _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ O0 I00 1=1,NN LRN T = LRN(T)__ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ LRW ~ I = LRW( ) -I- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ LGNT = LGN(TI) CT3 = 1.00 - LRNT C 14 = LRNT + 3.0 _ _ _ _ __ _ _ _ _ _ _ __ _ _ _ _ _ _ rC I= DLor((?.00/-,C T4 *CT3*LRw(I) CT? —= LGNT*L-GW( I-) _ _ _ _ _ _ _ _ _ _ _ _ _ SI = HO)T*(LRNI+1l.00) + A CALL FMAKFR(SIFST) 00 I00 J=1,NN I-_ _ R N.) = L R -N (, J ) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ LRWJ = LORW( J) CJ1 =.5Dn*LRWJ*LRWI__ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ T-7TJ'-=HTr TiiLrTI;Tf InwT Ujui T' A STJ= HO)T*LCNT*(LRNJ-1.00) + A T_ KP = HnF*(LRNJI*C13 - L14) + A _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ TP =HOF*(LRNJ*Ci3O A1) CALL FMAKFR(TI.J,FSTJ) CALL FMAKFR(TKPFTKP) CALL FMAKER( SKPFSKP) _ _ CALL FMAKFR(TJFTJ) __ _ _ _ _ _ _ _ _ _ _ _ 0, = FS!() FTJ(?) __ 0 = 0*0 + FST(1)**2 + FT.J(l)**2 K CALL (rFN( 0,R,NPTG,(;MlJ) CALL (;lMST(FSTJSFTTJG1)/H ___ _______ ________ CALL rIMST(FSKP,JFTKPGllKP) 00 90 NF=1-,47 368 UNCLASIFIE

UNC~LASSFE ~~THE UNIVERSITY OF 8525-1-F CALL FST(SITJIP(NE),JP(NE),FS1,FTJFSTI) MICHIGAN-I C~ALL F-ST( Sl,),TIJt IP(NE) 9JP(NE),IFSIJIFT1JgFSTIJ) CALL FST(SKPTKPIP(NE),JP(NE),FSKPFTKPFSTKP) MIJ V! M lt9p -T(1,MM) -= LRWJ*(C12*FSTIJ(.MM)*GlIJ(M) + CIl*FSTKP(MM)*GlKP(M)) - - &+ CJ1*WFS1!(MM)*G21(-M) 75 T(2,MM) = CJ1*FSTI(MM)*GMIJ(2,M) SIGN = 1.DO DO 76 MM=2,8,2__________________________ TVWf-lW5RT =Y.-*(LRWJ*(CI2*FSTIJ(MM)*(GIIJ(MM1)+SIGN*GlIJ(MPI)) C+ CIl*FSTKP(MM)*(GlKP(MM1)+SIGN*GlKP(MPl))) + CJI*FSTI(MM)* C(G21(MMl')+SIGN*G21(MP1)))T T(2,MM) = CJ1*FSTI(MM)*.5D0*(GMIJ(2,MM1)+SIGN*GMIJ(2,MPI)) 16 S IGN = -ST Wn~o 55 TR=Y EO(IR9l) = T(IRpl) + T(IR,,2) - T(IR,3) FwrrR92F= T ( IR,4) + -TT1W94-) FO(IR,3) = T(IR,6) - T(IR,7) tQfIR,p4) = T(IR,8)J + 11IR99) IF (NE.NE,4) GO TO 79 E T~TTi E Q (1I R, ITZTT TVP 1 EO(IR,3) = EOUIR,3) + EQ(IR,3) lY DOU g0 MMI,-~-4 KM = KK(.MMNE) lq0 fRAllJK9PJJ = MAltIlFKMj + tWI1KMM)WISZ a5 CONTINUE ___ 100 CONTINUE WE-TUWNN IJ SE ENTRY 'SETUOIQ TO CHANG~E QUADRATURE CONSTANTS FOR REAL PART ENTRY SE-TOFO,(JNLRNNIRWNLGNNLGW) REAL*8 NLRN(20)tNLRW(20),NLG-,N(20)i,N LCW(20) Do 300 I1,1 J _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ LRW(I) = NLRW(I) LGN(1) = NLGN(I) 300 IGW(I) = NLGW(I) NN =J R ET U R N__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 369 UNCLASSIFE

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F A., Li, ai., i all as above. It has been demonstrated that these quadratures do converge but the value of N needed to assure 7 digit accuracy in this situation has not been determined. CELL (U) The subroutine CELL performs the bookkeeping function of telling the integral calculating subroutines where in the matrix to store the results. Enlargement (Bordering) Method -- - I- - - --- -— Cl I -- - --- - -- (U) the form The n x n matrix M, which is to be inverted, is partitioned in A M =.C BT D where A columns. (U) is of order m (1 < m < n) B and C have n - m rows and m D is order n - m. If A -1is known then the following identity gives M If A is known then the following identity gives M BT p B^ -1 B D _ -1 — 1 A A-T F-1 CA-1 +A B F CA - F CA F-1 F where -1 T F = D - CA B this process may be written T -1 T 1) E = A BT T 2) F = D - CE 3) compute F1 3) compute F 370 UNCLASSIFIED

UNCLASIFIE ~~THE UNIVERSITY OF MICHIGAN 8525-1-F CELL SUBROUTINE CELL(IJKKK, L19LJ) r% V AA IkA Ai Al# a If- j- a v R A- IL s m a R- % U! ENS ~ KK( ~4941 911141,LJ 141 K=4 IF ( I * EQ. J) K=3 LI I(1 ).! a a w.. IJ I) =, LI (3)=I+l LJ(3)=J+l L I (2 )=1I.. Z-'ML —r-. - - tLJi(2?)J+1Y L I (4)=1+1 0 5 9 il- lk - I ILJ 14) =,I nO 1 L = UpK... I- - - - - N=Lj(L) &I I M 7*M- I N=?*N-1 I N=O DO 2 IK=1,N T M= IS-N+M _ _ _ _ _ _ _ _ _ 4(K(4,L )=TM+1 IF (M *EO*- KKw-r49*L)=KK(29L) 1 CONTINUE RETURN END 371 UNCLASIFIE

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F T T -1 4) G = E F -1 -1 5) H= F CA 6) K = A-1 + ETH then K -G -1 MI L-H F-1 In particular if m = n - 1 so that B and C are vectors and D -is a single -1 element, then F is also a single element and F = 1/F. Thus if A and -1 A are known it is easy to add one more row and column to the inverse. The process can be repeated to add as many rows/columns as needed. Refinement Method (U) Given a matrix M and some approximation to the inverse (M ). -1 n Then a better inverse (M ) + may be found from the following n+1 -1 R =(M ) M - I n (M ) =(M ) - R(M-1) n+1 n n this can be repeated until the improvement at each step is satisfactorily small, as when RE [R(M-1) 1 n, ij < TOL all j, i RE M1 ij ______ 372 UNCLASSIFIED

I- ~~THE U N CLASFE UNIVERSITY OF MICHIGAN 8525-1-F 1 rr pr% r &-l 7-1 - 10.1 1 - I C PACF nnni SIPRCUT IIE ECIPS (CCr'JX9DCINV) I PPL IC IT CCM PLE X* 16 ( A-H,0-Z) CIMENSICN CCI'TX(31,cCCNVf31 C, C RCUTINE TO INVERT A PATRIX BY ENLARGEMENT (BOARCERING) C ENTRY 'ODCeS6 FSTABLISHFS LOC OF MATRIX IICCPTX'3 AND ITS INVERSE C (IOCINV) ANC ALSC INVERTS THE 2 X 2 SUB-PA1P1*.e C 8OT1H PATRICIES-ARE SYMMETRIC WITH THE LOWER TRIANG. AND OKAG. C STORED PACKEC OY RCWS. C, EACH CALL CN CC1T -ACCS CNE NEW ROW (& COL) TC INVERSE. C THEF CCNTENTS CF THE CRIGINAL MATRIX ARE NOT CH4ANGEC. C, Oil CCPTX( I)*DCMTX (3P CCMTX( 2-)*OCMTX121 CCINV(l) = CCPTX(31/CTI OCINV(3)= CCPTX(1)/CTl CCINV(2) = -CCMTX(2)/DT1 N= 2 RETURN C ENTRY CCI%T NC = (N*(N+1)/2)41 CALL VPPRCC (CCPTX (NC ),9CCINV, CC INV(NC),N) NCK = NC +NK - 1 CTI = CCPTX(NCK41) CC 100 I=KCNCN 1CC DTI = OT1 - CCPTX(I)*DCINV(I) D7TL i.DC/DT1 -DCINV(KCK*1) = CTi IJ = 1 CC 2CC J=NC,NCN CC 2C0 I=NC,J DCINVIIJ) = CCINV(IJ) + OCCNV(I)*OTi*OCINV(J) 200 Ui IJ + 1 Oil =-O1`i CC OO1 I-NClphCN OCINVII1) = CCINV(I)*CT1 3C0 CCNTINUE N = N+ 1 RETURN END 373 UNCL~ASSFE

U N LASSFIE I THE UNIVERSITY OF MICHIGAN 8525-1-F tdwPPRCr 04-171-AR 1 QI11.44 PAGE 0001 __ Ok'ICLTIKE pFRCC(P'1tP2,2P9NqT) I PPL IC IT CCOPLFX*16 14-Ht Mt -Z I __-'IWF NSICK P I (1I), f-2( 1 ),P ( I ),T ( I r ol ANF M?- ARF N X N SYPMMFTRIC STCREC WITH4 CIEP TRIONG. AiD) DIAGO C PACNEC BYV QrWS. C- 1F TS N X N PEGLL.AR fPIX STCRFEC EY PON.S c T IS SCRATCO- VECTCOP CF AT LEAST N ELFMEKNTS NTJ = 1 _ _ I J S = 1 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ r IC 0 1 CO I,' T I i s_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ IJI I = S C U7~z J=1,N __ ( J) =PI(IJ) IF (J.hF.I) CC TC 2C IJ I _ _ = J _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ I SS= I; hI j!J * IJif 7'5 liJt = JI.I'c CALL Vf'PRCC(T,"'?,FP(NIJ),N I IUs = Iis 4+ 1 ICO NIJ = NIJ * N PETUPN FND 374 UNCLASIFIE

I PMPRflfl 04-.17-68 19 *aIt 4R PAGF Afl0l I I 7 v 9 - 'La NJ a %.Ii %J L- A. -IF - %# %P &L& SUBROUTINEPMPRCD(VECvPTXvPROO.NvNl)_______ IPPLICIT CCtJPLEX-*16 (A-HPMPO-Z) DIMENSION VEC(l)*,MTX(1),PRCD(l1) C C TFIS ROUTINE CCOFUJES THE PRODUCT CF A VECTOR AND A SYMM. MATR.IX m! C C PROC VEC*MTX VEC AND FRCC AR-E VECTCRS'OF LENGTI- N C C MTX IS N X N S-YPM. PTX STORED PACKED BY RCWS FCP ENTRY VMPROC, ALL N ELEMENTS OF PROD ARE COMPUTED (%) C,) U) -n mB CAD C FCR ENTRY PlVPRCC CNLY FIRST NI -ELEMENTS ARE CCMPUTED C NP = Ni IF (Nl.GTeN) NP=N G C T C 3, C FNTRV VPPRCD(VECPTXvFRCDN) NP = 3 IJS I DC 1CC I=1,NP S =.CDo 14 IJS 141 = 1 155s = 0 DC 75 J=1,f' c- =S + VEC(J)*!IIX(IJ) I F (Jo.NE.)I GD TO 20 [41 = 1I =S I 20 14 14 + 141 15 141 1 JI + ISS PROC.1II) =S 100 1JS = 145s I RETURN END z P01 cm 0d z0CA r

UNCLASSIFIED I THE UNIVERSITY OF MICHIGAN 8525-1-F lFF INF (4- 1 7-A R 10 11 52 PDAGf d1N SLBRCLTIKNE RFFIKNE(XVT.K.TQLMXI IP'FLICIT RFEL*8 (A-IO-ZPIMENSICK x(I).gv{ltT(l! C PFFiNE TIF N X h MTX 'VI TC BE!hE INVERSF CF THE MATRIX *X# UNTIL C THE COPRECTICN TC EACH ELEMENT IS LESS THAk "TOL.o C 071T IS SCRATC- PTX CF AT LEAST K X (N+II LCCATICNS C MA^X-I IS TFF MAXINUM NUMBER CF ITEIATIONS ALLOWED C *'1 AKC "0' ARE SYNMFTRIC PATRICIES blIT LOWER TRIANG. AND OIAG C STCREC EV RChS, PACKED. C All FLEMENTS tE CEL PREC CMPX LXCEPT: 'TOI * CRL PREC REAL AND C 'N$' '*XI 1 - INTEGER C - I IC 2 1 IS = 2*^*K IT = KNl* I NK = K + 1 IN = ^^ +: -10 CALL WPPPCC (vX, lT IS+l )I CC ICC I=I-IS,Ih ICC T(ill = 1(1 - I.CO II =! rC 2CC 1=1t CALL PMPRCr(T((I- 1 I*^+l),V, Il ),^,I?CC I I = II 1 I 1 1 K s C rC 3CC = 1, I___'_ IF (V(I).EC. 0.CO) CC TC 170 IF (CABS(T( 1)/V(I).GT. TCL) K K 4 ] 1 170 VflI = VII - T(Il 3CO CCKTINLE IF (K.EQo.C RPETRN IC C C.+ I IF (IC.GT. IMXI RETURN 1 GC TC 10 F^C 376 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F IM R(M~1) IL n ij < TOL all i,j IM (M1) J Where TOL is the maximum allowable relative error in any element. BRGEN (U) Each integral is of the form 1 b = f dt F (t) K(it) p(t) J [(t) sin 0] r m F(t) is calculated for each integral, K.(t) is the trial function, 0 is the angle of incidence. The subroutine BSLJ computes J (x). n (U) The integration is done numerically using a midpoint rule. The interval is halved until 5 digits of agreement between successive tries is achieved. BSLJ N ( 24k J (x) = (x/2)m (m + k) m k! (m + k) k=O0 This series is summed until the individual term in absolute value is less -12 than 10. CRMAKR (U) This subroutine simply performs the matrix multiplication of b and T involved in obtaining the final output. and T involved in obtaining the final output. _______377 UNCLASSIFIED

THE UNIVERSITY OF MICHIGAN 8525- 1-F 1 RRGEN S(IRRntJTTNE BRGEN(N,,THRRIRRII)__________ TMPLTICI REAL*8 (A-HO-Z) COMMON FMMICCIRXKOAALPHTT DATA SI.L-/S/.2D-4/gTGL/l.D-5/ DIMENSION TT( 100),BRI (2,10),BRI I(2,10),F(5),D( 1092),DD(1O92P, F FKT(2),FV(6) STH =DSIN(TH) ____ CY Th ~MMW DO 100 I =1,N N DO 100 J=1,? RRT(JIl) = 0. __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ DO 200 I=1,NM1__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ DT~T = TT (I +1 )-TT I) S = SI __ _ __ _ __ 198 NI = DT/S +.5 FNI = NI H = DT/FNI HT = H/2. DO 199 J=I1,2 DO 199 JJ=1,10 199 DD(JJl,J) = 0. Don 201_K=1,NI _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ FK =K - _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ DDT = FK*H - HT _ _ _ _ _ _ _ _ _ T TT(I) + DOT CALL FMAKER(TF) FKI(2) = H*DDT*F(1)/DT FKI(1) = H*F(l)-FKI(2) TP = F(2)*CThCT FV(l)_=_.5*DCOS(TP) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ FV(2) =.5*DSIN(TP) FV(3) = FV(1)*F(3) FV(4) = F V (2)* F (3 ) FV(5) = (FV(1)+FV(1))*F(4) _____________ FV(6) = (FV(V2) +FV (2))*F (4)TP = F(1)*STH _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Rj? = RSLJ(TPM) RJ3 = BRSLJ(TP,?M+1) 00_20? J=1,2__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ DO 203 JJ=1,,4 TP = FV(JJ)*FKI(J) __ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ DD)(JJJ) = DD(J.JJ) + TP*gJl 203 DD)(JJ+6,J) = DD(JJ+6,tJ)- + TP*BJ3 nDD(5,,J) = DD(5,PJ) + FKI(J)*BJ2*FV(5) 20? nD(6,j) = DD(6,J) + FKI(J)*BJ2*FV(6) IF (S 1- 5-) 75,75 7,76 _ _ _ _ _ _ _ __ _ _ _ _ _ _ __ _ _ _ _ _ brfl 204 J1,2 DO?04 JJ = 1,10 /1)4 r)J,),J)JJ =DDBJjvjJ 205 S S*, 5* I F (S2 S) 198,198,211 7 T DO 210 3 =1,? 378 UNCL~ASSFE

THE UNIVERSITY OF MICHIGAN 8525-1-F -f.0 — 'I CC cr c c cc, C Il 379 UNCLASSIFE

UNCLASIFIE THE UNIVERSITY OF MICHIGAN 8525-1-F 9 s L I1 FPINCIT1F)N RSLIJ(XM) __ ATA THL /1.0)-12/ __ P F T1 I IRf F TUJRN XT =X/?. = A XIT**m TIF-(-M') -7 7_5_7 7 4'_' 74 00 10nn 1, 100 C =C/Fl 75 XT -XT*XT F T = T 9*YT[UF T* ('F4FVJ-T _RSLJ= PSLJ + C I~ 7 NT T KIi)AM,(JIILF(,8JL 101 CR1NTTUNI ~0 ~WV ATSN7J 380 UNCL~ASSFE

UNCLASIFIE THE UNIVERSITY OF MICHIGAN 8525-1-F CRM AKFR SlIBROUTINF CRMAKR (CR!ICR I1,BRI,RRI I TI,NNFACTOR) TMPLcT-rFWEAK*-RAHfv0ZiT DIMENSION CRI(2,100),CRII(2,100),BRI(2,100),RRII(2,100),TI (2,5050) IDIAG =0 ILFGTH = 1I60 KLFGTH =0_____ ILEGH = LEGTH + 1 ___IF (ILFGTH - NN) - 1-70,9170,240 IDIAG = InIAG + ILEGTHJ= SIIMIA = 0.00O __S(IMRR =.00 IPO Ji = Ji + 1 KLEGIH= KLH11H + 1 ISIMRA TI(lI)*RRI(1,J)- TI(2,Il)*BRI(2,PJ) + SUMRA — = TT-f1~p11 1RT2FpJ.7TT~TT(2I)*BR I (1,J) + SUMI A SIJMRB= TI(lI)*BRII(1,J) - TI(2,I)*RRII(2,PJ) + SUMRB SIJMIR = ~IT YTYVTTThBI(,)+ SUMIV 11 (1-IDIAL,) 200,210,ZLO?00 I = I + GO TO 220u;?I0 I = I +KLEGTH __ 77n IF (J - NN) 1 80Yp230-2 30 230 CRI(1,ILEGTH) = -FACTOR*SUMIA URI(2,ILtGH) = t-ACTUR*51UMRA CRII(1,ILEGTH)= -FACTOR*SUMIB CRII(2,ILE F) ACTOR*SUMRB 5=-S GO TO 160 240 RETURN E~ND 381 UNCL~ASSFE

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F IX ACKNOWLEDGEMENTS (U) Experiments, analysis, interpretation and theorization are necessarily interrelated in a program of this complexity. It is possible therefore to indicate the gross catagories of specialization only for the individuals who took part in these studies. (U) The program was under the supervision of Prof. Ralph E. Hiatt, Head of the Radiation Laboratory. The Principal Investigator for the SURF radar cross section investigation was Dr. Raymond F. Goodrich; for the SURF plasma re-entry environment investigation, Dr. Vaughan H. Weston; for the short pulse studies, Dr. Ralph E. Kleinman and Prof. Herschel Weil. The Program Manager for all of these studies was Burton A. Harrison. (U) Dr. Thomas B.A. Senior was responsible for the analysis and inter pretation of the surface field data and for the synthesis of the SURF radar cross section formulas. He was assisted in this work by Dr. Ralph Kleinman, Leon P. Zukowski and Bruce C. Vrieland. (U) Eugene F. Knott was responsible for the experimental progranm in surface field measurements and radar backscatter measurements. He was assisted by David W. Brandenburg, Larry L. Brown and Ernest C. Bublitz. (U) The re-entry plasma experimental program was carried out by Thomas M. Smith who was assisted by Ernest C. Bublitz and Chester Grabowski. In the theoretical studies under the re-entry plasma program, Dr. Weston was assisted by John J. Bowman, David M. Levine, and Prof. Chai Yeh. (U) In the theoretical SURF studies, Dr. Goodrich was assisted by Dr. Chi-Fu Den and by Dr. Shun-Jen Houng. Working with him on the computer program for the radar section of the rotationally symmetric metallic body were Dr. Robert F. Lyjack and Assistant Professor Frank Stenger, who aided in the analytical work, and Thomas L. Boynton and Peter H. Wilcox who were 382 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F responsible for the programming. Harold E. Hunter carried out supporting computational studies and the hand check of the computer program. (U) Dr. Kleinman and Professor Well were aided in the short pulse study by Nicholas G. Alexopoulos, Jose R. Moron-Borjas, and Jon A. Soper. (U) The above named investigators were assisted by technicians and students. The preparation of memoranda, reports and manuscripts was carried out under the supervision of Mrs. Claire F. White. (U) We would like to express our appreciation for the assistance of Dr. Paul J. Schweitzer of the Institute of Defense Analyses in consulting on the program for the cross section of the rotationally symmetric metallic body; for the cone-sphere backscatter data made available to us by James H. Pannell of MIT-Lincoln Laboratory; and the constructive suggestions of Dr. Sidney L. Borison and Dr. Soonsung Hong of the same organization. We wish to acknowledge the expediting assistance and practical suggestions of Capt. James Wheatley, Contract Monitor for the Space and Missile Systems Organization and the technical critiques of H.J. Katzman and his associates, Dr. William Botch, Dr. Eugene C. Brouillette, Dr. Fred F. Meyer, and Edward N. Skomal of the Aerospace Corporation. An economy to the government and a savings in time to the project was effected by Kurt E. Golden of the Aerospace Corporation/El Segundo who loaned the Radiation Laboratory a "coated cone" model for use in the plasma experiments. _______383 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F X SUPPORTING TECHNICAL STUDIES (U) During the third phase of SURF, four technical reports were written. They are as follows: 1. "Some Facets of Surface Wave Behavior" by T.B.A. Senior, BSD-TR-67 -142 University of Michigan, Radiation Laboratory Report No. 8525-1-T UNCLASSIFIED. 2. "Physical Optics Applied to Cone-sphere-like Objects, " by T.B.A. Senior BSD-TR-67-143 University of Michigan Radiation Laboratory Report No. 8525-2-T UNCLASSIFIED. 3. "Diffraction by the Concave Surface of the Paraboloid of Revolution," by S. Stone, BSD-TR-67-143 AD 818382 University of Michigan Radiation Laboratory Report No. 8525-3-T. UNCLASSIFIED 4. "Surface Currents Induced by a Plane Wave on a Parabolic Cylinder with a Focal Length Comparable to the Incident Wavelength" by S-J Houng and R.F. Goodrich, University of Michigan Radiation Laboratory Report No. 8525-4-T. UNCLASSIFIED The first of these reports deals with the types of surface waves excited by incident microwave radiation on metallic surfaces. It discusses plane waves, traveling and Goubau waves and creeping waves. The second report discusses techniques for estimating the scattering behavior of an object for which the 'effective' dimensions are large compared with the wavelength. The third and fourth reports deal with the surface currents induced on curved surfaces. 384 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN. 8525-1-F REFERENCES Blore, W.E. (1964) "The Radar Cross Section of Ogives Double-backed Cones, Double-rounded Cones, and Cone-spheres," IRE Trans. AP-12, pp. 582-590. Brekhovskikh, L. M. (1960) Waves in Layered Media, Academic Press, New York, Chapter IV. Castellanos, D. (1966), "Notes on Electromagnetic Scattering from Rotationally Symmetric Bodies with an Impedance Boundary Condition, " The University of Michigan Radiation Laboratory Report No. 7741-1-T. Clemmow, P.C. (1966) The Plane Wave Spectrum Representation of Electromagnetic Fields, Pergamon Press, New York. Collin, R.E. (1960) Field Theory of Guided Waves, McGraw-Hill Book Co., Inc. New York Chapter 11. deRidder, C.M., and D.P. White (1966) "Plasma Effects on Body Cross Sections of Slender Vehicles," Project Report PPP-60, MIT Lincoln Laboratory. Erdelyi, A., M. Kennedy and J.L. McGregor, (1954) "Parabolic Cylinder Function of Large Order," J. Rat. Mech. and Anal., 3, pp. 459 -485. Fock, V.A. (1965) Diffraction by Convex Surfaces, Pergamon Press, New York. Golden, K.E. and T.M. Smith (1964) "Simulation of a Thin Plasma Sheath by a Plane of Wires, " IEEE Trans. on Nuclear Science, NS-11, pp. 225-230. Goodrich, R.F., E. Ar, B.A. Harrison, S. Hong, T.B.A. Senior and S.E. Stone, (1965) "Radar Cross Section of the Metallic Cone-sphere: Final Report (U) BSD-TR-66-112, The University of Michigan Radiation Laboratory Report No. 7030-5-T SECRET. Goodrich, R. F., B.A. Harrison, E.F. Knott, T.B.A. Senior, and V. H. Weston, (1966) "Investigation of Re-entry Vehicle Surface Fields, " (U) Quarterly Report No. 7741-3-Q, BSD-TR-66-355, The University of of Michigan Radiation Laboratory. SECRET 385 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F References (Cont'd) Goodrich, Goodrich, Goodrich, Goodrich, R.F., B.A. Harrison, E. F. Knott, T.B.A. Senior, V.H. Weston and L.P. Zukowski (1967) "Investigation of Re-entry Vehicle Surface Fields, " The University of Michigan Radiation Laboratory Report No. 7741-4-T SECRET R.F., B.A. Harrison, E. F. Knott, T.B.A. Senior, T.M. Smith H. Well and V.H. Weston (1967b) "Investigation of Re-entry Vehicle Surface Fields, " The University of Michigan Radiation Laboratory Report No. 8525-3-Q, SECRET. R.F., B.A. Harrison, R.E. Kleinman, E. F. Knott, and V.H. Weston (1967c) "Investigation of Re-entry Vehicle Surface Fields," (U) The University of Michigan Radiation Laboratory Report No. 8525-2-Q, BSD-TR-67-232, SECRET. R.F. and F. Stenger, (1967), "Movable Singularities and Quadrature, " To be published. Hong, S. (1967) "Asymptotic Theory of Diffraction by Smooth Convex Surfaces of Non-constant Curvature, " The University of Michigan Radiation Laboratory Report No. 7741-2-T. Hong, S. and V.H. Weston (1965) "A Modified Fock Function for the Distribution of Currents in the Penumbra Region with Discontinuity in Curvature, " Radio Science, 1., pp. 1045-1053. Ivanov, V.I. (1960) "Shortwave Asymptotic Diffraction Field in the Shadow of an Ideal Parabolic Cylinder," Radiotekhnika i elektronika 5, No. 3, pp. 393-402. Ivanov, V.I. (1963) "Diffraction of Short Plane Waves on a Parabolic Cylinder," USSR Computational Mathematical Physics, 2, p. 225. Johnson, C.C. (1965) Field and Wave Electi Co. Inc., New York, pp. 325-333. rodynamics, McGraw-Hill Book Keller, J.B. (1956) "Diffraction by a Convex Cylinder," IRE Trans, AP-4, pp. 312-321. 386 UNCLASSIFIED II _M

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F References (Cont'd) Kennaugh E.M., and D. L. Moffatt (1965) "Transient and Impulse Response Approximations," Proc. IEEE, 53, No. 8, pp 893-901. Kennaugh, E.M., D.L. Moffatt, R.C. Schafer, (1967) "Research into the Scattering of Electromagnetic Energy from Highly Conducting Bodies," Final report on AF 19(628)-4002, prepared for AFCRL, The Ohio State University, ElectroScience Laboratory, Bedford, Massachusetts. Keys, J.E.,and R.I. Primich, (1959) "The Radar Cross Section of Right Circular Metal Cones-I," Defense Research Telecommunications Establishment, Ottowa, Canada Report No. 1010. Kleinman, R.E. and T.B.A. Senior (1963), "Studies in Radar Cross Sections - XLVIII Diffraction and Scattering by Regular Bodies - II: The Cone," The University of Michigan Radiation Laboratory, Report No. 3648-2-T, Knott, E. F. (1965) "Design and Operation of a Surface Field Measurement Facility, " The University of Michigan Radiation Laboratory Report No. 7030-7-T. Levy, B.R. and J.B. Keller, (1960) "Diffraction by a Spheroid," Canadian Journal of Physics, 38 pp. 128-144. Logan, N.A. (1959) "General Research in Diffraction Theory, Vol. II," Lockheed Missiles and Space Division Technical Report No. 288088. Malinzhinet, G.D. (1959), "Excitation, Reflection and Emmission of Surface Waves from a Wedge with Given Face Impedances," Soviet PhysicsDoklady 3, pp. 752-755. Mitzner, K.M. (1966) "An Integral Equation Approach to Scattering from a Body of; Finite Conductivity," NOR 66-344. Poeverlein, H. (1958), "Low-frequency Reflection in the Ionosphere- I," J. of Atmospheric and Terrestrial Physics, Vol. 12, pp. 126-139. _____ 387 _ UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F References (Cont'd) Rheinstein, J. (1966) "Backscatter from Spheres: A Short Pulse View," Technical Report 414-MIT, Lincoln Laboratory. Rice, S.O. (1954) "Diffraction of a Plane Wave by a Parabolic Cylinder," Bell Sys. Tech. Jour., pp. 417-504, 33. Ross, R.A. (1965) "Radar Cross Section of Rectangular Flat Plates as a Function of Aspect Angle, IEEE Trans, AP-14, No. 3 pp 329 -335. Schweitzer, P. (1965) "Electromagnetic Scattering from Rotationally Symmetric Perfect Conductors, " MIT-Lincoln Laboratory, PA-88. Senior, T.B.A. (1965) "Analytic Numerical Studies of the Backscattering Behavior of Spheres, " AF 04(694)-683, The University of Michigan Radiation Laboratory Report No. 7030-1-T Senior, T. B.A. (1966) "An Approach to the Determination of the Surface Fields on a Coated Cone," The University of Michigan Radiation Laboratory Memorandum 7741-522-M. Senior, T.B.A. (1967) "Physical Optics Applied to Cone-sphere-like Objects," The University of Michigan Radiation Laboratory Report No. 8525-2-T. BSD-TR-67-182 UNCLASSIFIED. Senior, T.B.A. and L.P. Zukowski (1965), "The Interpretation of Some Surface Field Data, " The University of Michigan Radiation Laboratory Report No. 7030-8-T, AD 374993, CONFIDENTIAL, 107 pp. Siegel, K. M. (1959), "Far Field Scattering From Bodies of Revolution, Appl. Sci. Res. 7, pp. 293-328. Siegel, K. M, R. F. Goodrich and V.H. Weston (1959) "Comments on Far Field Scattering from Bodies of Revolution," Appl Sci. Res. 8, pp. 8 - 12. Smith, T.M. and K.E. Golden, (1963) "Surface Impedance of a Thin Plasma Sheet, " TDR-269 (4280-10)-1, Aerospace Corporation, October ____ 388 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 8525-1-F References (Cont'd) Smith, T.M. and K.E. Golden (1965a) "Radiation Patterns of a Slot Covered by a Simulated Plasma Sheet," IEEE Trans., AP-13, No. 2 pp. 285-288, March Smith, T.M. and K.E. Golden (1965b), "Radiation Patterns from a Slotted Cylinder Surrounded by a Plasma Sheath," IEEE Trans., AP-13 pp. 775-780, September Soules, G.W. and K.H. Mitzner (1966), "Pulse in Linear Acoustics," Report ARD 66-60 R, Nortronics (Northrop), Part II by Mitzner alone contains the material referred to. November. Stratton, J.A. (1941) Electromagnetic Theory, McGraw-Hill Book Company, Inc., New York, Chapter 8. Stroud, A.H. and D. Secrest (1966) "Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs, New Jersey. Tamir, T., and A.A. Oliner, (1963) "Guided Complex Waves: Part II, Relation to Radiation Patterns," Proc. IEEE, Vol. 10, No. 2. Wait, J.R. (1960) "The Electromagnetic Fields of a Dipole in the Presence of a Thin Plasma Sheet." Appl. Sci. Res. No. 8, pp. 397-417. Weiss, R.F. and S. Weinbaum, (1966) "Hypersonic Boundary-Layer.Separation and the Base Flow Problem," AIAA Journal, Vol. 4, No. 8. Aug. Weston, V.H. (1963) "Theory of Absorbers and Scattering," IEEE Trans. AP-11. Whittaker, E.T., and G.N. Watson (1927) Modern Analysis, Cambridge University Press, pp. 247-354. Wolff, E.A. (1966) Antenna Analysis, Wiley and Sons, Inc., New York (Traveling Wave Antenna Chapter). _______389 UNCLASSIFIED

SECRET SECRET Security Classification DOCUMENT CONTROL DATA - R & D (Security classification of title, body of abstract land itidexrlnd annotation niu.t be entered when the overall report Is claesilted) 1. ORIGINATING ACTIVITY (Corporate author) 20. REPORT SECURITY CLASSIFICATION The University of Michigan Radiation Laboratory, Dept. of SECRET Electrical Engineering, 201 Catherine Street, 2b. GROUP -- Ann Arbor, Michigan 48108 3. REPORT TITLE Investigation of Coated Re-entry Vehicle Cross Section (U) 4. DESCRIPTIVE NOTES (Type of report and Inclusive date) [ Final Report 5. AUTHOR(S) (First name, middle nliltial, lat name) [ Raymond, F. Goodrich, John J. Bowman, Burton A. Harrison, Eugene F. Knott, Thomas B. A. Senior. Thomas M. Smith and Vaughan H. Weston,,-. - - -- -.- o 0.,. o. oi U- As 6. REPORT DATE 7~. TOTAL NO. OF PAGES 7b. NO. OF REFS January 1968 391 53 Sa. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) F 04694-67-C-0055 8525-1-F b. PROJECT NO. C. ih. OTHER REPORT NO(S) (Any othcer 'number thot may hbe asslned' this report) d. 10. DISTRIBUTION STATEMENT In addition to security requirements which apply to this document and musl be met, it may be further distributed by the holder only with specific prior approval of SAMSO, SMSDI, Air Force Station, Los Angeles, CA 90045 I I..W I II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Hq. Space and Missile Systems Organization Air Force Systems Command 3. ABSTRACT Norton AFB California 92409 (S) This is the final report on Contract F 04694-67-C-0055, an investigation of re-entry vehicle radar cross section, the third phase of a program designated Project SURF. The objective of the SURF program is (1) to achieve the capability to determine the radar cross section of metallic and coated re-entry vehicles which are sphere-capped-cones in shape, or modifications of that basic shape, (2) to determine the effect on radar cross section of the plasma re-entry environment and (3) to study methods for countering short pulse discrimination of these re-entry shapes. Parts (1) and (2) of this program are based upon the interpretation of surface field data obtained on models illuminated by radar in a specially designed facility. Radar backscatter measurements and computer programs are used to check theoretical conclusions. This final report discusses the work carried out in the fourth quarter of this contract and such formulas for radar cross section as were developed and which extend the results previously reported. fnn rFORM 1A7 - vw I NOV 65 I - I JI SECRET S LrgSecret.. Scrc ri t V ti ( ' i ' i c. t ioIn

SECRET CLASSIFIED Securitv Classification I L 14. LINK A LINK B LINK C KEY WORDS RO R 0 L E | WT ROLE | WT ROLE | WT 1 Radar Cross Section Cone-Sphere Re-entry Vehicles Abosrber Coatings Surface Field Measurements Plasma Re-entry Environment Short Pulse Discrimination L. L I L. L L. I lft SECRET CLASSIFIED So tcurh lv (C l.stfi; t,III