THE UNIVERSITY OF MICHIGAN A RADC-TNASTIA Document No. STUDIES IN RADAR CROSS SECTIONS XXXI - DIFFRACTION BY AN IMPERFECTLY CONDUCTING HALF-PLANE AT OBLIQUE INCIDENCE by T.B.A. Senior February 1959 Report No. 2778-2-T on CONTRACT AF30(602)-1853 PROJECT 5535 TASK 45773 Prepared for ROME AIR DEVELOPMENT CENTER AIR RESEARCH AND DEVELOPMENT COMMAND UNITED STATES AIR FORCE GRIFFISS AIR FORCE BASE, NEW YORK

Qualified requestors may obtain copies of this report from the ASTIA Arlington Hall Station, Arlington 12, Virginia. ii

THE UNIVERSITY OF MICHIGAN 2778-2-T STUDIES IN RADAR CROSS SECTIONS I "Scattering by a Prolate Spheroid", F.V. Schultz (UMM-42, March 1950), W-33(038)-ac-14222 UNCLASSIFIED. 65 pgs. II "The Zeros of the Associated Legendre Functions Pmi') of Non-Integral Degree", K.M. Siegel, D.M. Brown, H.E. Hunter, H.A. Alperin and C.W. Quillen (UMM-82, April 1951), W-33(038)-ac-14222. UNCLASSIFIED. 20 pgs. II "Scattering by a Cone", K. M. Siegel and H.A. Alperin (UMM-87, January 1952), AF-30(602)-9. UNCLASSIFIED. 56 pgs. IV "Comparison between Theory and Experiment of the Cross Section of a Cone", K.M. Siegel, H.A. Alperin, J.W. Crispin, Jr., H.E. Hunter, R.E. Kleinman, W. C. Orthwein and C. E. Schensted (UMM-92, February 1953), AF-30(602)-9. UNCLASSIFIED. 70 pgs. V "An Examination of Bistatic Early Warning Radars", K. M. Siegel (UMM-98, August 1952), W-33(038)-ac-14222. SECRET. 25 pgs. VI "Cross Sections of Corner Reflectors and Other Multiple Scatterers at Microwave Frequencies, R.R. Bonkowski, C.R. Lubitz and C.E. Schensted (UMM-106, October 1953), AF-30(602)-9. SECRET - Unclassified when appendix is removed. 63 pgs. (S). VII "Summary of Radar Cross Section Studies under Project Wizard", K. M. Siegel, J.W. Crispin, Jr., and R.E. Kleinman (UMM-108, November 1952), W-33(038)ac-14222. SECRET. 75 pgs. VIII "Theoretical Cross Section as a Function of Separation Angle between Transmitter and Receiver at Small Wavelengths", K.M. Siegel, H.A. Alperin, R.R. Bonkowski, J.W. Crispin, Jr., A.L. Maffett, C.E. Schensted and I.V. Schensted (UMM-115, October 1953), W-33(038)-ac-14222. UNCLASSIFIED. 84 pgs. IX "Electromagnetic Scattering by an Oblate Spheroid", L. M. Rauch (UMM-116, October 1953), AF-30(602)-9. UNCLASSIFIED. 38 pgs. X "Scattering of Electromagnetic Waves by Spheres", H. Weil, M. L. Barasch and T.A. Kaplan (2255-20-T, July 1956), AF-30(602)-1070. UNCLASSIFIED. 104 pgs. XI "The Numerical Determination of the Radar Cross Section of a Prolate Spheroid", K.M. Siegel, B.H.Gere, I. Marx and F.B. Sleator (UMM-126, December 1953), AF-30(602)-9. UNCLASSIFIED. 75 pgs. iii

THE UNIVERSITY OF MICHIGAN 2778-2-T XII t"Summary of Radar Cross Section Studies under Project MIRO", K.M. Siegel, M.E. Anderson, R.R. Bonkowski and W. C. Orthwein (UMM-127, December 1953), AF-30(602)-9. SECRET. 90 pgs. XII "Description of a Dynamic Measurement Program", K. M. Siegel and J. M. Wolf (UMM-128, May 1954), W-33(038)-ac-14222. CONFIDENTIAL. 152 pgs. XIV "adar Cross Section of a Ballistic Missile", K. M. Siegel, M.L. Barasch, J.W. Crispin, Jr., W.C. Orthwein, I.V. Schensted and H. Weil (UMM-134, September 1954), W-33(038)-ac-14222. SECRET. 270 pgs. XV "Radar Cross Sections of B-47 and B-52 Aircraft", C.E. Schensted, J.W. Crispin, Jr. and K.M. Siegel (2260-1-T, August 1954), AF-33(616)-2531. CONFIDENTIAL. 155 pgs. XVI r"Microwave Reflection Characteristics of Buildings", H. Weil, R.R. Bonkowski, T.A. Kaplan and M. Leichter (2255-12-T, May 1955), AF-30(602)-1070. SECRET. 148 pgs. XVII "Complete Scattering Matrices and Circular Polarization Cross Sections for the B-47 Aircraft at S-band", A. L. Maffett, M. L. Barasch, W.E. Burdick, R. F. Goodrich, W.C. Orthwein, C.E. Schensted and K.M. Siegel (2260-6-T, June 1955), AF-33(616)-2531. CONFIDENTIAL. 157 pgs. XVIII "Airborne Passive Measures and Countermeasures", K.M. Siegel, M. 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)-2531. SECRET. 177 pgs. XIX "Radar Cross Section of a Ballistic Missile II", 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. 189 pgs. XX "Radar Cross Section of Aircraft and Missiles", K.M. Siegel, WE. Burdick, J.W. Crispin, Jr. and S. Chapman (WR-31-J, March 1956), SECRET. 151 pgs. XXI'"adar Cross Section of a Ballistic Missile III", K.M. Siegel, H. Brysk, J.W. Crispin, Jr. and R.E. Kleinman (2428-19-T, October 1956), AF-04(645)-33. SECRET. 125 pgs. XXII "Elementary Slot Radiators", R. F. Goodrich, A. L. Maffett, N.E. Reitlinger, C.E. Schensted and K.M. Siegel (2472-13-T, November 1956), AF-33(038)-28634, HAC-PO L-265165-F31. UNCLASSIFIED. 100 pgs. iv

THE UNIVERSITY O.F MICHIGAN 2778-2-T XXIII "A Variational Solution to the Problem of Scalar Scattering by a Prolate Spheroid", F.B. Sleator (2591-1-T, March 1957), AF-19(604)-1949, AFCRC-TN-57-586, AD 133631. UNCLASSIFIED. 67 pgs. XXIV'Radar Cross Section of a Ballistic Missile - IV The Problem of Defense", M.L. Barasch, H. Brysk, J.W. Crispin, Jr., B.A. Harrison, T.B.A. Senior, K.M. Siegel and V.H. Weston (2778-1-F, To be Published), AF-30(602)-1853. SECRET. XXV "Diffraction by an Imperfectly Conducting Wedge", T.B.A. Senior (2591-2-T, October 1957), AF-19(604)-1949, AFCRC-TN-57-591, AD 133746. UNCLASSIFIED. 71 pgs. XXVI "Fock Theory", R. F. Goodrich (2591-3-T, July 1958), AF-19(604)-1949, AFCRC-TN-58-350, AD 160790. UNCLASSIFIED. 73 pgs. XXVII "Calculated Far Field Patterns from Slot Arrays on Conical Shapes", R.E. Doll, R. F. Goodrich, R.E. Kleinman, A. L. Maffett, C.E. Schensted and K.M. Siegel (2713-1-F, February 1958), AF-33(038)-28634 and 33(600)-36192; HAC-POs L-265165-F47, 4-500469-FC-47-D and 4-526406-FC-89-3. UNCLASSIFIED. 115 pgs. XXVIII "The Physics of Radio Communication via the Moon", M.L. Barasch, H. Brysk, B.A. Harrison, T.B.A. Senior, K.M. Siegel and H. Weil (2673-1-F, March 1958), AF-30(602)-1725. UNCLASSIFIED. 86 pgs. XXIX "The Determination of Spin, Tumbling Rates and Sizes of Satellites and Missiles", M.L. Barasch, W.E. Burdick, J.W. Crispin, Jr., B.A. Harrison, R.E. Kleinman, R.J. Leite, D.M. Raybin, T.B.A. Senior, K.M. Siegel and H. Weil (2758-1-T, To be Published), AF-33(600)-36793. CONFIDENTIAL. XXX? "The Theory of Scalar Diffraction with Application to the Prolate Spheroid", R.K. Ritt (with Appendix by N. D. Kazarinoff), (2591 -4-T, August 1958), AF-19(604)-1949, AFCRC-TN-58-531, AD 160791. UNCLASSIFIED. 66 pgs. XXXI "Diffraction by an Imperfectly Conducting Half-Plane at Oblique Incidence", T.B.A. Senior (2778-2-T, February 1959), AF-30(602)-1853. UNCLASSIFIED. 35 pgs. v

THE UNIVERSITY OF MICHIGAN 2778-2-T PREFACE This is the thirty-first in a series of reports growing out of the study of radar cross sections at The Radiation Laboratory of The University of Michigan. Titles of the reports already published or presently in process of publication are listed on the preceding pages. When the study was first begun, the primary aim was to show that radar cross sections can be determined theoretically, the results being in good agreement with experiment. It is believed that by and large this aim has been achieved. In continuing this study, the objective is to determine means for computing the radar cross section of objects in a variety of different environments. This has led to an extension of the investigation to include not only the standard boundary-value problems, but also such topics as the emission and propagation of electromagnetic and acoustic waves, and phenomena connected with ionized media. Associated with the theoretical work is an experimental program which embraces (a) measurement of antennas and radar scatterers in order to verify data determined theoretically; (b) investigation of antenna behavior and cross section problems not amenable to theoretical solution; (c) problems associated with the design and development of microwave absorbers; and (d) low and high density ionization phenomena. K.M. Siegel vii

THE UNIVERSITY OF MICHIGAN 2778-2-T SUMMARY The exact solution is obtained for the problem of a plane wave incident at an oblique angle on a half-plane of large but finite conductivity. The usual approximate boundary conditions are applied and these lead to coupled WienerHopf integral equations from which to determine the currents excited on the surface of the sheet. The resulting expressions for the field components are found to be entirely different from those which would have been obtained by applying the technique used for the derivation of three-dimensional solutions in the case of perfectly conducting structures and, indeed, not one component is given to the required accuracy by this technique. viii

THE UNIVERSITY OF MICHIGAN 2778-2-T INTRODUCTION In this paper attention will be confined to diffracting structures which are two-dimensional in the sense of being composed of cylinders of arbitrary cross section whose generators are all parallel to the z axis of some coordinate system. If a structure of this type is perfectly conducting, it is possible to deduce the solution for a three-dimensional incident field from the solution for a twodimensional field and, in particular, the solution for a plane wave at oblique incidence can be obtained from that in which the plane wave is normal to the z axis. The method is based upon the fact that any solution of the two-dimensional wave equation gives rise to a solution of the three-dimensional equation on re-ikz sinj placing the propagation constant k by k cos ( and multiplying by e If the particular solution considered represents the solution for the electromagnetic problem in which the incident field is a two-dimensional plane wave, and if it is modified in the above way and then taken to be the z component of an electric or magnetic Hertz vector whose other components are zero, a solution of the threedimensional problem is produced. The corresponding incident field has either H z or Ez zero (depending on whether the two-dimensional field was E or H polarized), and the two fundamental fields so generated can be combined to give the solution for any incident field. The method has been described in detail by Clemmow (Ref. 1), and was used by Senior (Ref. 2) to determine the field of a dipole in the presence of a half-plane. 1

THE UNIVERSITY OF MICHIGAN 2778-2-T When the structure is not perfectly conducting the method is no longer applicable and the question then is whether an analogous technique can be developed to treat such cases. To attempt an answer to this an obvious approach is to tackle a particular problem in the hope that its solution may indicate a general transformation and apart from the problem of a circular cylinder (for which the solution is almost trivial), one of the most simple is that of a plane wave incident at an oblique angle on a half-plane of large but finite conductivity. The present paper is devoted entirely to a consideration of this problem. The crux of the analysis is the determination of coupled integral equations for the electric and magnetic current distributions excited on the surface of the half-plane. These are of Wiener-Hopf type and can be solved to give expressions for the currents in terms of the "split" functions which characterized the solution for normal incidence (Ref. 3). The field components are then given as integrals over the currents, and some ramifications of the results are examined. 2 THE INTEGRAL EQUATIONS Consider a thin semi-infinite sheet of large but finite conductivity occupying the half-plane y = 0, x > 0 of a rectangular Cartesian coordinate system (x,y,z). A plane wave is incident in a direction making an angle a with the positive x axis and an angle 7r/2 - 3 with the z axis; 3 = 0 then corresponds to incidence in the plane perpendicular to the diffracting edge. 2

THE UNIVERSITY OF MICHIGAN 2778-2-T The actual form of the incident field matters little as regards the analysis, but in order to simplify the comparison of the results with those for a perfectly conducting sheet the field is taken to be a quasi three-dimensional plane wave which is E —polarized and whose components are given by -ik(x cos acos B +y sina cos 3 + z sin3) E i= (-cos asin 3cos 3, -sin sin jcos j, cos2 8)e (1) -ik(x cos a cos +y sina cos 3 + z sin 3) Hi=(-Y sina cos (, Y cos a cos B, O)e (2) where Y = 1/Z is the intrinsic admittance of free space and a time factor eit is suppressed. The above field is that which is obtained by modifying a twodimensional E-polarized plane wave in the manner described in the previous section. If? is the reciprocal of the complex refractive index of the material comprising the sheet, the boundary conditions to be applied can be written as _E-(nE- n= 7 Zn^_H, (3) where n is a unit vector normal drawn outwards from the sheet. On the upper surface n is in the positive y direction, and equation (3) then gives Ex= ZHz, Ez=- ZHx Similarly, on the lower surface Ex=- ZHz, Ez =ZHx As a consequence of these conditions, the tangential components of both the 3

THE UNIVERSITY OF MICHIGAN 2778-2-T electric and the magnetic vectors will be discontinuous on crossing the sheet, and it is convenient to regard these discontinuities as due to the presence of electric and magnetic currents in the sheet. We therefore write y=+0 y=+0 Ez - I1(x', z'), Hxl = Iz(x', z') y=-0 y= -0 y=+0 y=+0 Exl =I3(x', z), E =I4(x', z') y= -0 y= -0 where I2 and I4 are the electric currents and I1 and I3 are the magnetic ones. I1 and 14 are perpendicular to the diffracting edge whilst I2 and 13 are parallel, and when the field is normally incident (3 = 0) I3 and I4 are identically zero. The electric field at a point (x,y, z) can be expressed as a surface integral in the form 1 Q l \ ikp E (xy, z)= 4 ikZ(nAH)-(n^E E)V -(n. E )V ep dS (4) _Er A-y P (see, for example, Ref. 4, p. 467) where n is a unit vector normal drawn into the volume contained by the surface S. The differentiation is with respect to the coordinates of the observation point and p is the distance to a variable point (x',y',z') on S. The surface S is made up of two sheets which envelop the diffracting structure, together with a cylinder of infinitely large radius centered on the z axis and meeting the two sheets at x = co. The integration over the cylindrical 4

THE UNIVERSITY OF MICHIGAN 2778-2-T portion is easily shown to produce the incident field and accordingly equation (4) can be written i O1 OD 0kp \ ^ y=+.O E(x,y,z) = E(x,y,z)+ -1 jikZ(n H) nEV -(n* E)V p dx'dz' 4-^ A - 47P - - o 3 ~)' ~ ~ y=-O (5) On the upper surface of the half-plane n H =(Hz, 0, -H ) -A\- z x from which we obtain y=+0 n H -A- y=-0 - (I4, 0, -I2). Similarly, ikp Y-+O (n _X E e P A P y=-O and ikp a a e _ 3r _i 3 - I9 3 3 e~(- 3 - I I 3 ay. 3 ax 3 - ay eP (n.E) = iEz- ej ( Ey-V0 | = k a'z' ax' ax' ay' z P and if these are inserted into (5), the equation becomes E(x,y,z) E(x,y,z) + 14 o S ikZ(I4, 0, -Iz) - -I3 8 -I1 ai I aY 4ir- a y 3ax -I iz 111) 0 -o -i 312 4 ( e dx' dz' k \az' ax',aX' aya aZ) p g(6) At normal incidence the fields and currents are independent of z (and hence z') and, in addition, there is no current 4; the terms in the third group of the above integrand then disappear. 5

THE UNIVERSITY OF MICHIGAN 2778-2-T An assumption is now made concerning the dependence of the field upon the coordinate z. The diffracting structure itself is two-dimensional and since all -ikz sin3 the components of the incident field involve z only through a factor e it seems reasonable to assume that this dependence will also apply to the total field. A consequence of this is that -ikz'sin/3 Ip(X', z) e il I (X') for p = 1, 2, 3, 4, which makes it a trivial matter to carry out the integration with respect to z' in equation (6). To evaluate the z' integral it is sufficient to consider -ikztsin3i ek dz -ikzsin e-ik(z'-z)sin3 e ik 00 00 -co P -oo P where, of course, p =/(x'x)2+y2+ (z'-z)2, and if we put z'-z= Q sinh with Q = (x-x) +y, the integral becomes -ikz sin ikQ(cosh~ -sinfsinh Y) e e d -00 -ikz sin 3 o ikQcos cosh(y+tanhi sin3) -ikz sin3 (k =e \ e dWe rie H (kQcosf{). — 00 The nature of this result shows that the assumption about the z dependence is self-consistent and allows us to suppress the coordinate z throughout the subsequent 6

THE UNIVERSITY OF MICHIGAN 2778-2-T analysis. Equation (6) can then be written as E(x,y)=_Ei(x,y)- kkZ(I4,0, -I2)+i -I3 +iksin3 Il, I 0 + ( kstinIa H(1a (kQ cos ) dx' (7) The above integrand involves -a4, and if 14 were zero at the edge of ax' x1 4 the half-plane, integration by parts would enable the derivative to be replaced by I4 a. It is known that in the case of an H-polarized plane wave at normal incidence the magnetic current perpendicular to the edge vanishes at the edge (Ref. 3), and for simplicity it will be assumed that the same is true at oblique incidence. If this behaviour is not postulated at the outset, the solution of the integral equations becomes more complicated, although the analysis can still be carried through. The details are given in Appendix I and from these results it follows that 14(0) must be identically zero. Equation (7) now gives rise to three scalar equations for the currents: ~~~-4 Zy - 3 0 (8) 7

THE UNIVERSITY OF MICHIGAN 2778-2-T Ey(x,)-E (y) -Ex yiI3 a +ksinI-iZsm32 - a H()kQcos)dx y y 4 3 5' 2 ay ax 0 (9) 0 E,(x,y)-Ei(x,y)= 1 { tkZcos -iI ayi +i Zsin3 ) H (kQcos3) dx'. 0 (10) Integral equations for I2 and I4 can be obtained from these by letting the field point approach the half-plane successively from above and from below. By taking the limits of equations (8) and (10) as y — + 0 and using the fact that ( lim + lim H (kQcos3) dx I (-X') -aH (k Q cos 0) \y —+0 -0/ P ay 0 0 we have i 1a Ex(x,+0)+ E (X, -0)- 2EX(X, 0)= - kZl4 + k x2 +xiZ six 2 ax 1 + i Zsin 2 aa } H0 (klx'-x| cosj3) dx' E (x,+0) + E(x, -0) -2Ez (x, 0) = kZcosI2 + o + i Zsinnp Ix a HH) (k|x'-x cos) dx' 4 ax sn I } (1) and since E(x, +0) + E(x, -0) = Z I4 Ez(x,+0) + Ez(x, -0) = - j Z I2 8

THE UNIVERSITY OF MICHIGAN 2778-2 -T it follows that -ikxcos a cos 2 I4(x)+2Ycos a sin 3cos e = 1 4 k ea ~ ~ Z k2 ax2 0 +isin/3Iz a H()(k Ix'-x cosB3) dx' (11) -ikx cos a cos c 7 I2(x)+2Y cos2 e = cos2 + 0 +isin3 I4- a H ((kx'-xl cos ) dx', (12) which are coupled Wiener-Hopf equations for the determination of the electric currents I and 14. To obtain the integral equations for Ii and 13, either of two procedures can be followed. The first of these has its origin in the observation that the currents I and I4 whose equations have already been found are both electric currents, and the starting point for the derivation of equations (11) and (12) was the expression of the electric field at a point (x,y, z) in terms of a surface integral over the currents. Consequently, it is to be expected that if the magnetic field were written as a surface integral analogous to that in equation (4), the same analysis as the above would lead to equations for I and I3. The second method is equivalent to this, but avoids the necessity of going back to the expression of the magnetic field as a surface integral over the field 9

THE UNIVERSITY OF MICHIGAN 2778-2-T components. If equations (8) and (10) are differentiated with respect to y before the field point is allowed to approach the surface of the sheet, equation (9) can then be used to determine Hx and Hz. If this second method is adopted, the fact that Ez ) Hx= i -~ (z - y ) Hz =i k \ y ax gives, by suitable differentiation of equations (8), (9) and (10), Hx(x,y)-Hi(x,y)= kYI1 (+k2 x2 - i 1 + 0 + i YsingI3 I- i H t(kQcos 3)dx', (13) oo Hz(x,y) -Hi(x,y) = - kY cos2 3 I 3-i 4 -iY si I H (kQ cos dx x z (14) Comparison with the equations for Ex and Ez reveals the expected symmetry between the electric and magnetic fields and currents. Indeed, equations (13) and (14) correspond to (8) and (10) under the transformation 10

THE UNIVERSITY OF MICHIGAN 2778-2-T E - H ZH -- -YE I1 -— 4 ZI2 - -YI3 13 - Iz ZI4- -YI1 and this duality enables us to write down immediately the integral equations for I1 and I3 by reference to the equations for I2 and I4. The results obtained in this manner are identical to those which would have been found if the first of the above methods had been employed, and are I (x) -2 sin a cos cBe.= i1( 1 + O 0 + isinj 13 a- H (k Ix'-x cosB) dx' (15) I3(x) = -- kcos2 I3+isingIl \ H(k x'-x cos3)dx' (16) 3k 2 3 1 x H 0 0 (c. f. equations 11 and 12). To complete the duality between the electric and magnetic quantities, it will be observed that the additional transformation A-mp is required. The absence of the inhomogenous term in equation (16) is caused by the fact that H is zero. z 11

THE UNIVERSITY OF MICHIGAN 2778-2-T 3 THE SOLUTION OF THE EQUATIONS In view of the similarity between the equations for I1, I2, 13 and 14 it is sufficient to restrict the analysis to just one of the coupled pairs. Let us therefore consider equations (11) and (12) and attempt to cast them into forms suitable for solution by the Wiener-Hopf method. Take first equation (11). This holds only for x >0, but if a function 04(x) is introduced such that 04(x) is zero for x >0 and is equal to the right hand side of (11) for x<O, the equation can be written as I4(x)+04(x) + 2Ycos sin cos B /(x)= 2 S 4 (1+ 2 ) + 2 X -OD + i sin3 12 H H() (klx'-x cos J) dx' (17) where IZ and 14 are defined to be zero for x <0 and -ik x cos a cos 3 (e for x >0 F (x) = [O for x 0. In this form the equation is valid for all x. The application of a Fourier transform to equation (17) can be justified by a study of the growth orders of the functions V (x), I2(x), I4(x) and 04(x) for large (x1. 12

THE UNIVERSITY OF MICHIGAN 2778-2 -T The Fourier transform of, for example, 4/ (x) is c eo -ifx 4/ (~) = 1 e Y/(x)dx and from the definition of VJ(x) it is immediately obvious that 1/ (f) is regular in a lower half-plane of the transform variable'. For the currents Ip(x), p = 1,2,3,4, the assumption that -ik x cos a cos 3 I (x) W e P as x - co implies that the transforms I (9) are regular in the same region and, p finally, by using the asymptotic expansion for the Hankel function when x is large and negative we have that p(r) is regular in an upper half-plane. All these regions of regularity overlap and within the common strip it is permissible to apply a Fourier transform to equation (17). By inserting the Fourier integral representation of the Hankel function, the right hand side of equation (17) becomes - 1 2 {klx) ( + I a 2 +isin Iz(x') a X d gdx' 27 J K k2i4(x2, 2 x Jkco - x2 13

THE UNIVERSITY OF MICHIGAN 2778-2-T which can be written alternatively as p_ ( ~ ~ ~ ~~ i x 1 2 _ ~ —- 4- (k - )I4()- sin3 I2() d where r = k2 cos2/- ~ 2 and 6 is a straight line path from -co to oD lying within the strip of regularity of the integrand. Application of a Fourier transform now gives -04(~)=f Ycosasing3cos sin ( k2 t'2) -_ s in_ I( 4 fi(g +kcos a cos ) k r 2 (18) and similarly, from equation (12), Y cos2I_ sing k___ 4 + 02 - in - I4()+ (+ kcos2 I i( + k cssacos3) /P (19) These are sufficient to specify Ig(;) and I4(j). The obvious way in which to attempt the solution is to eliminate I2( ) and I4(A) in turn and thereby obtain equations for each current separately. Unfortunately, however, the resulting equations are not capable of being treated by the Wiener-Hopf technique. An essential feature of the technique is the separation of the terms into two groups having overlapping regions of regularity, and in the process of eliminating a current it is necessary to multiply one or other of the functions 02(t) or 04(g) by a quantity which destroys its regularity in the upper half-plane. A consequence 14

THE UNIVERSITY OF MICHIGAN 2778-2-T of this is that coupled equations of the above type are, in general, not capable of being solved, and it is only by virtue of a particular symmetry existing between equations (18) and (19) that I2( ) and I4( ) can be determined. The clue to the method of solution came from an attempt to guess a relationship between the currents. If it is assumed that I-2(g) = f(g) T4( ), then f(g) must be such as to reduce (18) and (19) to essentially the same equation. Since both I2(g) and I4(9) are regular in a lower half-plane, f(g) must also be regular in this region, and from the way in which 7 enters into the factors multiplying the currents it is clear that f(g) must be chosen to satisfy (k2- 2)f-k os2 sin ) f The resulting values of f(g) are _ kand ksins k sin tsin13 and - _ _ but neither of these is sufficient to bring into agreement the remaining terms in equations (18) and (19). The failure of the method is not very surprising since it presupposes an extremely close connection between the currents. On the other hand, the method does reveal a simple relation between the coefficients of Ia( ) and I4(t) in equations (18) and (19), and suggests that a profitable approach would be to use the values for f(g) to derive equations for certain linear combinations of the currents. 15

THE UNIVERSITY OF MICHIGAN 2778-2-T IEf (18) and (19) are multiplied by k sin,3 and g respectively, and then subtracted, we obtain { I2()-k sin( 4() ( +- -)+JZY ( cos k-kcos a sin23)cosg =ksinf34(_ )- 2(;), r 7T i(g+kcosacos3) 2 (20) and since the manipulation has not affected the regions of regularity, this is an equation for I2(~)-ksinB j34() which can be solved by the Wiener-Hopf technique. For this purpose, let kcos _ K- ()'2cos+ - - - / rf3 7 K+(g) where K+(f) and K_(g) are regular in upper and lower half-planes respectively. Bearing in mind the definition of P, it is apparent that K+(W) and K_(4) differ from the "split" functions given by Senior (Ref. 3) only in having k replaced by k cos 3, and accordingly their analytical expressions can be obtained from that paper. If equation (20) is multiplied through by cos B K+(g), it can be written as I2(9)-ksinT4()K (~ - Ykcosaos2 Kt(-kcosa cos 3) -I 4 K( ) - Yko i( +kcos a cos ) f/Ycos213 ( r os - cossn -)k cos a s K( ) k os K(-k cos a cos3) = _ -Z_ Ycos' — 7Yr^T ~i( +kcos acos 3) + k sin 304( )- z02() cos 3 K+(9). 16

THE UNIVERSITY OF MICHIGAN 2778-2-T The left hand side is regular in a lower half-plane, whilst the right hand side is regular in an upper half-plane, and, moreover, these two regions have a common strip. It follows that each side must be equal to a function which is regular throughout the whole plane and its growth order as | | - oo then shows it to be at most a constant. Hence K+(-kcosacaosf) A' g I2()-ksin I4(g )=-i Ykcosacos2 K kco cos + 2 4 r ( +kcos acos f)K K ) K_() (21) where A' is independent of g. To determine I2(g) and I4(t) individually, another linear combination is considered which introduces the second value of f(g). Multiplying equation (18) by g, equation (19) by k sin 3, and then adding, we have {ksin3I2(')+ I4()\ (+ ) +j YsinB cos kcos co i - i( +kcos cos)cos = - g04(g) -ksin 302(4), (22) which is an equation of the Wiener-Hopf type for ksin3I2(w) + g-I4(). The factor multiplying these currents can be written 7? C (cos B + k os L_() kcos< rkcosp L+(y) where L+ ( ) and L ( ) are regular in an upper and a lower half-plane respectively, 17

THE UNIVERSITY OF MICHIGAN 2778-2-T and since L ( ) and L_( ) differ from K+(<) and K_( ) only in having 7cos 3 replaced by COS, their expressions can also be deduced from the formulae given by Senior (Ref. 3). If equation (22) is multiplied through by kcos L+( ),its c7 \/kcos3+ + terms can be rearranged to give ksin3 I( )+ I4( ) /kcos - L( )tY k sin2incos3 L+(-kcos ccos ) 7r 7 i( t+kcos acos3)) /kcos 3(l-cosa) JT= y k sin(cos2 rL+ (kcos(+cosa)' () - Y (k cos (+ C-os a) r 7 i( g +k cos c ) a jkcosB3+ L+(-kcosaccos 3) ] _ _ _ kos, L+() -ksin.acos 4()+ksin(3()I.cos _ k cos 3(1-cos a) J7 //k os 3+ (23) and by the same argument as before each side of this equation must be equal to a function regular throughout the whole S plane. Using the fact that for large 1I1, L+(1) and L _() are 0(1), the right hand side of (23) then shows that the analytic function is at most a constant, and hence, ksin3I2())+ sI( =Y sin asin 3cos3 L+(-kcos acosB) ksingIk cos)+ cos3 \kcos3 (l-cosa)(kcos-g )L()YB' kos+ (24) /kcos(-g L_(g) 18

THE UNIVERSITY OF MICHIGAN 2778-2-T where B' is independent of. Expressions for I2(9) and I4(9) can be obtained from (21) and (24) by eliminating each current in turn. The resulting formulae can be simplified to some extent by defining new constants A and B such that A' = i Cl Ycos,3 A+K+(-kcosacos3)t B' = i 2 Y sin2a sin, cos2 k cos- B tlr~ 7 >1 -cosa and this leads to the equations (Y +k sin23) I(g) = i Y s (k cos a cos ) + A( +k cosa cos P V2 Ir ( s +k cos a cos 3) K_( ) + i y Yk sin2a sin2 cos23 kcos kL+(-kcosacos3) +B(g+kcosacos3)t lr 7 \/(1-cosr)(kcos3- -S) ( +k cosa cos p) L_(9) (25) ( +k sin23)() = -i Y k sinf3 cos t K+(-kcos cos) +A( +kcosa cosp) V (t +kcoscacosp) K () + i JI Y s- sin^2 sinf3 cos2J k cos {kL+(-kcosacos)+B(!+kcosacos)4 7 1(l1-cosa)(kcosj- ) (+k cosacos3) L_(:) (26) As they stand, the above equations do not have the required regularity in that the expressions for Iz(M) and L(r) have poles at the points + = +iksin3 which lie in the region where the transforms must be free of singularities. If the poles are eliminated, however, two conditions are obtained and these serve to specify A and B uniquely. 19

THE UNIVERSITY OF MICHIGAN 2778-2-T The equations for I2(9) and I4(Y) each give rise to the same conditions on A and B, and it is therefore sufficient to consider only equation (26). For I4(J) to be regular at s = ik sin3, the right hand side of (26) must be zero at this point in order to balance the corresponding factor on the left, and the condition for this is A = e i e isin2a sin cos T cos K-(iksinp) B i in K(-kcocos 7Y l-cos a L_(iksin 3) cosr cospj+isin3 iP/2 + ie I sin2casin 3cos | cosp L+(-kcosacosf) K.(ik sinn)'7 1-cosa cosa cos + i sinl L_ (ik sin 3) (27) Similar analysis applied to the point K = -ik sinf gives -i3/2 K_ _ ______in___-k cosa cosp) A = -i e 2 1 sina sin i cos, cos K_(-ik sisin) B si (-k cosa cos) V l-cosa L_(-ik sinp) cosa cos3 - isin3 -iI/2 1 2s o /cos, L+(-k cos cos3) K_(-ik sinf) - ie - sinasin3cos3..... - - 7^ dv l C1-cosa cosa cosj -isin L_ -ik sin3) (28) and from (27) and (28) it is a simple matter to determine A and B. The values obtained are A = si(c ios2 a cos S Q() - sin 3P(3) K+(-kcos a cos P) (cos2 cos + sin 3)P() + 2 sin2a sin3 cos, / cos: L+(-k cos a cos 3)1 Vl-C(os (29) 20

THE UNIVERSITY OF MICHIGAN 2778-2-T B = ______ —-— 1 [{isin a Q(3) - cosacosf P(3)4 L+(-kcos a cos 3) (cos2a cos2P + sin2P) P(3) sin2la cos + (30) where p() Q(3) = ei1/2 K_(iksinf) + e /2 K(-ik sin) L (ik sing) L_(-ik sing) (31) If the expressions for I2() and 4(9) are now examined, several facts are immediately apparent. Since all the split functions are 0(1) for large I + 0, it follows that I() = o(l'), 4( = o(| l-3/2) and hence I2(x) - constant, I4(x). x1/2 as x - 0. This shows that the electric current perpendicular to the edge is zero for x = 0 (in accordance with our previous assumption), whilst the electric current parallel to the edge is finite there. The same behaviour is found* in the case of * In Reference 3 it is incorrectly stated that K+(.), etc. are 0( ||-1/2) for large I. This does not affect the derivation of the solution given therein, but does invalidate the statements about the behaviour of the currents for small x. 21

THE UNIVERSITY OF MICHIGAN 2778-2 -T normal incidence (3 = 0), and it will be noted that a consequence of the finite conductivity is the removal of the current singularity at the edge. When 13 = 0 equation (29) gives A = 0, and equations (25) and (26) then reduce to i2 () = i 2 y K+(-kcosca) ( +k cos a) K (9) and I4(=) 0 which agree with the known solution for normal incidence. The only other case of interest is that in which 7 = 0, but this is most conveniently considered in the next section. It is now time to turn our attention to the magnetic currents I1(g) and I3(). These have to be determined from the integral equations (15) and (16), which are of similar form to the equations (11) and (12) already discussed. Although the correspondence is not complete because of the lack of symmetry in the incident electric and magnetic fields, it is sufficiently close for us to omit the details of the solution. By the same method as was used for equations (11) and (12) it is found that (2+k2sisn21) I- ( ) = -i S/k sina sin2 cos22 { kL+(-kcosa cos) +C( +kcosacosf)t ( +k sin P3)?) =I -i-2 k simsmi cos j3 ( ^ ___ ____ - vi (g +k cosacosf3) L_( ) - i il-? sina cosf / k cos {K+(-kcosacos) +D( +k osacos)j (1 -cosa)(kcos13- ) ( +k cosa cosj) K ( ) (32) 22

THE UNIVERSITY OF MICHIGAN 2778-2-T (+k2 sin )I(r = i g si2 sin2 co ( kL+(-kcosacosB)+C(t+kcosaCOs3) } 3r (S+kcosa cosp)L_( ) - i I- ksinc sin3 cos3 ( k cos \l~ T K_. 1 -ncos)(kc - gK-(-k cosa cos)3)+D(g+k cosa cos3)l qnfi^ t) ( S+kcoscrcos,) K (0) 1V ^ 1- (33) (c.f. equations 25 and 26), where the constants C and D have the values 1 [ti sin3 Q'(3) - cosa I L-kcosacos3 P') L(-koscos (cos2a cos23 +sin2P3)P'(J3) + 2 - cosa cos K+ (-kcosa coso) 1 -coscr (34) sin3 I cos3 V -cosa [ {i c oacos Q'() - sin3 P'(j)} K+(-k cosacos) (cos2a cos2J + sin2f)P'(p) + 2 sin, cosp / 1-cosa cos3 L+(-k cosa cosl3) (35) with P'(13), Q'(,) = e / K_(ik sin3) ei/2 L_(ik sin3) K_(-ik sin,3) L (-ik sinj3) (36) The remarks made about the electric currents I(S) and I4() apply also to the magnetic currents I3(S) and I1 () 23

THE UNIVERSITY OF MICHIGAN 2778-2-T 4 THE FIELD COMPONENTS Since the electric and magnetic fields can be written in terms of the transforms I (5) alone, explicit determination of the currents IT (x) is unp P necessary. From equations (8), (9) and (10) we have, by inserting the Fourier representation of the Hankel function and restoring the z dependence, E EX 4 Z k2 I r2 sz in (S -I3() Yl 0d'*6le~~~~~~~~~~~ ~(37) ---- E iY - IT(H ) I Y 42 2 k3 - y y Ei 4F75r I3(s) - k sin- Z[sinLL(g k 1( JIyI (38) E = 1 ZP kcos2f3 IL() - z gsin_3 --- - Y z z - 4 I1 2 r 4 lyl ) d (39) with eix+i l P - ikz sin3 and if these are expressed in the form E = E i+ () cos 0d, 2-iri J g2+k2sin2f c+kcosacos3 (40) the components of I () are ex(=) 1 J sin sin cos jkcos (1 +cosa)(kcos+g) k -kcoscos B(+coscos) k L(Y) k2 3 SK+(-k cosa cos3) +A( g+k cosa cosp) k-. osinp k2acos2-_g2 K_(%) 24

THE UNIVERSITY OF MICHIGAN 2778-2-T Y sin sin/ cos3 kL+(-kcosacosP) +C( +kcosa cos3) I Y L (_ ) Y+ s ksin3 kcosg(l+cosac) K(-kcosaccosf3)+D(g+kcosacosf3) I7y V kcosp- r K_(S) (41) =2g 2+k2 sin2[ s sinp in-kc 1 [kL+(-kcosa cos3) +C( +kcosa cos3) y ) k2O1cos2l2 L_() 2 -1 I cosg(l+cosa)(kcos3+g) i kL+(-kcosa cosl3)+B( 9+kcosocos3)I 7 IYI k (42) () _ — I sin sin23 cos]k kcosj(1+cosa)(k cos3 - ) ^kL+(-k cosa cos3)+B( +k cosa cosj), (M) = - 1 sinasin2pcosBp s~kcosB(l+cosa )(kcosB-L)) 7 L (t) _ kS /cos2 -2 K+(-k cosa cos,3) + A( g+kcosa cos3) K_(s) + Y k sina sin2j cos,3 IlY] kL+(-kcosa cosj3) +c( s +kcosa cosB) s) $K+( -k cosr cosp) +D( g +k cosa cosj3) + -; LY IYI I k on.sI14 K " J, Nv. t.I X. I, % 1Ikx k cosp - g K(9) (43) The corresponding equations for the magnetic field can be deduced from (37), (38) and (39) by using the duality referred to in Section 2, and if we similarly write 25

THE UNIVERSITY OF MICHIGAN 2778-2-T 21ri _^\ _W(_ ) cosp t2 +k2 sin2f g+kcosacosp 0 d_ (44) (c.f. equation 40), it can be shown that = - k SK (-k cosa cos3) +D(g +k cosa cosf3) Jkcos3(l +cosa)(kcos3p+) +! -1- )(tK () + k2 _Slna sin2 cos2 kL+(-k cosa cos3) +C(g+k cosa cos3) iC+2sos2in si2 L (s) ~-k^cosg2-^2L (5) IYsT K+( -k cosa cos3) + A( +k cosa cos3) K () 1 Y k sina sin2 cos Ik cos3(l+cosa) kL+(-k cosa cos,) + B(g +k cosa cos3) 7 IYI kcos3 - g L_(9) (45) HY> = g2+k2sin2 P 1 2cos2 _ 2 K (g) [;K+(-k c os cos3) + A( +k cosa cos3) Y -7 Y lyl \ cosl(1+cosoa)(k cosf3+) I K+( -k cosa cosBp) + D(J p+k ncosa cosp.)j k (4.6) t- = - r sin3 Jkcos+(l+cosa)(kcosf+S) IK+(-kcosa cos,) +D(+ k cosa cos3) ~~~~~~~~z'~~~~K (i) + k - sin sinB cosp kL+(-kcosacosg) +C(f+kcosacosJ3) V2cos23 - 2 L_( ) 26

THE UNIVERSITY OF MICHIGAN 2778-2 -T Y k i SK+( -k cosa cos3) +A( ~ +k cosa cosp) T+ --- k sing lyl K () - ~ 2 sirn sin cos3 kcos3(l1+cosac) kL+(-k cos cos) +B(+k cosa cos3) Y7 YI kcos3 -X L_(X) (47) An exact evaluation of the integrals in (40) and (44) would appear to be impossible, but for most purposes approximate values are sufficient and these can be obtained by the method of steepest descents. The present paper, however, is more concerned with the formulas themselves; and of particular interest is the extent to which they differ from the ones predicted by the technique described in Section 1. If the technique were valid for imperfect conductivity, the field components would be given by E = (i i sing, Ui sg U cos2) k ax k ay = Y a au H _k a\ y ax / where U is the modified two-dimensional solution, and hence E = Ei R cosg p d~ (48) 2i J Jk2cos2j-3S2 t+kcosa cos3 H = Hi _ _ _ S cosf 0 dS (49) -27i J cos2-S2 +kcosa cosf 27

THE UNIVERSITY OF MICHIGAN 2778-2-T with R = (gsin FYI r sing, kcos2I)2 K4(-kcosacosf3) {1-y (l+cosa)(kcos3+) ) <. r ( y Ol K+(-kcosac os)) +cos(kcosP3+ ) - [y[I K/( k y kcost It is immediately obvious that this differs from the true field, and with many of the components the discrepancy is at least O(y). On the other hand, the boundary conditions themselves only reproduce the physical conditions to O(y), and if this accuracy were achieved by even two of the components found by the above technique, they might be sufficient to describe that portion of the true solution to which a physical interpretation can be attached. To determine whether such components exist, it is necessary to expand E (S) and Y(9) as series in h. Using the expressions for the split functions (Ref. 3), we have K+() = kos+ 1_ ) k2cos2f-2 (r sin _ _ + 2- (log cos 1) kcos3 ( r k V kcos)(/ irk 2 + 0 (2 log7 ) (50) L+() = \i (, 1k -r -sin +o(2) (51) cost ir Tk2cos2f - 2 2 kcosf ) where the expansions have been carried out under the assumption that e is finite. Substitution into the equations for A, Be C and D gives 28

THE UNIVERSITY OF MICHIGAN 2778-2-T A = 7 sini3 { 1 -cosa (log 2cs3 1) - /1 +cosa (ir-a) + (1log)% r cosl B = -~ ~~ 1 I 1 + 7 log cos3 _ i) - _ (Tr _) l+cosa cosf (I+cosa)cos3 1 cos sinacos3 + 0(721~og )4 C = -2-' log Ncos] -1- (,r-a) cota+O(1log)} IT JCosS3 ( 2 ( J D Q= _22L (ir-a)sin2f 1 +O((log7)%, sina /cosp and it is now a simple matter to compare the two sets of field components. From a study of equations (40) and (44) it is seen that because of the factor ( +k sin 23) in the integrands, the components H and E are the only ones for z y which the technique is likely to succeed. The former is identically zero according to equation (49), but from equation (44) the true field has z = 2 sina (+ksin2) +1 1 - cosa ZP ( \k2jcos2j_-2 X IYI kcos3(kcos3 -g) [( +k ocosag osos3) og? 3 -1 - k (r -a) cosa c osI 2 1 - cosa + ( - +sin-1 c ) kos -~ +O( 7log kcosos 3 kcos + 29

THE UNIVERSITY OF MICHIGAN 2778-2-T which is of order ~ log. For the component Ey a similar analysis shows that the value predicted by the technique is again in error by terms 0(7 log') and, in fact, this is the same with all the field components. In consequence, the technique which is so useful for treating perfectly conducting bodies fails completely when the surfaces are imperfectly conducting. The above discussion is based upon a comparison of the expansions for g(g) and {(B) with those for 2+k2sinfg R and S2+k2sin2 S k2cos2 _ - 2 4COS -2 and the implication is that discrepancies are automatically reflected in the corresponding expansions for the fields. Since the proof is not quite straightforward, a few words of explanation are in order. The expansions for the split functions were derived under the assumption that g is finite, and this restriction applies to all the subsequent formulae. In equations (40) and (44), however, the variable of integration S takes values from -oo to oo and if the expansions for 0(J) and &A(g) are merely integrated term by term, the expressions which are obtained are incorrect. The difficulty can be overcome by using the method of steepest descents to approximate the integrals in (40) and (44). Almost the entire non-exponential portions of the integrands can then be removed at the saddle point Y = o and included in this are the functions (9) and _(). Since go is, of course, 30

THE UNIVERSITY OF MICHIGAN 2778-2-T finite, J ( ) and WM( ) can now be expanded as before and the individual terms generate corresponding terms in the expansions for E and H. A direct consequence of this is that all the field components contain terms in'7 log? in addition to those involving powers of 7 alone. 5 CONCLUSIONS In this paper the solution has been obtained for the problem of a plane wave incident at an oblique angle on a metallic half-plane. The analysis requires the solution of coupled Wiener-Hopf integral equations for the Fourier transforms of the four current distributions, and the resulting expressions for the fields are exact, subject only to the (physical) approximation implied by the impedance-type boundary conditions. The solution has applications to the coastal refraction of radio waves, but this topic is reserved for future consideration. It has been shown that the technique commonly employed to determine oblique incidence solutions for perfectly conducting bodies cannot be used when the conductivity is finite and, indeed, all the field components produced in this way are in error by terms O(~ log'). It would be extremely valuable if a technique could be developed for treating finite conductivity, but unfortunately the analysis has not suggested one. From the solution given in this paper it is clear that such a method would have to call upon the normal incidence results for both polarizations, with A replaced by V cos3 in one case, and by: sec3 in the other. However, 31

THE UNIVERSITY OF MICHIGAN 2778-2-T the presence of the factor % 2 + k sin23 in the expressions for all but two components and, in addition, the occurrence of the complicated constants A, B, C and D, make the existence of a technique very unlikely. REFERENCES 1. Clemmow, P.C. 2. Senior, T.B.A. 3. Senior, T.B.A. 4. Stratton, J.A. 1951 1953 1952 1941 Proc. Roy. Soc. (A) 205, 286. Quart. Journ. Mech. Appl. Math. 6, 101. Proc. Roy. Soc. (A) 213, 436. Electromagnetic Theory. New York: McGraw-Hill. 32

THE UNIVERSITY OF MICHIGAN 2778-2-T APPENDIX: PROOF THAT I4(0) = 0 If it is not assumed at the outset that 14(0) is zero, the equations from which to determine I2(') and I4(S) are -0 ( =) J= Ycosasin3cos3 _ is I4(0)+ k2 2) I () -_sin 4 r i( +k cosc cos3) k/Vrr /' kr 4 z (Al) (= Y cos2 _ isinf 14(0) _ ssin 4() + ( + kcos2) 2(9 ) 7r i(5+kcosacosP3) JT r r kco (A2) which differ from equations (18) and (19) only by the single terms involving I4(0). As in Section 3, the first step is to derive new integral equations for those linear combinations of IZ() and I4(') which are suggested by the functions f(S). If equations (Al) and (A2) are multiplied by ksin3 and' respectively and then subtracted, a Wiener-Hopf integral equation for ~ I-2() - ksin IL4(g) is obtained. What is more, this process has also served to eliminate 14(0) and consequently the solution is identical to that given in equation (21). The second equation for a linear combination of I2( ) and I(S) is found by multiplying (Al) by' and (A2) by ksin3. On adding the resulting equations, we have {ksin3 It2() + 4(} ( + ) i 2 5sin2 k I 4.kk4 F + fi Y sinp cosp kcos3+g cosa = - 4() ksin3 () Na vi('W+kcosa cosf3) 33

THE UNIVERSITY OF MICHIGAN 2778-2-T which can be written alternatively as I 2+k2i2 1 ksinl I(t) + SI4() (kcosl3+) - i 4 (0) ksp'kcos3- L_() J2~ kcosp- J + i4(0) os +k sin2 L(kcos3) 2 kcs L+(k3cosp) 42 2 kcos2 - Xr 7 i( +kcosacos3) Jkcosj(l -cosa) k= _ r kY (k cos skcos+ cosa) L+() 47ffi7 i(9+kcosarcos3) os k cos3 +. - k sin2 cos L+(-kcosacosl) k kcos(1 -cosa)- i4(0) kcos3 %2+k2sin2 3 Lcos+) L+(kcosB) = 2-(~ kcos3 -S kcos3+ 2 kcos f - it4(g)+ksin pL(-)} kcosk ckcos+~ L+(). This is now in the form required for a Wiener-Hopf split. The left hand side is regular in a lower half-plane, whilst the right hand side is regular in an upper half-plane, and since the two regions overlap, each side must be equal to a function regular throughout the whole S plane. The growth orders of the two sides as J| -- co then specify the function as B" + C", where B" and C" are independent of 9, and hence 34

THE UNIVERSITY OF MICHIGAN 2778-2-T ksinp I2(%) + I4(g) i fiy k 2 sin2a sing cos3B ir'2 %+kcosacos f L+ (-k cosa cos3) 4k cosj(l -cosa)(k cos- 5)L (Y) _ i I4(0) 4 -2 g2 +k2sin23 k2cos2-_ Z2 s kcos, L+(kcosj) 1 2(kcosf3-S) L_(s) -1 + B"+C" (kcos 3+ ) kcosI-Z L_(%) (A3) From equations (21) and (A3), I () can be eliminated to give an expression for I (S) alone. The equation obtained in this manner is 4.2 2 2 (' +k sin 3)I(4S) + iY Tr - i -TY k2 cosa sin3 cos2f3 k sin2a sinlcos2l f7 (I K+(-kcosa cosj3) ( +kcosacos3) K_ () k cos3- L+(-k cosa cos3) I cosr)(kcos,3-%) (s+kcosa cos3)L (S)' % - i 1(0) _ a7 % 2+k2in2p k2 Cos2 -_ 2 / k cos, I 2(kcos3 -t) L+(k cos 3) L (%) - 1} + (B"+ C" 6 )_(kcosf3+ ) k cos3 - L_(j) from which it is seen that I4() = o(|I [3/ ).'4 _ A'ksin_ K_(The Fourier transform The Fourier transform relationship now gives I4(x) - x1/2 4 as x - 0 and accordingly the electric current perpendicular to the edge is zero there. 35

UNIVERSITY OF MICHIGAN II 1111111lil1 11111 11 1 3 9015 03525 0441