THE UNIVERSITY OF I'M I C H I G A N AFCRC-TN-57 -791 ASTIA Document No. 133746 STUDIES IN RADAR CROSS SECTIONS DIFFRACTION BY AN IMPERFECTLY CONDUCTING WEDGE by T. B. A. Senior Scientific Report No. 2 UNIVERSITY.OF MICHIGAN RElPORT No. 2591-2-T CONTRACT No. — AF-19(604) 949 -Octtber'1'957 Approved _ I_____ K. M. Siegel Project Supervisor The research reported in this document has been made possible through the support and sponsorship extended by the AFCRC under Contract AF-19(604)-1949. It is published for technical informa tion only and does not naecessarily represent recommendations or conclusions of the sponsoring agency,

THE UNIVERSITY OF MICHIGAN 2591-2-T STUDIES IN RADAR CROSS SECTIONS I Scattering by a Prolate Spheroid, by Fo VO Schultz (UMM-42, March 1950), W-33(038)-ac-14222, UNtJLASSIFIEDo I The Zeros of the Associated Legendre Functions pm (/o) of Non-Integral Degree, by K. M. Siegel, D. M, Brown, -H Eo HuiAter, Ho A, Alperin, and C. Wo Quillen (UMM-82, April 1951), W-33(038)-ac-14222, UNCLASSIFIED. III Scattering by a Cone; by Ko Mo Siegel and H. Ao Alperin (UMM-87, January 1952), AF-30(602)-9, UNCLASSIFIED. IV Comparison Between Theory and Experiment of the Cross-Section of a Cone, by K. M, Siegel, Ho Ao Alperin, Jo Wo Crispin, Jr,, H. E. Hunter, R. E. Kleinman, W. Co Orthwein, and Co Eo Schensted (UMM-92, February 1953), AF-30(602)-9, UNCLASSIFIED. V An Examination of Bistatic Early Warning Radars, by Ko Mo Siegel (UMM-98, August 1952), W-33(038)-ac -14222, SECRETo VI Cross-Sections of Corner Reflectors and Other Multiple Scatters at Microwave Frequencies, by Ro Ro Bonkowski, Co R- Lubitz, and Co E. Schensted (UMM-106,, October 1953), AF-30(602)-9, SECRET-UNCLASSIFIED when Appendix is removedo VII Summary of Radar Cross-Section Studies Under Project Wizard, by K; M. Siegel, Jo WO Crispin, Jro, and Ro Eo Kleinman (UMM-108, November 1952), W-33(038) -ac-14222, SECRET. VIII Theoretical Cross-Section as a Function of Separation Angle Between Transmitter and Receiver at Small Wavelengths, by Ko Mo Siegel, H. A. Alperin, R. Ro Bonkowski, Jo WO Crispin, Jr,, A. Lo Maffett, Co W, Schensted, and Io VO Schensted (UMM-115, October 1953). W-33(038)-ac-14222, UNCLASSIFIED. IX Electromagnetic Scattering by an Oblate Spheroid, by L. Mo Rauch (UMM-116, October 1953), AF-30(602)-9, UNCLASSIFIEDo X Scattering of Electromagnetic Waves by Spheres, by Ho Well, M. Lo Barasch, and T. A. Kaplan (2255-20-T, July 1956), AF-30(602)-1070, UNCLASSIFIED. XI The Numerical Determination of the Radar Cross-Section of a Prolate Spheroid, by K. M. Siegel, Bo Ho GereP, I Marx, and Fo B. Sleator (UMM-126, December 1953), AF-30(602)-9, UNCLASSIFIEDo ii

THE UNIVERSITY OF MICHIGAN 2591-2-T XII Summary of Radar Cross-Section Studies Under Project MIRO, by K. Mo Siegel, M. E Anderson, R. R. Bonkowski, and W. C. Orthwein (UMM-127, December 1953), AF-30(602)-9, SECRET. XIII Description of a Dynamic Measurement Program, by K. M. Siegel and J. M Wolf (UMM-128, May 1954), W-33(038)-ac-14222, CONFIDENTIAL. XIV Radar Cross-Section of a Ballistic Missile, by K. M. Siegel, M. L, Barasch J. W. Crispin, Jr., W. C. Orthwein, Io Vo Schensted, and H. Weil (UMM-134, September 1954), W-33(038)-ac-14222, SECRET. XV Radar Cross-Sections of B-47 and B-52 Aircraft, by C. E. Schensted, J. WO Crispin, Jr., and K. M. Siegel (2260-1 -T, August 1954), AF-33(616)-2531, CONFIDENTIAL. XVI Microwave Reflection Characteristics of Buildings, by H. Weil, R, R. Bonkowski, T. A. Kaplan, and M. Leichter (2255-12-T, May 1955), AF-30(602)-1070, SECRET XVII Complete Scattering Matrices and Circular Polarization Cross-Sections for the B-47 Aircraft at S-band, by A.L. Maffett, M. L. Barasch, Wo E. Burdick, R. F. Goodrich, W. C. Orthwein, Co E. Schensted, and K. M. Siegel (2260-6-T, June 1955), AF-33(616)-2531, CONFIDENTIAL. XVIII Airborne Passive Measures and Countermeasures, by K. M. Siegel, Mo L, Barasch, J. W. Crispin, Jr., R. F. Goodrich, A. H. Halpin, A. L. Maffett, W. C. Orthwein, C. E. Schensted, and C. J. Titus (2260-29-F, January 1956), AF-33(616)-253|, SECRET. XIX Radar Cross-Section of a Ballistic Missile - II, by K. M. Siegel, M. L. Barasch, H. Brysk, J. W. Crispin, Jr., T. B. Curtz, and T. A. Kaplan (2428-3-T, January 1956), AF-04(645)-33, SECRET. XX Radar Cross-Section of Aircraft and Missiles, by K. M. Siegel, W. E. Burdick, J. W. Crispin, Jr., and S. Chapman (WR-31-J, 1 March 1956), SECRET. XXI Radar Cross-Section of a Ballistic Missile - III, by K. M. Siegel, H. Brysk, J. W. Crispin, Jr., and R. E. Kleinman (2428-19-T, October 1956), AF-04(645)-33, SECRET. XXII Elementary Slot Radiators, R. F. Goodrich, A. Lo Maffett, N. Reitlinger, C, E Schensted, and K. M. Siegel, (2472-13-T, November 1956), AF-33(038)-28634; HAC-PO L-256165-F31, UNCLASSIFIED. iii

THE UNIVERSITY OF MICHIGAN 2591-2-T XXIII A Variational Solution to the Problem of Scalar Scattering by a Prolate Spheroid, by F. B. Sleator (2591-1-T, March 1957), AF-19(604)-1949, UNCLASSIFIED. XXIV Radar Cross Section of a Ballistic Missile-IV (to be published). XXV Diffraction by an Imperfectly Conducting Wedge, by T. B. A. Senior (2591-2-T, October 1957), AF-19(604)-1949, UNCLASSIFIED. iv

TH.E UNIVERSITY OF MICHIGAN 2591-2-T TABLE OF CONTENTS List of Studies in Radar Cross Sections Page PRO Preface Summary Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter References Appendix vi I Introduction I Peters Method HI Formulation of the Electromagnetic Problem IV The Adaptation of Peters' Method V The Difference Equations VI The Relation between 0(r, 0) and F(c) VII The Conditions on H( J ) VIII The Solution for Infinite Conductivity IX The Solution for Finite Conductivity: General Theory X The Chosen Function H ( ) XI Some Elementary Examples XII The Half-Plane XIII Wedges of Angle 3ar/2 and 2r/3 XIV Some Ramifications XV Conclusions viii 1 5 7 9 12 17 23 27 33 41 47 49 54 61 64 67 68 V

THE U NIVE RS ITY OF MICH I GAN 2591 — T PREFACE This paper is the twenty-fifth in a series growing out of studies of radar cross sections at the Engineering Research Institute of The University of Michigan. The primary aims of this program are: 1. To show that radar cross sections can be determined analytically, 2. A. To determine means for computing the radiation patterns from antennas by approximate techniques which determine the pattern to the accuracy required in military problems but which do not require the unique determination of exact solutions. B. To determine means for computing the radar cross sections of various objects of military interest. (Since 2A and 2B are inter-related by the reciprocity theorem it is necessary to solve only one of these problems) 3. To demonstrate that these theoretical cross sections and theoretically determined radiation patterns are in agreement with experimentally determined ones. Intermediate objectives are: 1. A. To compute the exact theoretical cross sections of various simple bodies by solution of the approximate boundary-value problems arising from electromagnetic theory. vi

T-H E UNI VERSITY OF M I CHIGAN 2591-2-T B. Compute the exact radiation patterns from infinitesimal solutions on the surface of simple shapes by the solution of appropriate boundary-value problems arising from electromagnetic theory. (Since 1A and 1B are inter-related by the reciprocity theorem it is necessary to solve only one of these problems). 2. To examine the various approximations possible in this problem and to determine the limits of their validity and utility. 3. To find means of combining the simple-body solutions in order to determine the cross sections of composite bodies. 4. To tabulate various formulas and functions necessary to enable such computations to be done quickly for arbitrary objects. 5. To collect, summarize, and evaluate existing experimental data. Titles of the papers already published or presently in process of publication are listed on the preceding pages. The major portion of the effort in this report was performed for the Air Force Cambridge Research Center under Air Force Contract AF-19(604)-1949. K, M. Siegel vii

THE U NI VERS ITY OF MIC H-IGAN 2591 2-T SUMMARY The problem of the diffraction of a plane electromagnetic wave by an imperfectly conducting wedge is solved subject only to the physical approximation implied by the usual impedance-type boundary conditions imposed on the faces of the wedge. The method is based on one which was originally proposed by A. S. Peters for the treatment of a problem in hydrodynamics and leads to a difference equation for a function related to the Laplace transform of the field with respect to the radial distance from the edge. This is solved exactly to give an expression for the total field valid for any angle of wedge. Several particular cases are examined and some ramifications of the theory are discussed. viii

THE UNIVERSITY OF MICHIGAN 2591-2-T CHAPTER 1 INTRODUCTION During the last few years considerable attention has been focussed on diffraction by structures whose conductivity is assumed to be infinite. In practice, however, the behaviour of actual metallic bodies may differ appreciably from that found theoretically in the ideal case, but attempts to take the conductivity into account have not met with great success. One problem which is of some importance is diffraction by an imperfectly conducting wedge. The first satisfactory treatment of the effect of loss in the wedge faces was carried out by Raman and Krishnan (Ref. 1) who employed the ingenious device of multiplying by the appropriate Fresnel reflection coefficient that part of the two-dimensional Sommerfeld integral solution (for perfect conductivity) which gives rise asymptotically to a plane wave reflected from the wedge. This provides for the loss effect on the reflected plane wave, but though the solution reproduces some of the observed diffraction phenomena, it violates the reciprocity condition concerning the interchangeability of transmitter and receiver. Moreover, the method is inadequate theoretically in that it does not give the solution of the previously specified boundary-value problem. In recent years more rigorous approaches to the problem have been taken by Jones and Pidduck (Ref. 2) and Felsen (Ref. 3), and the two methods are similar in many respects. Both adopt the usual approximate boundary conditions at the wedge faces and, using Green's 1

THE UNIVERSITY OF MICHIGAN 2591-2-T functions, express the solution as a perturbation about t appropriate to a perfectly conducting wedge; this process leads to an integral form for the loss correction due to the finite conductivity. Jones and Pidduck obtained a formal expression for the correction term when the incident field is that of an electric or magnetic line source parallel to the apex of the wedge, but considered in detail only the case of an incident plane wave. The integral was further simplified by assuming the point of observation to be far from the apex and well within the shadow region, but even then its form was not such as to be amenable to numerical calculation. Felsen (Ref. 3), however, has shown that a reduction of the perturbation integral can be achieved without approximation in the case of an electric current source, the result being valid for arbitrary positions of opurce and observation points and for wedges of open angle less than r. A series representation of the correction term is given and this is rapidly convergent when the source or point of observation is near the wedge apex. For plane wave diffraction observed at large distances, an asymptotic calculation of the solution yields a set of plane waves reflected from the wedge faces with reflection coefficients appropriate to the material of the wedge, together with a cylindrical wave which appears to emanate from the apex, and this is analogous to the result obtained by Raman and Krishnan. It should be noted that both the methods just described involve the assumption that the solution can be expanded as a series in ascending powers of 4, where This the reciprocal of the complex refractive index of the material comprising the wedge. Unfortunately this may 2

THE UN I VE R SITY OF MI C H I GAN 2591-2-T not be justified and in the particular case of a metallic half-plane it has been shown (Ref. 4) that only for an H-polarized incident field, (magnetic vector parallel to the edge) can the exact solution be so expanded, that for the other polarization containing a term a log. Accordingly, any treatment of the more general problem based upon this type of expansion is of doubtful validity, especially when the incident field is E-polarized. In order to verify whether such an expansion is possible, it is necessary to solve the problem subject only to the (physical) approximation implied by the assumed boundary conditions, In the mathematical sense the solution is then exact, but the existing methods in diffraction theory are insufficient for its derivation. Whilst the Wiener - Hopf technique is capable of treating the half-plane, the corresponding integral equation is not susceptible to solution in this manner when the two surfaces of the diffracting structure are no longer parallel. If the structure is perfectly conducting the Sommerfeld method, based upon multi-valued solutions of the wave equation, applies equally well to the wedge and the half-plane; the procedure, however, is of a largely intuitive nature and does not lend itself to problems with mixed boundary conditions. Finally there is the method proposed by Kontorovich and Lebedev (Ref. 5). This is ideally suited to the wedge problem and when the conductivity is infinite the resulting integral equation can always be inverted directly using a special type of transform relation, whilst its logical nature suggests that it may also be applicable to an imperfectly conducting wedge. Unfortunately, this proves to be impossible since the integral equation is no longer capable of direct inversion. Moreover, to seek a solution by successive substitution is equivalent to expanding the unknown function as a series in powers of A, and even this technique is limited to the case of H-polarization

THE UNIVERSITY OF MICHIGAN 2591-2-T by the way in which tenters into the equation. In his quest for alternative methods the author was led to consider that developed by Peters (Ref. 6)for the exact solution of a problem in hydrodynamics. The basis of the method is the expression of the differential equation and the boundary conditions as a difference equation for the determination of a regular function whose real part represents the velocity potential, and although the method has been extensively criticised (the integral solution obtained by Peters is not convergent), it can be modified to give the exact solution for diffraction by an imperfectly conducting wedge. 4

T H E" U NJI V E -R S I T Y OF M I C H I G A N 2591-2-T CHAPTER II PETERS METHOD.' -;..,.-'' - In reference 6, Peters derives the solution of a mixed boundary-value problem for the equation (v2-4) 0 = 0 in a sector, the problem arising in connection with water waves on a sloping beach. In termsof polar coordinates (r, 0, z) the method enables solutions to be'found for the two-dimensional equation rr + + 1 -k2 0 <1) r r in a sectori 0 <Q(' nder the boundary conditions = \ r P for 0 O (2) and r-= 0 for = r (3) where X is a real constant and the suffices denote differentiation. The first condition is that for a free surface whilst the second is for a barrier and expresses the fact that there is no motion normal to it. Solutions of (1) representing progressive water waves are found by prescribing the required behaviour at the origin and at infinity. The important stages in the method are (i) application to the differential equation of a Laplace transform with reppect to the radial distance r; (i) change of the transform variable from s to p to yield Laplace's equation for a function 3(p, 0); (iii) identification of yp,0 ) with the real part of a complex function f(t) regular in 0 < Im. w <,, where wp i4; (iv) expression of the boundary conditions as conditions upon f(.) and their reduction to forms suitable for the use of an image' technique; and (v) derivation and subsequent solution of a difference equation for. f(.. 5.

THE UNIVERSITY OF MICHIGAN 2591-2-T Although specifically developed for a discussion of water waves on a beach, the method can be applied to most problems which involve the solution of (1) in a sector of arbitrary angle 7; boundary conditions other than (2) and (3) can also be admitted, -but the reality of these is essential for the success of the method as it stands. The diffraction of an electromagnetic wave by a metallic wedge is a similar problem to that considered above and it is natural to attempt a solution in an analogous manner. Instead of (1) we now have the wave equation, but the change in sign of k2 affects the method only as regards the transformation from s to p. The boundary conditions are also of the general form (2) but differ in that X is no longer real. This appears to prevent the conditions from being cast into forms suitable for imaging, but the difficulty can be overcome by treating separately the real and imaginary parts of the function )b(p, 0) and with these modifications the method serves to provide an exact solution of the diffraction problem. 6

THE UNIVERSITY OF MICHIGAN 2591-2-T CHAPTER II FORMULATION OF THE ELECTROMAGNETIC PROBLEM The wedge is defined in terms of the cylindrical polar coordinates (r, 0, z) by the equations 0=0 and 0=, where ~ is the open angle of the wedge. For simplicity the incident field is taken to be a plane wave normal to the apex of the wedge (z axis) and at an angle a (0 a T r ) to one of the bounding faces, and if the magnetic vector is assumed to lie entirely in the z direction (H-polarized), it can be represented by ikr cos(0-a) (4) i (r, 0) = eikr () where the affix tit denotes the incident field. M. k s. units are employed and a time factor eikt suppressed. Since the diffracting structure extends to infinity in the z direction, the use of the plane wave (4) implies that the problem is two-dimensional, and hence the solution can be expressed in terms -of a function 0(r, 0) signifying the z component of the magnetic vector in the total (incident plus scattered) field. The wedge itself is regarded as composed of a homogeneous material whose complex refractive index is e V I o. kcEo where E and cr are the permittivity and conductivity of the material and o is the permittivity of free space. Clearly 0 arg y. /4 with the two extremes corresponding to a pure dielectric and a perfect conductor 7

THE UNIVERSITY OF MICHIGAN 2591-2-T respectively. At the faces of the wedge the usual approximate boundary conditions are imposed. These are p -ik r r0 = O for 0 = 0 (5) and pe +ikN r, = for = T (6) (for their deriviation see, for example, Griinberg, Ref. 7) and the problem now confronting us is the exact solution of the wave equation J + L A ( + - ^+ 1 =,0 (7) rr r r r ( ) under the conditions (5) and (6), together with certain additional conditions (to be discussed later) at the origin and at infinity. 8

THE UNIVERSITY OF MICHIGAN 2591-2-T CHAPTER IV THE ADAPTATION OF PETERS': METHOD Consider the Laplace transform 00 -sr 0* (so,0) e P pr. 0 dr (8) o at the outset s is regarded as a real parameter and the transform is supposed to converge for s ~9, where e is a small positive number. If rp(r, 0) — 0 as r —-0, application of the transform to equation (7) gives (k + s2)* +3s P*S +*+ cd) ~0 that is, ~~__ __ 2 * I0 __ (9) ([k2 + s2) + (sp,*) + -0 (9) 3s ds <s for 0 < 0 < and s At first sight this does not appear to be simpler than (7), but when s is replaced by s =k sinh p (10) it is found that the functionl (p, 0) defined as (p, 0) - k cosh p * (k sinh p, 0) (11) satisfies Lkpp'' + -0 (12) Thust /(p, 0) is a harmonic function of p,0 in the semi-infinite strip 04 < d 0< sinh- E/k p < Co. Furthermore, from equations (5) and (6) the boundary conditions on the function are -0 cosh2 p + i f ('p cosh p - / sinh p) = 0, (13) 9

THE UNIVERSITY OF MICHIGAN 2591-2-T 4 cosh2 - i ( cosh p - sinh p)= 0, (14) for 0 and 0 and 0 = r respectively. As noted in Chapter II the technique which is adopted is to separate 8L (p, 0) into real and imaginary parts. Let (po ),1 (pi ) +i 2 (p 0) where and 2 are real functions of the two real variables p and 0. These functions individually satisfy equation (12) and are subject to the boundary conditions p1 2 2 2 8) 1 cosh p- (a9 + b -I )coshp- (ali2 +bb) sinhp} =0, (15) 29 cosh2 p + { (a -b 92 ) coshp- (a/ -b4 2) sinhp] =0, (16) at 0 = 0, and 4 cosh2 p+ { (a /2+bl)) coshp- (a/2+b'l1) sinhp o0, (17) 9 p p I cosh2p - f (al, b -bp2) coshp- (al -b-'b2) sinhp} -0, (18) cosh P p - at 9 =, where a + ib with a and b real. Following Peters, p and 0 are now regarded as the real and imaginary parts respectively of some complex variable o. We shall write p + i8, (19) but it must be emphasized that the complex notation is not the same as that which is implicit in the problem. Although the same symbol i is used in both cases to denbte' F-1, no confusion should arise since the two notations are never simultaneously:emplo1 Indeed, the new'i' is introduced solely for the purposes of the analysis and its presence implies that the analysis is carried out in a three-dimensional space which is in some respects analogous to the Riemann space used in the Sommerfeld method for treating 10

THE UNIVERSITY OF MICHIGAN 2591-2-T diffraction by a perfectly conducting wedge. Let f(w) and g(w) be two functions regular in the strip S: 0 < 0 <', < sinh- ~/k < p<o. The harmonic functions'1 (p, ) and2a (p, 0) can then be identified with the rbal parts of f(u) and g(w) respectively, so that / (p, ) Re. f(w) (20) and:2 (p,0) Re. g(tw), (21) and in terms of f(w) and g(o) the boundary conditions (15) - (18) become Re. i cosh2 wf'(w) - { (ag'(w) + bf'(u) cosh w - (ag(w) + bf()) sinh ] 0 (22) Re. [i cosh2 wg'() + {af'(w) - bg'(w)) fosh - (af(t) - bg(w)) sinh w=0 (23) at 0 = 0; Re. [i cosh2 (w-i ) f'( ) + I(ag'() + bf'(w)) cosh(w-i ) -(ag(o) + bf(c)) sinh (w-i P )}] 0 (24) Re. [i cosh2 (w-i ) g'(w) - {[af'(w) - bg'(w)) cosh (w-i') - (af() - bg()) sinh (w-i?)] =0 (25) at 0 =, where the primes denote differentiation with respect to w. The problem remaining is therefore the determination of two functions regular in S and satisfying these conditions. 11

THE UNIVERSITY OF MICHIGAN 2591-2-T CHAPTER V THE DIFFERENCE EQUATIONS The form of the conditions (22) - (25) enables us to analytically continue the functions f(to) and g(wc) across the lines Im. t=0 and Im. wt T. The way in which this is carried out may be illustrated by considering equation (22) in which the expression in square brackets maps the lower boundary of the strip S (that is, the positive real axis of the w plane) into part of an imaginary axis. Consequently, the expression can be continued across the real axis using Schwarz's reflection principle, and for t in S we have* i cosh2 wf'() + {(ag(w)) + bf(w)) cosho - (ag() + b(w)) sinh w = i cosh2 wf'(tu) - {(ag (t() + bf'()) cosh w - (ag(w) + bf(t)) sinh ol. (26 Similarly, from equations (23) - (25) i cosh2wg'(w) - (af'(w) - bgt)). coshto - (af(T) - bg()) sinh j = i cosh2 wog'(w) + (af(w) - bg(w)) cosh w - (af(u) - bg(w)) sinh w, (27o i cosh2 (tif) f() - (ag'(t) +bf'(t)) cosh (w-i I6) - (ag(t) +bf(o)) sinh (w+i)] = i cosh2 (w+i T) f'(w+2i T) + (ag' (w+ 2i' ) + bf'(t+2i )) ) cosh (toi r) - (ag(t+2i f ) + bf(w+2i r)) sinh (u+i ) ] (28, i cosh2 (+i ) g'(w)+ {(af'(w) - bg'(o)) cosh (t+i T) - (af(w) - bg(w)) sinh (wi ) i cosh2(w ) g'(+2i ) (29; - (af'(o+2iT) -bg'(wo+v2it)) cosh(tiaT) - (af(tZ2iT)-bg(t+2ir)) sinh (t+inT) Together these provide the analytical continuation of f(w) and g(t) from the strip S into t] We write w = p-iO for the conjugate of w, f(w) for the conjugate of f(w); f(w) is f(w) with conjugate coefficients and f(u) is f(o) with to replaced by o, so that f(w) = f(o). 12

THE UNIVERSITY OF MICHIGAN 2591-2-T wider strip - rt4 Im. < 3 After some re-arrangement, the above equations can be integrated directly, giving rise to four equations involving f(O), g(w) and their conjugates: |icoshw-b3 f(w)-ag(ow) i coshw+b} f(U)+ag(w) +c (30) (itcoshw-b} g(w)+af(w) ticosh w+b\ g(u) -af(w) +c (31) i cosh (o-+iy ) + b} f(o+i2ir ) + ag(w+2i ) = i cosh (to+i ) - b' f(w) - ag(w) + c3 (32) {icosh(o+i ) +b| g(w+2i' )-af(w+21i6 ) = i cosh (w+i t ) - b] g() + af() + c4 (33) where c, c, c3 and c4 are arbitrary constants. Elimination of the conjugate functions then leaves [ cosh (x+i) - b {i cosh w - b -a2 g(fo) - [ i cosh (wc+if ) b {i cosh u - b} - a2 g(a+h2it) = - a i { cosh(o+i r ) + cosh wt -2b f(tw) -a [i { cosh (w+i )+cosh wa +2b] f( 2i) + c - a(c + C3) + (icosh t - b) (c + c4) (34) 2 4 and i cosh (+i r ) -b] icoshw -b} -a2] f(t) i cosh (t+i )- bI i cosh w - bJ -a2 f(+2i) = a [i cosh (w+i ) + cosh w - 2b g(y) +a i { cosh (o i cosh + 2b g(w+2i ) + c + a(c c4) +cL oi + c 2 4 + (i cosh t - b) (c +c )c (35) 1 3 13

THE UNIVERSITY OF MICHIGAN 2591 -2-T If we now introduce the functions Fg(c) and G(w) defined by F(w) f() + i g() (36) Ggo) = f(w) - i g(x) (37) we have, on multiplying equatiow(35) by i and subtracting from and adding to, equation (34), + cosh ((w+if ) t \ + cosh F(o) = {-cosh(+i t )] - cosh w F(+2i f ) - c - ic + (ai + b - i cosh )(cc + ic + c + ic), (38) I 2 1 2 3 4 i{- cosh (+i T )} { I -cosh w G(w) 1 |T + cosh (w+i t )} 5+ cosh w) G(-+Z2i T ) - c + ic - (ai-b+i cosh w)(c - ic + -ic ) (39) 1 2 2 4 respectively, where = a-ib. These linear difference equations are necessary conditions which must be satisfied by the functions F(t) and G(o), but for our purposes it is sufficient to consider the corresponding homogeneous equations which follow from neglecting the arbitrary constants c,.... c4o The conditions upon F(w) and G(e) are therefore l+ cosh(w+i'P.) Y + cosh w F(w) - cosh W +i {-cosh w (+2i, (40) >- cosh(+i cosh i - + coosh ((it )T\ + cosh w (U((+2i). It is clear from the symmetry of equations (40) and (41) that if X(x, v\) is a solution of (40), a possible solution of (41) is G(w) = XS - ), although this may not be the required solution. Nevertheless, from (30) and (31), 14

T HE UNIVERSITY OF MICHIGAN 2591-2-T G(O) )= + sc6h F(w) (42) v - cosh w G+o = cosh to F(o) (43) G(l) =_ -' +cs G cosh w and thus the two cannot be chosen independently of one another. To specify (for example) F(o) automatically determines G(t) using equation (42) and for this. reason it is sufficient to restrict attention'to equation (40)0 To reduce it is a form amenable to solution we make the substitution _J et (44) which maps a strip of the w plane on to a sector of the J plane As it stands the transformation is multivalued, but a one-to-one correspondence can be established by inserting a cut into thej plane, the precise location of the cut being immaterial providing the strip 0 K Im. t < is mapped on to the sector 0 < arg < g Let F(o) = H(g ) (45) so that F(t^2i'68)H(5 ei ). Since ei-i -i i v + cosh(to+i ) -\ ) + e )+ =V 2 e and i~ -i^ -i. \V -cosh (+i )= - ~e - e )' 4-Te ), 2 2 j' ) Xl 2 where =v +i i2 and -i - (46) 1.' ~~~~~~~2 15

THE UNIVERSITY OF MICHIGAN 2591-2-T equation (40) becomes (j+C e +~C e e 5.)(3 +T ) H(j (47) = (f -Z e i)(_ - % e" )(; - C )( -1C,) H(_ e2 i). The solution of this equation is the crux of the problem now remaining. 16

THE UNIVERSITY OF MICHIGAN 2591-2-T CHAPTER VI THE RELATION BETWEEN 0(r,) AND F(j) Before proceeding to the solution of the difference equation it is necessary to consider the way in which 0(r, 9) is to be expressed in terms of F(w), Only by doing so it is possible to formulate in their entirety the conditions to be imposed on the required function F(w). It will be remembered that the Laplace transform 0* (s, Q) led us to a function (p, 0) whose real and imaginary parts were represented as the real parts of the functions f(w) and g(w) respectively. Moreover, f() = { F() + G() ), 2 g(w) = L G(w) -F(W)] 9 2 and if F(V) and G(w) are written as F() = A(p, ) + iB (p, 0), G() = C (p, ) + iD (pi 0), where A, B, C and D are real functions, then Re. f()) = (A(p, )+C(p, 0)} and Re. g(y) = 1 {B(p, )- D(p, 0) But. (p, ) =( -1 (p, 0) + /2 (p,0) Re. f(u) + i Re g(w), where the'i' is that which implicit in the problem, and hence 3/^ (P, ) = i A(p, 9) + i B(p, 0) + C(p, 9) - i D(p, 0)T = eF(w) +, G(). (48) By using equation (43) we now have 17

THE UNIVERSITY OF MICH I GAN 2591-2-T (pl) = Fw - F( coshuo Fp)} (49) 2 M Ll^ - cosh 3 which gives the functions (p, 0) directly in terms of F(w) This result is important in enabling us to pass directly from the complex notation of Chapter IV to that of Chapter III without having to separate F(x) and G(ws) into their real and imaginary parts. The naturei of the function 4(p, 0) is such that F(w) and G(w) can be taken over in their entirety and in going to equations (48) and (49) the analytical'i' has been displaced by the naturally-occuring'it of Chapter III The form of equation (49) suggests that it may be possible to devise a more straight-forward derivation which would avoid the necessity for introducing the functions 7 (p, 0) andL2 (p, 0). This is indeed so and the new method turns out to be very much more concise. However, owing to the two complex notations which are now simultaneously employed, the steps in the argument are more difficult to follow, and for this reason the original treatment is to be preferred. When a suitable expression for F(c) has been found we have, from equations (11) and (49), * (s, ),,., i/ (p, 0) k cosh p = 1 F(p+i0) _ + cosh (p-i) F(p 2 k cosh p ( - cosh (p-iO) J (50) where k sinh p = s. So far s has been regarded as a real quantity, but from the theory of the Laplace transform (3* (s, 9) is known to be a regular function of s in the 18

THE- UNIVERSITY OF MICHIGAN 2591-2-T half-plane Reo s Z) o Consequently, if s is complex, the right hand side of (50) is a function of the complex variable p which is regular in the domain corresponding to Re. s > E, and 0 then takes on the role of a parameter. The inversion formula for the Laplace transform gives (r, 0) - _i e * (s<90) ds, (51) where the path of integration is a straight line parallel to the imaginary axis and to the right of all singularities of A* (s, 0@) The transformation to the p plane is effected by s=k sinh p, which maps the whole of the s plane on to each of the strips (m-l/2) T Imo p (m+2) ra m 0, + 1 + 2, o O. but though the transformation is multivalued, the range of p cannot be chosen arbitrarily. The choice of strip is governed by the initial assumption that real s and p correspond, and this implies that the s plane is mapped on to — r/2 < Im. p < /2 (52) Since the right hand s half-plane corresponds to the right hand half-strip in the p plane, r, e- 4 i ekr sinh p 1 F(p+i0) - - s c ) F(p-i) dp, 4) 4i I' - cosh (p-iO) where the path of integration Q in the p plane is a loop from p oo -i /2 to p=oo +i r/2o If we now write Pli(/2 -7 ) mapping the above strip of the p plane on to a displaced and rotated strip of the complex V plane, then 0(r. 0) = - eik cos i( F ) + (53) +sin({+) F i) da r - sin(v +0) 2 19

THE UNIVERSITY OF MICHIGAN 2591-2-T where,)" is a loop path from 7) + ioo to/ ico whose precise form between these points is immaterial providing it lies above all singularities of the integrand. In general the transformation from s to 7v using s ik cos B' will introduce branch points into the - plane at ) m=, m = 0 + 1, + 2,... o and it is natural to take the resulting branch cuts parallel to the imaginary p axiso The transformation, however, may serve to remove branch points, a multivalued function in the un-cut s plane becoming single-valued in the un-cut zv plane. This will certainly be true if the function F i ( _ - 2)+ satisfies the following conditioin (I) In - - Reo 7^ w, FP iit -[, ] has only a finite number of poles and is otherwise regular and free of singularities for all 6s 04 8 0 T. Since the integrand of (53) is then single-valued in the un-cut V plane, deformation of, It outside the strip to which it was originally confined is possible and, moreover, necessary if the integral is to converge. From a study of the regions in which the path may approach infinity, we are led to choose the path L of Figure 1, and whilst it is convenient to take the end points to be 7 = 3 t/2 + ico and V -J/2 + ico, any path commencing at infinity in the strip r < Re. V < Zi and ending at infinity in the strip -~ 4 Re. V< 0 will do providing all singularities of the integrand lie below it. If it is further: assumed that 61 - sin(h +^ #l) 2' 2' J 20

THE UNIVERSITY OF MICHIGAN 2591-21-T then (r,) = -1 P(r,.6) = I ikr cos?/ e L F i (ff - 7) +0) 2 = - 1, L L ikr cos -) e F ir -v+7)+0) F i( - i[ d z (54) where L' is the path shown in Figure 1o L -27 2w s(-r) FIGURE 1 21

T H E U NI V E R S ITY O F M I C H I GA N 2591-2-T In view of the condition (I) the only1 signularities between the two paths are poles and hence (r,) = 4- - e F i( - v+ 0) d-v - i S(i) - [ J 2 R with R denoting the sum of the residues at the included poles. It now only remains R to make the substitutions p = 7+ r on S(r) and 3 = V) - r on S(-r) to give (r8) = e e-ikrcos {F [ -+e] -?F (0); [ -2 iR 0(55) where S(Q) is a steepest descent path through the origin, and this is a solution of the desired form. If the poles of F i -V + ) are chosen correctly, the term L will [ 2.R give the geometrical optics contribution and the integral can then be interpreted as the diffracted field. The expression for (r, 0) as a whole will represent a possible solution of the diffraction problem if a function F can be found satisfying our assumptions, and it will be shown that the conditions (I) and (II) are not only consistent with those imposed on F in the original analysis but also serve to specify a unique solution of the problem. 22

THE UNIVERSITY OF MICHIGAN 2591 -2T CHAPTER VII THE CONDITIONS ON H( Let us first review the way in which the difference equations (40) and (41) were derivedo The functions f(w) and g() defined by (20) and (21) are, by virtue of the wave equation (7), regular in the strip 0 < Imo w < ~ and consequently F(w) and Gts) are also regular in this strip. The boundary conditions on the faces of the wedge enable F(s) and G(w) to be analytically continued into a wider strip -' < Imo s) < 3', the values of the functions in the wider strip being given in terms of their values in 0 < Im.o t < t by the difference equations (40) and (41). These equations are directly analogous to the ones obtained by Peters (Refo 6) in his discussion of the water wave problem. 0O 2 * t2 Because of the relations (42) and (43) coimecting F(e and G() it is sufficient to consider only equation (40), and this can be convwerted into a difference equation for a function H( ) by means of the transformation n t44) he a plane the function HQ( ) must be regular in that region which corresponds to the strip S of the 23

THE UNIVERSITY OF MICHIGAN 2591-2-T w plane; it must therefore be free of singularities in >: l < exp i sinhl f O | < arg~ < and since ~ is small, this is essentially the sector 0 < arg <' with the inside and boundary of the unit circle excluded (Figure 2), The behaviour of H(5 ) for large j [ is specified by the form of 0(r, 0) when r is small. For a half plane ( 6 2ar) of infinite conductivity the admissible edge condition has received considerable attention and it is now acc-1 epted that no field component can have a singularity of greater order than r 2 as r —0. In itself, however, this does not serve to ensure uniqueness when the field is a two-dimensional H-polarized wave incident in the plane normal to the edge since all components are finite for r=0 This point has been brought out by Clemmow (Refo 8),who shows that the condition becomes sufficient if the field is made three-dimensional and, in fact, for an H-polarized wave inpident on a half-plane, H \ constant as r — 0. z For the actual case of a wedge or corner the work of Jones (Ref, 9) has more direct application and leads us to impose the same condition on the wedge as on the half-plane. Although Jones was concerned only with perfect conductors3 the presence of finite conductivity is unlikely to affect the general nature of the singularity3 and we shall therefore assume that i(r, 0) %d constant as r —- 0, (56) which implies H( g )'u constant as | ) — o. (57) 24

THE UNIVERSITY OF MICHIGAN 2591-2-T In passing it should be noted that the conditions: 0 remains finite and r/ar-,0 as r- 0 used by Pauli (Ref. 10) in his analysis of the solution for a perfectly conducting wedge are satisfied by a function whose behaviour is given by (56). This is also true of tlee condition: r 0 —40 as r —-0 assumed in the derivation of equation (9), The difference equation (47) is valid independently of the field incident on the wedge, but the precise form of the solution is, of course, governed by the incident field. For the simple case of the two-dimensional plane wave (4) it can easily be verified that if the incident field is taken by itself A}(pOt) = -2cosh p sinh p - i cos (0-a) and this influences the choice of H(9 ) To produce the required geometrical optics field (and, in particular, the plane wave 4), it is necessary that H(g ) should have one or more poles of suitable residues, a fact which is equivalent to specifying the behaviour of 0 (r, 0) for large r. In addition to the physical conditions described above there are three others which are more intimately connected with the final reduction of the expression for a (r, ). Two of these have been listed in Chapter VI and can be translated into the following conditions on H( f ) (I) with the exception of geometrical optics poles, H(_ ) is regular in -r/2 < arg C 3r/2 + K for all I, a fact which ensures the'partial' regularity in -Ir/2 + 0arg 3f/2 + forany 0, 0 < <t (II) (J + CI)( + 2 - ( e ) H 2= HL(_)-H( — 1 (58).25

THE UNIVERSITY OF MICHIGAN 2591 -2T where'p -i 0. The third condition concerns the residues at the geometrical optics poles and amounts to a restriction on the allowable zeros of H( i ). It guarantees that the solution is finite for al angles a at which the plane wave is incidentd but is most conveniently formulated after an examination of the perfectly conducting wedgeo

THE UNIVERS ITY OF MICHIGAN 2591 -2-T CHAPTER VIII THE SOLUTION FOR INFINITE CONDUCTIVITY When Y 0, 0, andT, reduce to + i respectively and the difference'1 2 equation (47) collapses to H( )- H( e l (59) This represents the equation for a wedge of infinite conductivity and the solution H(s ) = ho( ), say, must give rise to the Sommerfeld expression for B(r, 9). For large and small r the general behaviour of 0(r, 0) is independent of the conductivity of the wedge and hence the conditions to be imposed on ho(? ) can be deduced from Chapter VII~ They are 2i~' (i) ho(l ) n g 2i 9 (ii) ho(? ) is regular in (iii) ho(3 ) v constant as 1j \ 10^, and (iv) h ( ) has one or more poles whose residues serve to provide the geometrical optics fieldo With the exception of these poles, hob( ) is regular in -/2 arg /3 + 2 There is also the condition ho( )= ho (~ ) 60) (see equation 58), but this satisfied automatically by a choice of ho( ) in accordance with the above The solutions of equation (59) are of the type (1) ho) A(: ) (61) 27

THE UNIVERSITY OF MICHIGAN 2591-2-T where A is a single-valued function of its argument', and this leads us to propose the following expression for ho ( 6 Q Q hQ( ) + 1-e q 1r( (6Z) h( = -Q — 1- -62) I _-e-,/ i'e-qJ: -'Y/h Q, Q, qx and q are constants whose values have yet to be found and for this 2 2 purpose a knowledge of the incident field alone is sufficient. The expression given by (62) certainly fulfills the conditions -(ii) and (iii), and is of such a form as to enable (iv) to be satisfied. The constants will now be evaluated, From equations (53) and (62) we have, when = 0 p (r, a) _ _ 1 ) ikr cos -v - Q ~(r, 0) 4- 1,__ ee l c sin__ /.. 2n 4r i n -in/ 22/-+)/" 1~e R e 2' 2 z a Q +.1 - I. i ^ r -7) - Q 2 dv + (63) I e- lel 2' 1 e2 e -in/2(t/ — ) -0) where n = 2/. The poles of the integrand are at Re. V (4 + i O- lm9 Im.q i m -Z 2 Re. q; (64) n 2/ ~ n n n Re. - +-). r+O 2 Im. q ImoV = Re. q (65) wt\n 2-C nO + +2... n 2 with N ^ 0, +1, +2,.... and since the lncident field is 0'rg - e~ikr cos(O-a) (4) it is clear that the poles must occur for real values of 7), requiring that ql and q be pure imaginary. The geometrical optics poles therefore lie on the realV axis and will be included in a positive sense in any deformation of the path of integration As yet the path is confined to the strip 0 Re. o 4 wr and consequently only poles on this part of the axis can be absorbed 28

THE UNIVERSITY OF MICHIGAN 2591 -2T into the lower half -Vplane. The incident field (4) will be provided by poles of the form + (e0) + 2r, M 0 +1, 2 0 0 0 0. which coincide with the poles (64) of the first term in the integrand of (63) if 2 Im. q =2 (2N-M +1 Wr + 9 + (- ) ) n 1 n 4J, the lower sign may be discounted since q is independent of 0. Hence Im. q (2N Mn+n) P ++ 1 4 2 and the corresponding poles lie in the interval (0,r) if a - 2M~r < 9 a + T - 2Mr. Since 0 0<, a 2 2 Ir/n with n/, he only admissible value of M is M = 0 We therefore take q ~ i n / + \ (66) 1 2 VT 26 and if Q n (67) 2 ai 1 ilkr cos (0-a) the resulting pole at z - a has residue. ei os ( With this choice 2ri of Q and q, the third term in (63) provides no contribution to the incident field through the residue at its pole. The illuminated region of geometrical optics is 0 4 min (( +a, 2r/n)) (68) and that part of the range which has not yet been accounted for is covered by the fourth term in (63). If the above process is repeated for this term9 it is found that 29

THE UNIVERSITY OF MICHIGAN 2591-2-T q =i n_ ( -a) (69) 2 2. 2 and Q =no (70) 2 The poles of the first and fourth terms in the integrand of (63) then yield the required incident field over the entire range (68), whilst the second and third terms provide the rest of the geometrical optics field ----- a fact which can easily be verified. We thus have F i ( - ) n 1______ 2 in/:V -0-a) -in/2( V"- t) J v1 -e 1 -e (71) and the corresponding function ho( ) is h -n -/z-n/7 - n -e/2(/2 2 1- ei2+a) n/1 i (72) n inr/2 n -..... (73) n n/2 einr/ 4 cos n' + einl/2 ^" 2j e cos 2 This has the required properties and, in particular, satisfies the condition (60)o If any further poles had been included in the expression for ho(J ) there would have been residues additional to those of geometrical optics which would have violated the condition upon the behaviour of 0(ra 0) for large r. For this reason it is thought that the conditions (i) - (iv) in conjunction with (60) are sufficient to determine a unique 30

THE U NIV ER S I T Y OF MIC'H IGAN 2591 2 -T function h ( ). Obviously it would have been possible to arrange for the incident field to be provided by the poles of te second and third terms in the integrand of (63) and yet another approach would have been to impose the condition (60) at the beginning, thereby relating Q and q wii Q and q If desired the poles in the whole range -r z Re o 7) r could now be absorbed and the incident field would then follow from the ie resdue of the first term only With both methods, however, fhe final expression for h( f ) is identical to that of equation (72)o The main reason for giving a detailed derivation of h )( ) i is that this function is part of the complete solution of the difference equation (47). Nevertheless, hoj ) by itself does represent the solution appropriate to a perfectly conducting wedge and whilst this problem is hardly one for which the present method is well suited (for exampe,. te method proposed in Refo 5 is more simple and straight-forward, it is of interest to follow the solution lhrough to its conclusion. The geometrical optics field produced by he fnction h( f )I is of the expected form and need not be discussed, but for the diffracted field we have AD r"/J} nik- so f (r,{T X JsiO) e d: kr cos3 F 7 [ 1 1 4 Jg^ e o emlin//+Oa) + 91 (o~ 31

THE UNIVERSITY OF MICHIGAN 2591-2-T the terms of which can be combined in pairs to give rO) sin n __f D(0) 0ni J_ ero) _2 O_ Cos ^~~1 n. ^ at ^ 1 e ^ T cos 2 I + - do (74) n(+0 —.a) Co nr cos ~2- c 2 2 This is the well known Sommerfeld formula. Only in the particular case n 1 I (half-plane) is the integral capable of exact evaluation, and for other values of n it is necessary to use asymptotic methods assuming r large (see, for example, Ref. 10). It should be noted, however, that if /2 is an integer m, ( (r, 0) vanishes, which confirms that a wedge of angle- /m can be treated using a finite number of plane waves. This is true even if the wedge is imperfectly conducting, the complete solution being a set of plane waves whose directions can be obtained from image considerations.

THE UNIV ERS I TY OF MIC H I G A N 2591 -2T CHAPTER IX THE SOLUTION FOR FINITE CONDUCIVITY; GENERAL THEORY The singularities of H( j ) which are necessary in order to produce the required geometrical optics field are identical with those of hO and this suggests that some simplification can be achieved by writing H( h l h H g, L (75) where H ( ) also satisfies (47) h ) is the function derived in Chapter VII and since this represents the solution for a wedge of infinite conductivity, the above separation implies that H(? )I~ as - 0. The conditions governing the choice of H ( $ will be discussed in the next chapter, and for the moment we note only e requirement that H ( ) be regular in o Let h( > o) be a function wich is regular in the larger sector 0 arg < 2 for all. | >Oand whih satisfies ef) h( c,e) - ( )h( e o e) (76) i where c is some parametero Then h( e c )) satisfies +c e- 1P h(i ei ) ce ) -e. h% e, 8 )9 and hence, by multiplication, h( a.Oe is a solution of the difference equation (hh)c (5es r c ( ) in ee a g) ( f e,,c) (77) which is regular in 0 < arg g < i for all \\ > 0o 33

THE UNIVERSITY OF MICHIGAN 2591 -2-T It is obvious that the problem of finding a suitable function H ( ) is tantamount to the solution of two equations of the general form (77) and this fact leads us to consider (76) in some detail. The equation (76) is directly analogous to those obtained by Pe:test (Refs 6, 11) and while it can be treated by reduction to an ordinary difference equation, the more direct approach adopted by Peters is to be preferred. If the plane is transformed usingf = u1:/n and if the logarithms of both sides of equation (76) are taken the function log fh(i.c)1 =E(u) is seen to satisfy E(u) - E(ue2i) = log [ un-c, (78) and a particular solution of this equation can be found in the following manners Let the initial point of a half-linet be the origin 0, the positive direction along o be that direction which is away from 0, with positive side on the left when f, is traversed in the positive direction and let C make an angle - with the real axis of the complex u plane. Suppose that w(u) is a function regular in a region containing ( and such that X(u) 1 w w(t) dt (79) 2ri L t-u is regular in f < arg u < 2+r + -' o It is well known that if to is a point on. and 2i.r X(to), X(toe ) are the limits of X(u) as u approaches to from the positive and negative sides of respectively, then X(t) - w(to) + P(t 2~~ 34

THE UNIVERS ITY OF MICHIGAN 2591 2-T and X(toe ) )- wQto) + P(to, where P(to) is the principal value of the integral in (79) when u4oo A consequence of these formulae is that 2ir X(to) X(toe ) w(to, and if this fesult is applied to (78) it is seen that the integral 1i: i1 log r t /nc dt 2ri J t Ru/n+ o represents a function which is regular in 0 < arg u < 2r and which satisfies equation (78) at least when u approaches C from the positive side. In terms of the integral is.:-2ri;. n.' /n -c 2Wri ) +._ c n Ic o I 00 ct./n i nri I t 2/n -c 0 and hence, by making the substitution t 1/n I c.. 9 < xi e - L t D J log I 1 t I t 5 dt -J / Vq r /a ni 1g _ ____ log 1 -) dv I( )' " i. I W L(80) The function I( V ) here defined is regular in 0 < atrg <and exp { I( )J satisfies (78) at least whenj p -p + 0, where p is real and positive. Still more can be asserted. Since log 1 ( ) n -j 0 as v -— O, I(J ) can be analytically continued into a region which contains (0,'6 ) by moving the path of integration in (80), and the process will now be examined. 35

THE UNIVE R S ITY OF MICHIGAN 2591 -2-T With reference to equation (80) we first remark that | 1 - has a + im' branch point at the origin of the v plane and zeros at v - ce for all integer m, each of these giving rise to a branch point of log l-(). (81) In order to make (81) single-valued it is necessary to insert certain cuts into the v plane and for convenience we choose radial cuts joining the origin to + im~ v = - c e, the branch of the logarithm being defined as that for which -r 4 arg log < wo A valid representation of I( ) in the sector 0 < argg < j is provided by equation (80)o The path of integration is the half-ray from the origin making an angle arg (c ) with the positive real axis, and when argJ - /2 the integral can be written as co.cei'/2 n -co c \ V $ F (/)8v v +. ----') ---- log Ilf I dv (82) If 0 C arg c < w the integrals in equation (82) represent I( )in the half-plane arg c +'/ - r argJ < arg * + tl/ since they define a function which is regular in this region and which coincides with I( ) when arg t /o The analytic continuation of I( ) can be achieved as follows. If C is an arc of a circle of (large) radius R with centre at the origin in the v plane Rl o - log I - ) dv 0. 36

THE UNIVERSITY OF MICHIGAN 2591 -2 -T Moreover, for arg c + / - arg < arg c + /g, v -g log I - has no singularities in the sector arg c + \/ < argj < arg c +, and hence, if C is any ray in this sector, i f o ce C 0 and 1 (3) = -- 2 2ri -I log - 1 - ( 1_ C v-S \v dv + -c- I1~ - log{l -. n. dv 2?-i co, ce 12 v l for arg c + l/2 - Ir < arrg c +'/2. These last integrals, however, define a function of which is regular in arg c + t/ - r < arg <( arg c + r and which coincides with I(j ) when arg c + 2 - r < arg ( arg c +'2. Since analytic continuation is unique, it can be concluded that IU( ) is regular in arg c +'/2 - < argg < arg c + In a similar manner we can shift the path in the second integral of (83) and so find that I( ) is regular in a;rg c - r < argj < arg c +; if is in this sector (83) C + 1 log fl- ( 1 dv 2i Th v tin t e takei aoutess of trobution of te p sinuarie of e integrrio an be epeads, care mstple be taker. about t Ji-i-Ct-ibutioz of tbe caingula..ities of ihe integrands. fcv Iarnle, (84) 37

THE UNIVERSITY OF MICHIGAN 2591-2-T (81) is singular at v=0 v=ce and if C is rotated in an anticlockwise direction the effect of the singularity at cei must be considered. If C is a ray in the sector arg c + < arg_ < arg c 2 and if arg - r arg < arg c+, then wher where r is a loop path which circumscribes in a clockwise sense tl ir from ce to the origin. The integral along r is ________ $p ~1~_- cei J IT J log i(c dv -2i log (.. ) I)r QJ and hence, for the sector arg c - < arg < arg c + 2 9 ) -log ( + 4L f I log l-c,)C *-5 0 — r 1 ie branch cut dv. (85) + 1w i g ~ logl -(C) dv 2ri ~ v f D where the integrals represents regular functions in this sectorO By repeated use of this method the path C can be moved into a path C' along the line arg v - arg c + 0 osay) and D can be moved into a path D' along arg v = arg c + 0 - 2.0 If these two integrals do not now cancel, C' and D' must then form the opposite banks of a branch cut, and in the sector arg c + 0o - 2ar < arg a < arg c + 9 which is to be regarded as a sheet of a Riemann surface), the function I( has logarithmic singularities only with the exception of the one at the origin, these are situated on the circumference of a circle whose radius is I cI with centre at the origin. 38

TH E U NI VE RS ITY OF MICHIGAN 2591-2-T The above representations may be used (see Refo 6) to verify that exp [I( )} eh(S,c) (86) satisfies equation (76) and we can therefore write h(,c) exp [ I J W X log 1(V) dv]. (87) L 2 For small c9 v, h(J,c) rvg n/2 (88) and if the branch point atj =0 is excluded, the only singularities of h(j, c) are poles outside and on the boundary of the sector arg c - -r g argf A arg c + t; in the interior of the sector h(_, c) is regular and moreover, free of zeros. Although these results have been obtained for the case in which 0 < arg c < r, other values of arg c can be treated in a similar manner. For example, if -i < arg c < 0 it is still true that (87) satisfies equation (76) and is a regular zero-free function in arg c < argj <>+arg c +r I_ >0. Outside this sector, however, the behaviour is in direct contrast to that of h(_,Sc) for -n/ arg c > 0. For small Ij it behaves as 2 and this branch point is the only singularity, the points which were poles of h( 9 c) for arg c > 0 now being zeros. These facts are obvious when it is realized that the corresponding solution of equation (76) for arg c <0is {h(,cei)) ~ where h(,c) is given by equation (87)o The apparent discontinuity in the solution when Im. c changes sign can be traced back to the derivation of equation (80) in which it was assumed that the line lies in a region of the u plane where 1 ^Tn c - \ lo un39 c 39

THE UNIVERSITY OF MICHIGAN 2591 -2-T is regular. Since X has been taken to be the positive real axis the condition is violated when c is real and though such such values can be taken account of by a deformatioi of the path, 9 it is more convenient to exclude them from our consideration. If necessary, real values of c can always be treated as a limiting case. The expression of h(, c) given by equation (87) is merely a particular solution of (76), but is sufficient to enable us to determine a function )H( ) in accordance with the required conditionso For this purpose c must be identified successively with t and', where and - = Vz -1~,/1 n' 2 The branch of 1- 2can be chosen arbitrariy and taking that with positive sign we have, since 0 ~ arg 4' ~4, T/4 < arg N < /2 and ~/2 < arg L (<;/4 with > andl I | o1 the moduli equal unity when Y/ is rea., corresponding to a pure dielectric wedge. Sincet )! 1, the circle on which the main singularities of h( 9 / ) lie intersects the region 2, and the assumption of regularity in 2 dictates the choice of function. On the other hand, the corresponding circle for h(j ) is everywhere outside2 and the regularity condition in this case would have no effect, notwithstanding the fact that the 0singularities are zeros and h(j T) is regular for all J 1 O>0. 40

THE UNIVERSITY OF MICHIGAN 2591 -2-T CHAPTER X THE CHOSEN FUNCTION Hi(L The conditions upon the function H ( { ) can be deduced from Chapter VII and are (i) (ii) (iii) (iv) (v) (vi) (vii) H (_ ) satisfies the difference equation =( eL) ()+ - e^ )(L +t) (-+ + ) H) e ) I 2 ~2 -1 2 1 e (? - H e ). e(9) H ( ) is regular in: 0 <arg < ", \? l, In -ir/2 arg L_' 3r/2 + H H ( ) has, at most, only such singularities n inxr/2 as are cancelled out by e ho( ) Since ho ) is proportional to. - e (equation 73), the only allowable singularities of H ( ) ) are those arising n inff/2 -1 from a pole factor ( e- In order for the residue at the geometrical optics pole x + ac to produce the required incident field (4), it is necessary for H ( ) to have the value unity at this point for all a, 0 L a 4- o This can be achieved by normalising with the factor { H [ e12 } and if this is to be finite for all a in the range, H ( ) ) must be free of zeros in r/2 4 arg. _ /2 + Y o H( ^ ) - constant as 1~ I -oo H( )^l-as O-0. ( I. C> H ( )- H ( e2) 41

THE UNIVERSITY OF MICHIGAN 2591-2-T In seeking to derive a function HI g ) in accordance with these conditions we first observe that to multiply a solution of the difference equation by either n - n or f - n does not affect the satisfaction of the equation but does change the location of the singularities. This fact is most important and is made use of in the following way. In Chapter IX it was shown that equation (89) could be split up into two equations of the form (76) which can be solved in a relatively simple manner to yield a function h(, c) regular in 0 < arg g <' for allI l >0 and whose reciprocal is also regular in this region. When c = t the sector of regularity is S -r < arg K + v, where /4 < arg _ /2, and is bounded by polesat_ 1- eiimS. eim+l, m 0, 1, 2.... By it1 2 I 0 self this function certainly contributes to the solution of (89), but whilst the region of regularity is more than sufficient to satisfy (ii), it is not in agreement with condition (iii). However, the interfering singularities can be annulled by multiplying the function by n n n without affecting the satisfaction of the difference equation and the regularity is now in accordance with the conditions (ii) and (iii), When c s t the situation is somewhat different in that the bounding 2'singularities8 are here zeros, which allows h(,. ) to be used by itself. Hence, consider (j-t n T n, ) h(, Z ). ^ 1 1 i > 2 This reduces to e - ein/2 when Vi 0, and to comply with the limit condition 42

THE UNIVERSITY OF MIC HIGAN 2591-2-T n iMn/2 (vi) it is necessary to divide- e by, a process which is permissible in view of the identical factor contained in ho( ). The new solution is now n n _ _ _ _ _ 1 n _ _ l og2 j -e =n 9 n /n _ S ew | 2 / i i t -1 n + /( C)( 1 I t n. - -e t (90) 1/n,\} t l dt tl/n+t 1 2 2 which clearly satisfies conditions (i) (iii) Since h( _ - C ) is entirely free of zeros and h(3, 7t ) is free in - (< arg ( < - S +', the condition (iv) can be satisfied (but only by the smallest possible margin) by using the constant normalising factor J [ ei1a/2)t, and this gives H (n ) -l —' D(91) ft j [ / e )i/ which represents the required solution of the difference equation (89). For large (\ \, t log J 3 () j constant (see BRef 6) and hence H ( ) r constant as \ l - Oo Since the condition (vi) is obviously complied with, there now remains only the condition (vii) and it is shown in the Appendix that this also is satisfied. The analysis given there strongly suggests that (vii) is vital for the unique specification of H( S )o If any factor additional to those of equations (90) and (91) were to be included in the expression for H 1( ) it would have to transform into itself when? eilS 9 if, then,, it were to 43

THE UNIVERSITY OF MICHIGAN 2591-2-T appear in the denominator of HI( ) its own singularities would conflict with condition (iii) and if it appeared in the numerator the normalising factor demanded by condition (vi) would violate (iv)o For this reason it is believed that the conditions (i) - (vii) are not only essential, but also enable H (j ) to be uniquely determined. Whether or not this is true, they have certainly led to the true solution of the present, quite general, problem. With the exception of (vii) all the conditions have a physipal origin and whilst it has not been possible to attach d direct meaningful interpretation to (vii), it is obviously necessary for the reduction of the solution in the manner described in Chapter VI. The final check upon the determination of H(t ) and, indeed, upon the method as a whole, must come from an examination of the resulting expression for P (r, 0). From equations (73), (75), (90) and (91) we have F [ I )J n i(- -+e)]" 2. - n -in/2{ -) {1:' i/2 e A +0)/ j [ei)/2/) cos _ cos tn 2 r/ 1/n ^ \ t/n + t \ 0 1 exp 1_ _ 2ti J 0~.1 t-e~~S Y I~ (92) and the expression for 0 (r, 0) then is 0 (r, 0) - e L F [i (_. -v+@) 1 d + - - 4 4 J ikr cos e -+sin(V6+ 0) -X+ inglO F- i ( ",-v - 0) 1 d; i - sin(+ ) L 2 J (93) 44

T H E UNIV E RSITY OF MICHIGAN 2591 2 T where L is the path shown in Figure 1. It is now a simple matter to-verify that the boundary conditions (5) and (6) are fMfilled. For small values of e we make the substitutions/ A t 7- 0 in the first integral of equation (93). This eliminates 9 from F and displaces the path an amount 0 to the left, but since the integral is free* of singularities in the intervening region, the path can be shifted back to its original position. With the second integral in (93) the corresponding substitution is/- V7 + 0 and hence wedge to give Or ~ -iki- r - 0e 30 as required. When g -8 is small the appropriate substitutions in equation (93) are =/ _ _J'' -0), leading to an expression for p(r, 0) in the form I J eir cos~(/w ) F [ r +,J L _ -+ eikr o608( 4 F [i( - /- ) ]df (94) l-sin(r, +~ ) 2 This can be written alternatively as ~r, e) [ -r, / i,Je sn sl n eikr- os ( e) 4 L J +sin/i F [i(j - A+ )] dc, 1^' 1""~ST~"~""I"~"""e" The fact that 0 can be made as small as we please ensures that no geometrical optics poles are included. 45

THE U NIV E R SITY OF MICHIGAN 2591-2-T by using the difference equation (40) for FMy and when 00- we now have -c~l + ik rp 0 in accordance with the boundary condition on the upper face of the wedge. The only remaining conditions are for large and small r. For large r the radiation condition is satisfied by the solution for infinite conductivity and consequently, by virtue of the decomposition of H( 5, for all conductivities. And similarly with the edge condition. Since all the requirements are met, the uniqueness theorem for electromagnetic problems enables us to conclude that the solution which has been obtained is, indeed, the true and correct solution of the problem formulated in Chapter I ti 46

THE UNIVERSITY OF MICHIGAN 2591-2-T CHAPTER XI SOME ELEMENTARY EXAMPLES The degree of complication of the final expression for 0(r, 0) is determined by the form of H ( g ) and it is of interest, therefore, to consider H ( ) for certain particular values of wedge angle. From equation (90) we have h(,c) =exp 1 I log t -/ c dt Fo a2 t- In tIn +c and hence n-I'/ ___ = n J t -c a3. {,og h(5,)c log, - r ) r~~~~~~~ _t dt 6lb n-i =n n 2. i t -' n log c {t/ n... + - _ ri n-i _t-n t _ -1 0 t/n -1 t....... dt t:ln.c2 00 n-4 I/jn 1n- + I e ct dt (95 -~ 2n with the upper or lower sign according as arg c ~ 0 respectively. If' is of the form P/q (that is, n S /p) where p, q are integers > 1, the integral is capable of considerable reduction and, in some instances, of complete expression in terms of trigonometrical factors. The simplest cases are those for which p=1 and whilst the analysis is then superfluous in that {(r, 0) reduces to the geometrical optics field alone (see Chapter VIII) and is independent of h(,c) some trivial examples can be quoted. 5) 47

THE UNIVERSITY OF MICHIGAN 2591-2-T Thus, for q = 1 we have T'= r an h(, % t) = - 1~~~~~ h( 9 t ) = - 2 S giving = -tI) (- 12) F 2-t s1 E"+i 1/2 Similarly, when & =r/ (q = 2) J( ) = 1? - -,( ~R) V ~Ca B) ( + izl) (3 + ir ) 4 C - 1 7/ andwhen 3 (q =3),. _(5- ).(~-~)V ( e -e -/3) (f-f e-,/3) -W e-i/33) - e2 _ +1 The generalisation to a wedge of any angle /q is now obvious, and it can easily be verified that all these functions satisfy the conditions (i) - (vii) of Chapter X. 48

THE UN IVE RS I TY OF M I CHI GAN 2591 2-T CHAPTER XII THE HALF-PLANE When p 1 the solution becomes more complicated and this can be illustrated by taking the simplest wedge of this type, namely, the half-plane, for which p 2 and q = 1. For n = 1 equation (95) becomes ~3 S log h( c)\ = + c a + g + 1 l L 0 ggt )2J 2xi L J c 1 Clog(t-) -log(t+c) + (1. c } + 1) ~ 2 J j- 2 -c,( t -c)i I +C 4 -c~ cri wri n r + -I21 2ri (loge -logj) ( I -c I -_JL, a + +e log h(,c) - {log h( ) d =lo + 1 log l -g + 1 log + log log 2ai J+c ai o e2X + c where the lower limit of integration was chosen in view of the known behaviour of h(j,c) for large 1 1. Hence l3h s lsQ (ogg l-I+g h( c> - 3.......i glog e c where~~~~~~~~+ th oe ii fitgainwsch nveo h nw eaiu > exp - 1 2ri 5 logs 1g\ log 0 e -c e x e + c 49

1t 4 THE UNIVERSITY OF'MICHIGAN 2591-2-T with the upper or lower sign for arg c > 0 respectively, and from this we obtain ( ) = i -t) -(S.t2) 3'/ ( -T,)(-t + 2) -2r l -i l; S+t ) (S -;2) 1 ll log j el) log( ei l 1)^.XJ )-!- 3 2 exp- Zl log 1_(3_ _- ~ 1-X-'i-'2i Se x +t l 0 (96) eX -z i e + 2. The pole a i is cancelled out by the corresponding zero of ho(S ) and if this point is excluded the behaviour of J( ) is as follows: arg = S -r: pole arg = - i +Zr: RandN-Z - i: zero +27r: RandN-Z -: zero - +3r: zero - r: R and N-Z s+3w: zero 6 r: RandN-Z - g+44: pole where o = arg r t and'R and N-Z' means'regular and non-zero. Thus, with the exception of the pole at_ = i, J( ) is regular in S -Ir arg < - l 4r, thereby satisfying conditions (ii) and (iii) of Chapter X, and since it is also free of zeros in S< arg < - g +2r, which region includes (fr/2, V2 + r ), condition (v) is fulfilled. It is no more difficult to show that the remaining conditions are satisfied. The particular case of an imperfectly conducting half-plane can be treated by the Wiener-Hopf techniqe (Ref. 12) and the solution can be used to provide a check on the present method. For this purpose it is necessary to work through to the final expression for 0(r, 0). 50

THE UNIVERSITY OF MICHIGAN 2591-2-T The function H(f ) is given in terms of J( ) by equation (91) and since ho() = -i (i. co (i ~ )'. cos - 2 ( i I equation 73), we have (j) = i i(I/2+a) {(L -tC) -t2)}fI, / 1 log,' _ _-1 ___(i-_ _s____ 2 /2 f s( -. xt) (, ) 4 - 2(i Cos +i (j+ ) ( -C-2a)J ~' (S +T1) ( 1 +T1v2)2ri l (5 -zl)(..). -17r) log(J e - logj exp 1 2ri.. log \f ex I% )( F ex AY |Ve+?1 2 dx (97) Giving. r'i2 -V+0)] iL 2 -. 2 3iir/4 e { J f ei(1/2+a)l 1 -1 cos - c os 2 2 2 2 - sin( -V -0) i +sin(V -) } V f /1_2 - cos (-V -09 1'. r log t. I + cos ( -9) -+ sin (-ii () - 0)) /4-(v -o)/(2 exp 1 27r T/2-( V -0) 0 0 log (4-+cos x \ +Cos x dx (98) in expression for the total field now follows by inserting this result into equation (55). 51

THE, UNIVERSITY 2591 -2-T OF M I CH I GAN If V is replaced by r + 3, equation (98) becomes -1 F i (- Ir -.0+0. 2r 3ir/4 =1 - - e J 2 (J C i(r/e2+a)). + sin (0-0)' 1- + cos (00) I- 2 -cos (P-e) 1 log il - 2ri 3 _ + sin 3-0 2 2 V. + sin (3-8) K - sin (P-0) A 3/4 + (3-o)/(27r) r/? -(p-O) T h exp. 2 ~- T j log. 3ir/4 { J ei(/2+a)] cos a + sin 2 2 f " cos x -1 I + cos x \ ]dx e sin - 2 say, where sin L(3) = 2 | sin/2 V~ 11k +sin13]'2 I 1 1 I- +Os f 1.1 -yC- Cos (3J log'1 1/4 - 2/(r) +f -sing I V+ sin 3 exp 1 2[ r 1/2 — 30 log \ +cos x - cos x I dx. (99) Also i( 3r.- +o0) 2 = 1 I 2 q 3if/4 e J ei(T./2 3I-1 cos a - sin 3-O 2 2 h-sin(B-e) 2

THE UNIVERSITY OF MICHIGAN 2591 -2-T and hence, from (55), the diffracted field is given by D (r, ) 8= - Je S(0) 32+0) l(Ir/2+ a)] cos -- + sin - 2 2 ~ -sin (-e) cos a -sin 2 2 -ikr cos f3 e sin 2 d3 1=- I1 27ri ( ei"' ~~~~~~~~1. r/2 -- s ()J [ a 1- co os co-s -ikr cos(P-0) e do3. cos a + COS f (100) The geometrical optics terms are of the expected form and need not be considered. Equation (100) is identical to the solution obtained by the Wiener-Hopf method (Ref. 12) and this fact provides striking confirmation of the present theory. Indeed, L(B) is merely the'split' function L+(k cos 3) 1 introduced in that paper, with -iy/4 e j [ e1r/2)] equal to the complementary function L (-k cos a)} 53

THE UNIVERSITY OF MICHIGAN 2591 2-T CHAPTER XIII WEDGES OF ANGLE /2 AND /3 There is obviously no limit to the number of examples that could be worked out and for each wedge angle the solution can be exprected to reveal some new features. However, to illustrate the general nature of the results it is sufficient to consider particular wedge angles, and the two that have been selected are /3 2 /and 2 /3. 3r 4 When' = /2 (n4 /3) the integral expression for h( c) can be evaluated completely to give h4/3 4 3 4/3 e4 3 h(~ ~') x' ~ 4 _.. (~ +^^)(_ + i f y) and hIf % ~' (_'' i'-" * T^ -> e _ 4/3 C 4/3'2i/3 from which we obtain j(3 ) = -__(- -Z) ( -t)- ( -i'Ai)( -i 2) (101) J(f )= 4/3 e 2ir/3) 4/3 4 /3 e 2ir/3) g 4/3 - 4/3 e2ir/3) (~ 4 /3-> e )e -5 ae (r -> 2 2 With the exception of the poles provided by the first factor in the denominator ( and these are cancelled out by the corresponding zeros.of ho 0 ( ), J(3 ) is regular in -ir +g < arg < 7/2 - and since it is also free of zeros in S<arg _ < T/2 -, the conditions (ii), (iii) and (iv) of Chapter X are satisfied. Moreover, from equation (73), 4/3 2ir/3 ~43 _ 4-e ho. 3 44/ 2/3 ir/3 2a 2i.r/3 3- 2 ^. e cos r+e 54

TH. U N I V E Ra S I T Y OF MICHIGAN 2591-2-T id hence ) 4 3 -1 i [.ei('/2+a)] I 4/3 2/3 /3 ( I — 2 2 e ir cos 2 a 3 ig - C,) T -(. j _ c - it I ( -iT_e22ir/3 (4/3,4 4 1/3 3 4/3 P4/3 2ir/3 I 2 here j ei(/2) ( h + sin a)( - cos a) 2a sin - 3 (cos 4a -T) 3 ith 4/3 T= - (i2 2 4/3 +'2 ), iving F [i( -7 +0)]= 4. 2 3 {' - sin (V- e) v - cos (1J - 0)] ( L +sina) (v\ cos a) (cos 4a -T) sin 2n 3 3 COS (V =) -T cos c (V=)-cos 2 If this is inserted into (55), the equation for the diffracted field can be reduced > the form D (r, ) = 1 Jliri S( s(e) +sin /' — +' sin a - cosa / ( a os T c os. -+ +* - -T) cos ( r - 3) - T 3 4af (cos - - T) 3 sin 2a sin. 3 3 { c 4 +3) - os cos -(7+f3) -T 4 Cos (3r-13) - cos 2 J COS — 3os 3 (T + 9) - coo2a s 3 -ikr cos (-) d e do (102) 55

THE UNIVERSITY OF MICHIGAN 2591-2-T It will be observed that the integrand is composed entirely of trigonometrical factors and as a result the solution is not very much more complicated than that for the corresponding perfectly conducting wedge. When 27r/3 in = 3), the evaluation of h(i c) is more tedious and ultimately leads to the following expression for J(Q )I jJ~ Z- 2ir/3> 2ir3 I 4ir/3 4i/3 i/2 e3i. <- +Ce' i>/ 0j Zir/3( 2 e1/ 4i^ 3 ) 4ir 3 V 1 / 6 3 3 3 3- 2 r',,-21,2i3)( i^2i1t/3( 4 I/ 4i/) 3 ( 3 3 - (- 3 ), +L23 ) lo exp _____I log g /e3X a3( e32 ) --..I - 3x f3- 1 -I e 3 -1 With the exception of the poles arising from the factor ( + i) J( is regular in -r < <arg, 8r/3 - and isfree of zeros in < argo < 5r/3 Also hot( )' _j~-t___ 3 -2$33/e cos 2 - 2 and hence 56

THE UNIVERSITY OF MICHIGAN 2591-2-T 3/2 IV+ = i 2 2,+8 ); i(7r/2 + au) 1]1 j e~ *- sin (72 - ) _ 3 3 cos Y a - cos - (V.0) C 2s-j- 2 4-Z sin( V -0+ 2ir/3)3 Y. -sin( j -0- 2ir/3) 1/2 { +sin(V -O+ 27r/3) f{-sin(i.-O 2tr/3) 31 6 + sin () -) sin( V -0+ 27r3) {L +sin( -0- 27r/3) j 1" A... log t 1- 2 cos(V -0) - / 2 - C8( I -+ 2/ 3) J T Cos(t/ -0- 27r/3) 27ri g j1-.+ +cos( -0)} 5 + 1- +cosC( -+ 2/3)j i /I1 - + cos( / -0- 2ir/3) - ] 1 +sin(v -e ) f - sin('- -0) pt/2 -( -~ -e) 27r 0 v + sin(V -9+ 2r/3).. + sin( 7 -0- 2Ir/3) f 1/4 ( )/(2T) - sin( 2-0+ 2ir/3)5 I - sln( i- 3- 2Zr/3) log ( CO X - os x + 27/3 - Cs x - 2r3 3) l +cos x! V + cos x + 27/3 /\ + oos x + 2z/3 dx I =, n - sin(C -o9) I -sin(- -Ze+ 2r/3) I' \ -sin( -' -0- 27r/3 } cos 2 ( V -0) cos - a - cos ( -V ) 22 2 T(- W +ir+O) J ei('/2 + a)] 57

THE UNIVERSITY 2591-2-T OF MICHIGAN say, where T(3)= -6i^ 3 sn 3 sifl 2 r ^+s8iii )r -smO+27yS)l { -sim(3-2r)i 3 L. ) +sin(P3-27r/3)j lY -s8in(9-2r/3) \ r, I _C-sinL( 3+2 )3) l i/ i ) +sln(g3+2) i jI. -> +o/s-P ( 9 l-n - s(fi23) iI? +}cos(-2z/3)j 1 41 r -Coesp i-^2 -co8s(d+2)/3)} l-V - c os(-2z/3 i log 7-8s113i 27-sin($+2/).f t )-s.in3-27rg/3)i I 4 - t f+8sin Y+sin( +2T/3) Iq-(+si( n 3/- i exp ( /log i? / c X I!/, Y+cos x+2 i3 k, q-cos x-2A,3. dx With this definition of T(3), Jei(Ktl2-f 2i L+smal +sin(a+23. 1 ~ 3 sin 3 L/,.sinca- T A'- T (r)gr F i(- -+ o) F 2 ( -'01: ]sinm(-0')?!n-s(-6+2T/3 n(P- sin -s,-0-s) sin 3 fcos 3 a + sin 3 (0-) 2 2 2 )i+sin(-0)s (?+sinq3-+) qsin i-0-2r/30) T(e-3) L - T(e-a3) j l(2r/2 ++a i i 3in sin - (3-e) 2 cos 8 -- s8ina 3 (-0) I 3 58

THE UNIVERSITY OF MICHIGAN 2591-2-T and if these are inserted into equation (55) we obtain D r, 1 = {\ vr (4t -3) - 2cos - a cos (P-0) T(0-e) (r,0)= StO) cO8 3(-O) + cos 3 a. j.... e / +/2 S(O) cos 3(3-0) + cos 3 a Je e -ikr cosd! do ~Ir 3 8ri oC J'>'(4nl -3) - 2 cos 3 a cos3 (0) S\ ( * + sn _ _ -:. i _': /.. s - i- u Y0) tt + s in a u. + sin(+2t/3) { + sin(a-2Tr/3) S(O)~~~~~~~~~~~~~ sin 3 a cos 3 ((3-0) + cos 3 a T (e-3) T (ir +ao -ikr cos P d3. A final substitution of 3 in place of 0-0 then gives an expression for the scattered field in the form 0 ( =8Tif t8 3 \rL (42L, -3) -2 cos3 a cos. r 8tri S(0), Y+ sin a' Y i + sin(a+2rI3) A + sn(a - 22r/3) S(B)~~~~~~~~~~~,<,, sisnua-27/} (104) sin 3 a cos 3,3 + cos 3a T(p) e-ikr cos(,B-0) d T(ir +a.) The fact that H1(5 ) is more complicated for ) = 2 t/3 than for y = 37a/2 has led to a more involved solution and, indeed, equations (103) and (104) are typical of those wedge angles for which the expression for h( C, c) is capable of any reduction. 59

THE UNIVERSITY OF MICHIGAN 2591-2-T The function T(3) contains an irreducible integral which can be split up into a product of separate integrals of the type Y so log cosx dx (105) Jo' cos x and this seems to be true of most values of n of the form 2q/p, where p and q are integers. And only for these n is it possible to simplify h( c a) atll. Many wedges of angle = P/q have been examined and for all of them h(, c) can be written as a product of linear factors or as a product of such factors with integrals of the form (105). This suggests that if (105) were to be evaluated numerically for 0 < y < r it would cover all wedges of homogeneous refractive index 1/v\ and angle P/q, and would enable the appropriate h( c) to be calculated in a relatively straightforward manner. 60

THE UNIVER SITY OF MICHIGAN 2591 2 -T CHAPTER XIV SOME RAMIFICATIONS In the preceding chapters a method has been developed for the exact solution of second order differential equations of the form (7) under boundary conditions imposed on radial surfaces. The method has been illustrated by application to the two-dimensional problem of a plane electromagnetic wave diffracted by a metallic wedge, but differential equations other than (7) can be treated, as can boundary conditions more complex than (5) and (6). Indeed, it is obvious that boundary conditions containing derivatives of higher order than the first can be treated in a similar manner. Our main interest is centred on the electromagnetic problem and whilst only an H-polarized incident plane wave has been considered, it should be emphasized that the method can be applied to more general types of incident field. Thus, for an E-polarized plane wave having E given by equation (4), the boundary conditions (5) and (6) are replaced by P 9i(k/{) rp o (106) for 0=0 and 0 + iV)r =rO. (107) for 0 =', respectively. If, then, the difference equation is retained in its original form, the parameters t and' must be interpreted as (1 2 1 (1+ ) and 1 ( - -2) "I \ V - 61

THE UNIVERSITY OF MICHIGAN 2591-2-T and the formulae for H( ) and, (r, 0) now differ from those for H-polarization in having 1/ Y in place of v\ This result agrees with that found for an imperfectly conducting half-plane (Ref. 12) and is an important advantage attached to a (mathematically) exact solution of the boundary-value problem. In particular, it enables the solution for one polarization to be deduced from that for th0 bther (and opposite) polarization by means of the transformation YE-.ZH, H _ -E, q-1/A, a duality which is a direct consequence of the impedance-type boundary conditions and which corresponds to the invariance of Maxwell's equations under the transformation YE -ZH, H — E. If the plane wave is incident at an oblique angle the problem is no linger twodimensional, but the solution can be found from the above in the manner described in Ref. 13. By regarding modified forms of the two-dimensional solutions as singlecomponent Hertz vectors, two fundamental fields are obtained which can be used to determine the solution for a general three-dimensional plane wave, and hence the solution for any incident plane wave can be deduced from that given in Chapter X by performing only simple transformations. For an incident field produced by either an electric or magnetic line source parallel to the apex of the wedge, the solution can be derived by integrating the 62

THE UNIVERSITY OF MICHIGAN 2591 -2-T corresponding plane wave solution with respect to its angle of incidence a. If the incident field is that of a point source the analogous technique is to integrate the appropriate three-dimensional plane wave solution with respect the component angles of incidence, and this method was used in Refo 12 to determine the field diffracted by a perfectly conducting half-plane when excited by a dipoleo Finally there is the generalization to the case of a partially inhomogeneous wedge. If the two faces are composed of different materials having unequal refractive indices, the method can still be used although the simplification to the difference equation represented by equation (76) no longer holdso To conclude this chapter we shall examine briefly the solutions which have been obtained for particular wedge angles. In Chapter I reference was made to the dangers involved in attempting an approximate solution by assuming a series expansion in powers of, and for a half-plane the expansion of L(P) gives sin 3 (see equation 99). For large y, however, v ~ [ ^ r J L(13) =| 2sin4 [ l l - {1ot) L cosBI3 +log 2 a) +OQ ) j and the correspondence between the two polarizations now shows that a series expansion is not valid for E-polarization owing to the term in log v o For a wedge of angle r/4 it is obvious that a series expansion is always possible (see equation 102), and this,, surprisingly enough is true also for - /3 (equation 104)o No attempt has been made to expand the solution for arbitrary wedge angle,.but it seems unlikely tbatthe term in log \is peculiar to the solution for a half-plane.

THE UNIVERSITY OF MICHIGAN 2591-2-T CHAPTER XV CONCLUSIONS As developed in this paper the method is, in many respects, similar to that of Wiener and Hopf, of which it may be regarded as an extension; the use of different transformations, however, makes difficult any detailed comparison between the two. The crux of the Weiner-Hopf technique is the splitting of a function into two parts with overlapping regions of regularity (see, for example, Ref. 14) and as applied to the problem of an H-polarized plane wave incident on a metallic sheet the task is the determination of a function L +(J ) such that L( +J) L++-J) k + k where L+(J ) is regular in the upper half-plane of the Fourier transform variable. By treating thej plane as a Riemann surface, the above equation can be written as L+ 3 ) g18) >( e }. k l -^ kS ( 08) which is, to all intents and purposes, a difference equation for L+( ). The equation has a Operiodicitya 2, equal to the open angle of the redge, and gives the value of the function at the point J in terms of its value at the corresponding point on a lower sheet of the Riemann surface. Moreover, the right-hand side of (108) closely resembles the Fresnel reflection coefficient of the diffracting halfplane, a fact which is brought out more clearly on making the transformation kcos3. 64

THE UNIVERSITY O-F MICHIGAN 2591 -Z-T The essential feature of Peters~ method is the derivation of a function F(o) satisfying the difference equation F(W) t -~cosh a 1x -cosh(w+i y ) ~ (10-9) F(w<)+2i ) v +cosh J V o+cosh(+i) J the right-hand side of which is related to the product of the Fresnel reflection coefficients for the two faces of the wedgeo Equation (109) can be written alternatively as F__)_ -cosh F(W+i' ) V+cosh 9 (110) where t' is the open angle of the wedge (co fo equation 76), and the analogy with the Wiener-Hopf method is now obvious. The above comparison shows that the two techniques are of essentially the same type. The analysis involved in similar and in each case the Fresnel reflection coefficient plays a dominant role, notwithstanding the fact that the final solution contains the coefficient explicitly only in its geometrical optics terms. The method originated by Peters represents a new and powerful tool for the exact solution of diffraction problems and is particularly suited to such problems as the imperfectly conducting wedge where the physical discontinuities lie in inclined planes. By comparison the Wiener-Hopf approach and others (for example, that of Clemmow, Refo 8) based upon it are more limited in scope, being virtually restricted to cases which involve discontinuities in parallel planeso The present method has much of the elegance associated with the older techniques although its simplicity is somewhat marred by the necessity for introducing complex variables additional to those implicit in the problems and this is liable to cause confusion in converting conditions upon b(r, 0) into conditions upon 65

THE UNIVERSITY OF MICHIGAN 2591-2-T the required solution of the difference equation. With both treatments, however, most of the analytical difficulties are concentrated in the solution of equations of the form (108) or (109). Note added in proof: Dr. L. B. Felsen has drawn the attention of the author to a paper by Malyuzinec (Ref. 15) dealing with the radiation of sound by a vibrating wedge. In an appendix the solution is stated for a problem which is equivalent to the diffraction of an H-polarized electromagnetic wave by a wedge whose faces have different refractive indices. Although no details of the method are given (Malyuzinec refers the reader to previous papers by himself: Refs 16-19), a preliminary examination suggests that the results are in agreement with those which we have obtained. The author is indebted to Dr.'Herschel Weil, of the Radiation Laboratory, for carefully checking various parts of the manuscript. 66

THE UNIVERSITY OF MICHIGAN 2591-2-T REFERENCES 1. Raman, C. V. and Krishnan, K. S., Proc. Roy. Soc., (A) 116, 224 (1927). 2. Jones, D. S. and Pidduck, F. B., Quart. J. Math., 1, 229 (1950). 3. Felsen, L. Bo, Report on Symposium on Microwave Optics, Vol. 1, McGill University (1953). 4. Senior, T B. A., Ph. D Thesis Cambridge University (1954). 5. Kontorovich, M. J. and Lebedev, N. N., Jo P hys U.SS.Ro., 1, 229 (1939). 6. Peters, A. S., Comm. Pure Appi Math, 5, 87.(1952). 7. Grunberg, G., Phys. Rev,o 63, 185 (1943). 8. Clemmow, P. C., Proc. Roy. Soc., (A) 205, 286 (1951). 9. Jones, D. S. Quart. J. Mech, Appl1 Math., 5, 363 (1952), 10. Pauli, W., Phys Rev., 54, 924 (1938), 11. Peters, A. S., Comm. Pure Applo Math., 3, 319 (1950). 12. Senior, T.B.A., Proc. Roy. Soc., (A) 213, 436 (1952). 13. Senior, T. B.A., Quart.. Mech. Appl. Math., 6, 101 (1953). 14. Titchmarsh, E. Co. Introduction to the Theory of Fourier Integrals, Oxford: Clarendon Press (1937). 15. Malyuzinec, G. D., Acoust, J. (Russian), 1, 144 (1955). 16. Malyuzinec, G. D., Doklady Akad. Nauk,, 60, 307 (1948). 17. Malyuzinec, G. D., Doklady Akad. Nauk, 78, 439 (1951). 18. Malyuzinec, G. D,, Zhurnal Tekh, Fiziki,: (1951). 19. Malyuzinec, G. D., Dissertation Fian Izd, Akad. Nauk. SSSR, (1950). 67

T H E U N I V E RS I TY O:F M I C H I GAN 2591-2-T APPENDIX The chosen function H( ) is Hn n-z1n( ^ hj' H(s).= t h(i,t1)h( T) ~. n,,ate~ where h(,c) is given by equation (87), and for H1( ) to satisfy the conditions stated in Chapter X it is necessary that:J + } (n i + Z H ( H ( e ~ 5 v vj S v Ie O To prove the relationship we consider first h(,,c). This is defined in terms of h(,c) and if f is replaced by) in the integrand of the expression for h(?,c), we have ooc: log h(,c) i 3. log (- + - 0 1 OC I ri j 0 log 1-( l- )n t{(on} dy (Al) F — c-p le..2 log t1-( )J dv log ( ), +c + I(,c) where 1. I(i,c) =- - 0 nri", 0 ct2/n 2,t//- -1 t V1 ^2 log 1 - t n, 68

THE UN I VE RS I T Y OF M I CH I GAN 2591-2-T When c' the path of integratio n inequation (A2) can lie anywhere in the sector | n( S -) n } sa where i arg I i and if its inclination to the positive real axis is denoted by n / the integral defines a function which is regular in -' + A < arg f < A Since arg - / < 09 all the allowed values of lie below the path and this fact enables us to rotate the path in an anticlockwise direction so as to collect a contribution from the residue of the integrand at its pole t' n giving 1(3, f1l (^ n coe nA e) I/n -1 Ig * ) =- log I - ____) t\ dt. o J nn J f7 JT, i The integral now defines a regular function of3 in - + r +/ < arg<gr + A 1n and the transformation t ( s) reduces it to the form ooxaei =" 1elo- ( ( ds, ~A', S e' 0 where - < < - + But this represents a regular function of eir in < arg e +) < + r and hence, from equation (A1), it can be written as 1 {(\-27-/ 2A ~i. -e 1t~) l Jl~ /JI 0-e ) oJ 3 " cJ I Is. The remaining integral can be simplified by introducing a new variable x, where x = s' $ e -iar and since the new path lies in the sector (- + S ) we can now write.2 9 o 2i + I 1-u I log - ( il V 1, f-r t 0 Vn 1 r.L Ldx. (A3) 69

THE UNIVERSITY OF 2591-2-T MICHIGAN The case c -- can be treated in a similar manner. The path of 2 integration is initially confined to the sector rotated in an anticlockwise direction we obtain -nS,~r-)l but ifit is 1 2 -log 1- ( 2eir I - 1 nri in(A 2r) e 2 o tl/n -1 u 2 el,' -(, a log (1 t ] ~n (A4) where - % I( S, t <A< 2g log r - 8. The above sequence of transformations then gives 1 (2 eif. I J n + } +log h + t'I vi s O 0 it X2 e T 2: 2..i P 2 log [ (xeA (A5) a Hence, n log (1+ IC? ( +. it \ s _ %2 H1I )I _ log {. n inr/2 e +log h ( e.+ C0 1 ~i J0 2) + log 2 2 {h (I tI W.~ r~)> x2 2 2 ap-C logx- e }i,o~[_. (~ and since n we have _n... eir n 1 i 1-(' 2 ) -n inr/2 ( )( iI 2 I 19 2 r/ n t,"re a e").ir \n (e...). 6 J in _/ I 1i - e _. H (g) -W( ) H ( e ) ~~~~~~~~~~~~~~~i I " (A J. as 70

THE UNIVERSITY OF MICH I GAN 2591 -2-T where log 1W( )j} nlog (t2e 2)/ + i ( 2 l 3 1 2 l 2 In the complex x plane the branch cuts radiate from the orgin to +f'and if the path of integration is rotated through an angle ~r, contributions are obtained from the integrations along the branch cuts to and Ut e, giving co -OD 2.2eiT S = J + 27ri log x -2tri [logx] -o + / 2ei \~ = + 2ri log 0 An identical result would have been obta and using the fact that O O f0 0 we now have = ri log ( -2 Lined by moving the path in a clockwise direction, Thus log { W (. = n log (tCei/2) n log ( - 0, which confirms that H ( g ) has the required property. 71 (A7)

3 9015 (