ABSTRACT ON THE DIFFRACTION OF WAVES FROM A FINITE WEDGE by Daniel Richard Killoran The scattering of electromagnetic waves emitted by a line source and impinging on a wedge is examined by means of a combi- nation of the Wiener-Hopf technique and a modification of the Lebedev- Kontorovitch integral transform. The wedge is considered infinite in the axial direction but finite in the plane perpendicular to the axis. An infinite system of equations involving values of the unknown transform function and its derivative at special points is obtained, but the system is not solved. For the special case of a strip with the source at an infinite distance from the wedge, a simple assump- tion leads to agreement in the first order with the results of Sommerfeld, but produces disagreement in the second and subsequent orders. For the symmetric finite wedge, the nature of the variation of the cross-section arising from a change in the wedge angle is determined qualitatively. ON THE DIFFRACTION OF WAVES FROM A FINITE WEDGE BY Daniel Richard Killoran A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCT OR OF PHILOSOPHY Department of Physics and Astronomy 1963 TH ESl ”/771. .-»< w _. I . /¢‘//a [It’d ACKNOWLEDGMENT The author wishes to express his sincere appreciation for the encouragement and assistance given by Dr. Alfred Leitner in conjunction with the research presented in this thesis. ***>§<**>‘.¢***** ii II. III. IV. V. VI. VII. VIII. IX. X. XI. XII. XIII. XIV. XV. XVI. XVII. . Introduction . . . . TABLE OF CONTENTS Statement of the Problem. ..... . . . . . . . . . . The Integral Representation . . . . . . . . . . . . . . The Representation of the Scattered Field . . . . . . . The Representation of the Source . . . . . . . . . . . The Second Boundary Condition and the First Integral Equation............ The Third Boundary Condition and the Second Integral Equation . . . The First Integral for F1{H3 and the Function C i u}. . The Second Equation and the Function Aiu} . . . . . . The Wiener-Hopf Technique Applied to Flip} . . . . . The Infinite System of Equations for A1: -m}. . . . . . The Infinite System of Equations for B{ -m}. . . . . . The Series Expansion of the Solution. . . . . . .l . . . The Scattering Cross-Section for the Symmetric Case 8A a P- . . . . The case Of the Strip 0 O O O O O O O O O O O O O O O O O The Case of the Wedge . Special Value 3 of iii Page mO‘UON ll 13 21 24 31 32 33 36 38 39 44 THESI IEC I. Introduction This thesis examines the problem of the scattering of electro- magnetic waves emitted by a line source with the electric vector parallel. to the axis of the source and impinging on a perfectly con- ducting strip or wedge of half width b. The wedge angle 39 and the location of the source with respect to the wedge are arbitrary, and the wedge is to be thought of as having infinite length in the direction parallel to the axis of the source. The problem is treated by a combination of the Wiener-Hopi technique and a modification of the Lebedev integral transform [1]. An infinite system of equations involving the unknown transform function and its derivative is obtained, but the system is not solved. For the special case of a strip with the source at infinite distance from the wedge a simple assumption leads to agreement in the first order with results previously obtained [2], but produces disagree- ment in the second and subsequent orders. rt 01 W17 fir Cir Val II. Statement of the Problem Consider a wedge of infinitesimal thickness and of infinite extent along its axis (the z-direction). Let the length of one side of the wedge be 2 and the interior angle be 19. Place a line source of waves (:5) at the point (r0, o0) with its axis parallel to that of the wedge. Calling the z-component of the incident electric field Ui, then Ui=i1rH${k\[rz+rg-2rrocos(¢-¢o)j (l) where _l§ is the wave number and H3 is the Hankel function of the first kind (see Figure 1). We will use the abbreviation R= N/rz+ 173-21% cos(¢-¢o) (2) Then we have U=US+ianflkRJ (3) We seek a solution (U) of the 2 dimensional wave equation in circular cylindrical coordinates div grad Ut+ kZU = o (4) valid everywhere and satisfying the boundary conditions: eikr (a) U, Ui and US-> as r —-> oo . (Sommerfeld radiation r condition). (b) U = 0, rb. ((1) US is continuous for all values of 3. in O] a Ma ensu thEn III. The Integal Representation The representation chosen is the modified Lebedev transform: Fix} = fLnyl KyZXZ WW (5) where Ky{x} is the MacDonald function, or Hankel function of imaginary argument, defined by the relation n X in Hlyiix} = 2e 2 Kyix} (6) and the contour extends from a -i 00 to a + i 00. This modified transform is chosen in preference to the more usual transform fo3 = fLGiyl 1Y{Xl Y (W in order to simplify the representation of the source function. Moreover, if a residue series is extracted from this integral, each term will contain a MacDonald function which, upon returning to real wave number, will ensure that the solution satisfies boundary condition (a). We must extablish the conditions under which equation (5) may be inverted. One form of Lebedev transform theorem [1] states that: Theorem I. If fol = IL ny5 Iy{x3 y dy then i n Gay} = ID Fix} Kyix} £355- I7" THES] l [1.7- r ' the : we , Provided that: GZy} -is analytic in a strip of finite width containing the imaginary axis . 1 2' -decays at least as fast as lyl- expf '[Yl "121} as y—> s :i: i 00 where 5 lies in the strip. Fix} -goes to zero like some positive power ofx as x —)- 0. -is bounded by lxl eX as x—-> 00. L -is some contour from a - i 00 to a + i 00 lying entirely within the strip. It can be seen that G {y} is even. Then if we break the first integral up into an integral along the top half of the imaginary axis plus an integral along the bottom half, change y to minus y in the second integral, and recombine, we get 1 00 Fix} = ID GU} (1-31% -IY{X}) Y dy Using the definition of the MacDonald function 17 Luix} - ly{x} Kyix} = _2- Tsin y 11 (7) and the fact that the resulting integrand is even, we can again extend the integration over the original contour. Then Fle = ’71, IL val KYEX} sin Y" Y dy We may now redefine Giy} so that Giy} becomes -11 (II—{£3— sin y1T Using this new definition of C {y} , we have: \v—l th Theorem II. If FZXS = IL cfyg Kyixg y dy (8) “IT2 (I) then i iffy“ = [0 pm Kyixg-‘if (9) Provided that: Giy} -is analytic in a strip of finite width containing the imaginary axis. -grows no faster than IyI 2-exp{ Iyl 73 as y —-> a :I: i 00 where 3 lies in the strip. F{ x} -goes to zero like some positive power of x as x -> 0. . x -1s bounded by le e as x—> 00. L -is some contour from a - i 00 to a + i 00 within the strip. at 1'6 L1 tl IV. The Representation of the Scattered Field In order to use the Lebedev transform effectively, it is desir- able to consider the wave number to be imaginary, a method due to Oberhettinger [3]. Let where g is to be real and positive, so that the functions U, Ui, and US decay exponentially rather than algebraically at infinity. It will be necessary to use different representations in the three regions of the (r,q>)-plane (see Figure l) in order to ensure continuity. Let Us = IL(F1{ H3 C08p¢ + in p} Sinp¢) Kp{gr} pd}; (ll) in Region I Us '-' IL (M1 [MCCSHXH -¢} + 'MzsinHZv-‘b‘x )Kpfgr} udu (13) in Region II U = 3 IL (M1{H}C°3H {17+ 4’} ' Mai H} sinwf 1r+ id ”(“5er P-di-L (13) in Region 111 We note that we already have continuity for ¢ -> :1: It. Now we require continuity for <1) —-> ip . The integrands of the expressions on either side of the wedge must be identical, and the resulting set of equations gives M1 cos p {11- [3} F1 cos pf} (14) M2 sin HZ 17" [33 II F; sin M3 (15) FIjMV‘B 1-» H SGMLQmm '9 \\ O THESE V. The Rggresentation of the Source According to equation (10) given by Oberhettinger [4] i 00 2 f Kpi gr} angro; cos ”{fl'|¢'¢ol} dp. (l6) i1T o Ko ZgRg". The integrand of the above is even in p, so the integration can be extended over the entire imaginary axis. i 00 f . Kpigr} Kpigrog cos péw- I4) - (#0)} le. (17) oo -1 l in KoigRi = Since the integrand is entire, we may deform this contour to coincide with L, the contour used in the representation of Us' Then, using equation (6), we have ~ 2 in Hgfigpfis = 3;— IL Kpfgr} KH{grO} cos pfn - I¢ - iol} dii (18) THESi VI. The Second Boundary Condition and the First Integral Equation For the total wave functioni we now have U = f (F cos 4: + F sin ¢ +¥-Z—K { r cos {Tr - (43-4) )3) L l l4 2 H lTTp. p, g 0? H O . Kpigrg pd}; (19) We will assume for convenience that B> l¢ol (see Figure 1)- Then, by the second boundary condition we require that U = 0 for (b = g. . 2 0 — fL(F1COS}iB + FZSIan+ m Kpigrog COSuifl-Bi‘tbéflfil gr} pd}; (20) and also at 4) = - [3. 0 =f (F cosuB - F sian -2— K {gr (cosnin-B-433K {gr} pd}; (21) L 1 Z in}. P- 0 0 P Adding the above equations, and reducing the sum of cosines, we have 2 0 = fL(F1cos}i{3 + 177—}: Klli grog cos}i¢0 cosnin-fl))KP£g1:j}id}i (22) r52 Subtracting the equations, . Z . . 0: IL(F231n}iB 'prIgro.) Slnp¢o Slnp. {Tr-fl} )Kpigr} udp. (23) rb. Differentiating equations (11), (12), and (13) under the integral sign, and letting 4) -> :i: [3 .19. 84> ifi= IL ('FISlnipfl + FZCOSiHB) Kpigr} “.de (28) in Region I as “f (M sin in -fii - M cos {17- [3} )KI r} 2d (29) 94’ 413- L 1 H 2 P' p, g P- p. in Region II U . 3E5 l-fi 3 IL(’MISIHH{TI ' B} " MZCOSHI: TI - ‘3} ) Kfliglj )J'ZdIJ- (30) in Region III Requiring continuity for it = B. o = fL(-Fzsin}i}3 + cmosufi -M181np.£TT'(-3}+ MzCOSHU-B} ) ° Kptgr} +1de (31) Requiring continuity for 4) = -B. 0 = fL(Flsin}i}3 + cmosufi + Mlsinpffi - [3}+ Mzcosuifi " 53) ° Knigr} +£de (32) Subtracting (31) from (32) o = F 333E11— K r} 3 d (33) IL 1{ ll} COSH{fi-B} pig H H r>b ll THESI 12 Adding (31) and (32) _ sin 1T - HEW} sinnLTr- 43} K1”: gr} ”2 d” r>b (34) which will be referred to as the second integral equations for F1 and F2 respectively. TH E51 VIII. The First Integral for F; {p3 and the Function Ci B3 We must now determine whether equation (26) can be closed on either half-plane, and whether its integrand has any singularities. Letting d) = (3 in equation (19) U {6}: fL(F1cos}ifi + Fzsian+ 712:; Kpigroj cosn i" -B+¢03)- Kpigr} Hdtl (35) Letting o = - s in equation (19) U i #33 = fL(FlcosnB-Fzsin}iB+—E— Kpigrog cos}i Z fl-fi-¢og) . iflp Kpigr} pd}; (36) Adding and subtracting, cf equation (24), (25) Uifl} +2U£- (3} 41,566“? KH {gr} +1.1”, (37) UZfilz- Ué’fi} :IL Jail-‘3 Kpigr3 pd}; (33) Applying the inverse transform 2 (I) 11 .. Uifll + U {-{3} dr i sinpfir Jlil‘L}_ lb 2 Kpigr} ‘17 (39) and ____"2 0° Uffli - Ui-isl di- i sianr hi ”3: [b 2 Kp{gr} ‘17 (40) 13 THESI 14 Since the integrals on the right converge absolutely, and uni- formly in p. on the entire finite p. plane, the right hand side of equations (39) and (40) must be entire in p. Therefore, a fortiori, J1 and J3 must be entire functions of }i. The validity of this proof depends upon the applicability of the inverse transform. It can be seen, moreover, that the only condition of Theorem II not obviously satisfied is the restriction on the growth of JZ [.13. It can be shown, by a method presently to be exhibited, that the integrals on the right hand side of (39) and (40) have the asymptotic form (const.)’ Kpfgb} /}1 as l p. I")- 00. It then becomes apparent that J{ ~ ~ '% 1'— ip} ~Jziu‘i ., 1H1 eXp[|Hl 2} (41) as )Impléa) Examination of equations (24) and (25) then shows that szpi} sin}i[3 is entire d an (42) F, {p} cosnfi has but one singularity; at zero, possessing residue F1513» — Kozgro} as n—> o . (43) in}; However, since J1 and J2 do not decay at infinity on either side, the integrals (26) and (27) cannot be closed. We must modify the function in the integrand somehow to provide suitable decay. We will try to split the MacDonald function. Using the definition of the MacDonald function (7), equation (26) becomes 0 = IL Jiful— ‘I—‘i‘hgil— HdH - f LJlful— ‘I‘E‘Lg'l‘ “(11* (44) sinpm sinpnr TH ES’ 15 fl— I'DIZLVIE Late Figure 2, COMtOHV'S fa theft-Flume}, TH ES] 16 In the first integral we change the variable to —}1 and reverse the direction of integration. 0: thjli “3— Mpdp - f ”3.1ij m pdp (45) SlinT sinp'rr Now J, has no singularities; the pole at zero caused by the sine in the denominator is removed by the p in the numerator, and all the other poles caused by the sine are removed by the zeroes of 31, so we may deform the contour to coincide with the original contour L and combine the integrals. 0 = IL Li" H3“ J1£E3__ —Ip{gr} HdH (46) Slinl' Now define Cfu‘iE—g J1 {$193+}; hip} (47) So we have o = [L Gin} Ipfgrs no}. r oo , the integral converges at the upper limit. At the lower limit the U's may possess singularities, but by Meixner's [5] edge condition, the singu- larity can be no worse than U ii [33 —> (r-b)-%_ (52) which is integrable. Moreover, Kpigr} is an entire function of p , therefore the integral is an entire function of p . Then C£p73 is entire. A. Treatment of the local field For future reference, and in particular to show that Uffi} can be expanded as a power series in r , we must examine the expected behaviour of the solution at metallic boundaries for very small distances from the surface. In particular, we are interested in the solution near a corner. For the case we are considering (U = 0 on the boundary) U corresponds to the axial (2) component of the electricfield vector. Then let us assume that 6 where p is the distance from the edge or corner. Then the requirement that the volume integral of the energy density be finite over any volume, however small, and that it must approach zero as the volume goes to zero even at an edge gives us the conditions. fEipdp~IU2pdp ~ Inadldpzpz‘” 3Uz N - - (54) inpdp 1", f(-—9p) pdp ofpz(61)+1dpgp2(6l)+z IQUZ zé-z+z fpp 9 “/pr¢ p p p all of which must go to zero with p. Therefore 6>O. TH ES) 18 In other words, for this (Transverse Electric) case, the field U may not possess singularities at a corner, even if it is an edge. B. Asymptotic behaviour of Cf pi Since I_J_ is non-singular at r = b, it may be expanded in a series of ascending powers of (r-b). Expanding each power binomially, and collecting like powers of r, we obtain a series expansion of U :tfi in- powers or 3. This series will have an infinite radius of convergence, except for the special case ¢0 = :l: (3, which we ignore. Then we can write 00 n+0. Ui-(33+U {+3} = 2: an (gr) (55) n20 where a is some positive real number. Then 1 00 00 n+0. -l Cip} = 2 . 2 an I (gr) K {gr} d(gr) (56) 1T1 :0 gb 1* Since KP- decays exponentially at infinity, the integrals all con- verge absolutely and the inversion of the order of integration and summation is justified. Adoreover n 1 _ l Ix Hpi x} dx - (n+ p - l) x Hpix} Sn-1,).L-1{x3 (57) ' x HL-1£X}Sn, pix} where Sn’pflx‘j is Lommel's function [6]. Let 12 = x. Then 1 - _ 2' ' - .1 HH{1z) — .1" exp{-1 2 p} Kpfz} (58) f expii-g—nj zn Kpiz} dz = (n+p-l) z Kp[z} Snfi‘wrliizi (59) .. z expii €21} KH_1£z} sn,p{iZ} TH ES) (3 ‘i 1’ ‘1 .‘ ’14.! —— 19 Using the relations between Kpizj and Kpfliz} and a K/a z and the similar relation between S and its derivative, we have . TI’ . .[zn Kpiz} dz: e -177?)gKP{21‘—S'%M -Sn, gal—Ely} (60) But, according to Erdelyi, it al. [6] Sn,p{_iz} = smpiiz] + Aimn} Jp{iz} + Bin,}i] J_p{iz} (61) But when this expression is substituted into equation (59), the terms involving the Bessel functions become Wronskians, which are pro- portional to 1/2. Then the factor z makes these terms constants, and they will cancel out when we evaluate the integral at the limits. Therefore 00 CD 1T z 9 Sn p 3K 11 = _' _. _ __._L - __E Igb z Kpiz3 dz exp[ 1 2 n}(i) KP- a z Sn,p 6'): gb (62) But at the upper limit, Kpflz3 decays exponentially, while n-1 l sn’Hiz] ~z (1+0fz3) (63) Which is merely algebraic, so the expression in the brackets goes to zero at the upper limit. At the lower limit 00 a s {1gb} n = _- l} - n: P- Igbz Kpflz} dz expi 1n 2 (1gb)zKH{gb} 3gb (64) 9K {as} .. s {ing—L— n, P‘ 9 gb But n+1 . N (iz) l —-> Mini (MI); 2 (1+ 0 {F 3) as | (i I oo (65) THES' 20 This asymptotic series may be differentiated, so zn Kpiz} dz: const. Efié'gL} + const. m3 (66) P 3gb oo Igb QKlligbI 33b But 3 p Kpigb} as I p I —> oo (67) °° n K £133 Therefore jgb z Kpiz‘g dz 25—1.:J— as In)» on (68) From which it can be seen that CZfix—%—25b§ as I (1| -> oo (69) C. Closing the contour for the first integral equations We have already shown that 0 = IL C {p} 11* [gr] p dp r00 (70) Re p>0 But for :52 this expression decays exponentially, and therefore the contour can be closed by an infinite semicircle enclosing the right half-plane. Therefore the function C {p} is a suitable function for the application of the Wiener-Hopf technique. Similar considerations apply to D2 p3 [see equation (49)]. THESI IX. The Second Equation and the Function A {p} If, in equation (22), we change p to -p and change the direction of integration, we get sin TT o= f m- 113,05“, ,3 Kpigri i2 an r>b <71) But the only singularity present between 1/2 and -1/2 in the integrand is the simple pole of F1 at zero, which is canceled by the p2 in the numerator. So if the original contour lies between these limits, the contour L* may be deformed so as to coincide with the original contour L. Then we may subtract equation (33) from the resulting equation, a course suggested by the fact that the even part of F1 obviously contributes nothing to the integral. We conclude that -IL[Fx€-vi-Flfi31'3%EZ—irii Ktzgrz ,2 d» <72) r>b Unfortunately, the integrand of this equation exhibits an infinite number of singularities on both half-planes. We must, therefore, so modify the function that these disappear on at least one side. Let Bi’i‘j'Fl {11} = AU} - AL-ix} cos p (17-6) p sin p 17 (73) where we would like A {p ]to be free of poles on the left half-plane. Let AIB} be the discontinuity of across I3. Then from the .331 *3 ¢ manner in which equation (33) was obtained, it can be seen that Aft} - Alf-6% -21LFi{n}::ns*;"(, (3)131?ng dn (74) 21 THES) 22 Splitting the Kpigr} in the usual manner, Aial- AE-s}= -2IL(Fit-u1-Fiin})—}§;L}%-_§Tuzdu (75) or, in terms of A {p} AU? "Af'HI A - A - = - 1 d I {(31 1: Bl 11 IL sinim H{gr} PL 1* . (76) It follows from our treatment of the behaviour of the solution near a corner that A—> O as r —-> 0. We may apply the inversion formula (Theorem I) b mi?- AM" “'93 = 10 (Aipi- Ai-63)KH{gr} 9;- <77) Sinpfl’ Now if we identify -1 b dr Azii= 35- 10mm - Ai-fil)1-HZgr1 :— (78) then A i p} has no singularities on the entire left half-plane. Moreover, by a treatment similar to that accorded the function Cfp}, it can be shown that Aziiz—ILfi—fl as lpl—> oo (79) But K r 1_ b) ~ 1 _r_ “I“ (i 1‘” n (b asRep—> -oo (80) so if we split the integral (72) after substituting relation (73) we get 0 = ILA“) Kpigr} Pd H" IL Ai-HlKnigrlH dP- (81) Changing p to -p in the second integral and changing the direction of integration, we get .1 I M Q! 23 0 = ILAEM angr} (“1%) + IL* AER} Kpigr} (“111 (82) But A {p} is analytic in some finite strip containing the imaginary axis and L, so we may deform -L* into L. 0=f AIM K {gr} (“11* L ” r>b (83) But in view of relation (80), this integrand decays exponentially on the left half-plane (because E_>_l_)), so we may close the contour by the addition of an infinite semicircle enclosing the left half-plane. Therefore the function A E p}is suitable for the application of the Wiener-H021“ technicnie . Similar considerations apply to the function defined by the equations Bz-E} - Big} _ Fg£EI+ Fit-E: psinp'n‘ _ sinp(1r-(3) (84) and -l b dr Bipjz 31—d— f0 (A£B}+ Ai-8})I_H£gr} T (85) Therefore 0: [LBipI Kpigrhi dp r>b (86) in which Bip}. is analytic on the left half plane and possesses such decay as to make it possible to close the contour on the left. Big} is a suitable function for the apflcation of the Wiener-Hopf technique. The proof of these assertions follows exactly the same lines as that forAipj. TH ES) If 1 ['- ‘wnq X. The W'iener-ngf Technicwe Applied to F1 {J3 Corresponding to F1 we have the equations ILA {H} K}. I 81'} P dp r>b (33) and ILC {MI}. {gr} 1* C111 ro COSp { TT-fi-S} (87) 11?}; I“ Substituting (73) into the above oz.) -.- {mini —AL- pm” “BMW” psin'pfl 2 sin p17 4 iwp K}, {gro} COSp (1)0 cosp (1r - (3)3 (83) We must now separate this equation into functions which are analytic and possess at least algebraic decay on the right half-plane [plus-functions] and functions which are analytic and possess at least algebraic decay on the left half-plane [minus-functions]. It can be seen at once by equation (69) that the function G f p] does not approach zero at infinity on either half-plane. We must therefore divide the equation (88) by diffs} in order to make the quotient decay at least algebraically at infinity on the right half-plane. 24 TH ES) 25 For convenience in the subsequent work we define: e = 529 (89) it eH COSpB c05p( TT-fi) Pm: 3;" rm (mini)z (90) P = Res (132)”): -'—1_('€)-ncosn8 cos n(1T-B)f{n}, (91) n P-_>"n 217 - .2“. 6” COSp¢Qc05p(1r-I3) REM - 1P PIP} Ki‘igroi sinw (92) 2(- )n Rn = R“ RM = - 6; Knigrol cos n (>0 cos ntw-fi) (93) p.—>n 1 1111 a? Lim (JL’ “)2? {P} _ (in C08 n 9 cos n (Tr-B) Qn [JR—'91]. 172 ‘2 2 11' n E (94) .. I“ - S {HI = i 6 1-}.Zgroi cos p 4?.3 cos p.(1T (3) (95) ip r {p} (sin pv‘n' )7 -n Sn = é—E‘Jn— cos n 4’0 cos nB In E grog (96) _ 9 1r isPL - n Mn {p}- 57 -2—)’-— I, {11} cos p Bcos p(1r-I3> Af-p} W (97) Mn = Res Mnil‘l} p—>n _ -2 6 ” cos p 4; cos pf'rr-fi) NM‘ in fin} 18(ng (sininfv (98’ where n is a positive integer.. Rearranging equation (88) and using the above definitions we have P {83A {H}: [5&3 (3183+ Pitt} Af-H} + RIM) (991 II TH ES) 26 in which we have put the prospective "minus-functions." on the left hand side and prospective "plus-functions" on the right hand side. Due to the singularities of P i p] , the function on the left is not yet a minus-function. It can easily be shown that the function pr] Alp; has the proper decay on the left half-plane, and also in the strip 0A{‘P3 _ H Egigfiifi} C0389 9051* C 11-137 Aim} -B (P. f 211+ 13 sinpfi COSpCH-p) AI'H) -(1T-Q)€H . .EiH'I'l} COSpfiSlnp(1T-[3>Ai-p} (104) Therefore n , _ Mn: HT cosznfi A{ -n:( { :—::§}- loge€+Ian+ l] + (2}3-1T)tan up} (105) The function Rf p} must be treated differently depending on whether b>rO or b b The function Rf p] behaves properly in the strip, but at infinity on the right half plane, it approaches ~l b REp]~T(-1:—>[H[ asRep——>oo (106) from which it can be seen that R{ p} decays only if r0 > b. Subject to this condition, however, we can treat it as we have the previous func- tions. R{ p} has simple poles at the positive integers, which we remove by subtracting (107) from both sides of the equation. The case b > rO Since the difficulty arises from the undesirable behavior of the MacDonald function, we split it using equations (7), (95) and (98) RIHI=NU3 - 5&8} (103) where N L p} will become a plus-function and S E p }will become a minus function. In the strip, S{ p] decays exponentially because we have assumed (he < (3. On the left half plane, 5 g n} exhibits the behaviour 1 b "IHI 5U} j,— (‘r;) (109) Re p < 0 from which it can be seen that S a p} decays exponentially when Re p -> - oo . S ép} has simple poles at the negative integers and a double pole at zero. We subtract I1 TH ES) 29 00 I E 511.. + E129. .}. 2L0 (110) n=1 1" + n 14' I": from each side of the equation. In the above _ -2 NO—TT Ioigro} (111) and No = 3—32— IofgrOEEI-f-E-ggg - logee+1r21}] (112) In the strip, N f p} behaves the same as SZ p} , while on the right half-plane it decays like the inverse square of a gamma-function. However, Ni p] has double poles at all the positive integers. They are removed by subtracting oo 2 [———7Nn + N5 (113) -1 (P - n) 1* - n n from both sides of the equation. In the above II N' = -2 (-€)n cosn¢ cosn8l£gr} jig—8:3} n inf n! O n O Inflgro] (114) +loge€ -’V[n+1} - ¢Otann¢o+ (Tr-8)tan n8] and n -2(-€) Nn= W In'igro‘g cos n 4’0 cos n 8 (115) Then for the case b > r0 we have the modified equation P581A181+ Siu1=£;:} Gil-11+ P181AC-H} (116) + Nip} TH ES“ 30 We now have two equations; (99) for the case of b < r0; (116) for the case of b > r0. After subtracting the appropriate Mittag-Leffler series from both sides in each of these equations, we find that in each case we have a minus-function on the left, a plus-function on the right. .Since these functions are analytic in the same strip 0 < Re p < 1,. they both define the function zero. Hence finish}: 2° W. edgy + M10} n=1 p-l-n 1" P- a) I 0) CD (117) + 2 (gal-£322- + 2 MIL + z ——Rr_1n n=1 P. n=1 p-n n=1” £91221 and CI) Pf HIP-{1L} + 51113: 2 P-n—Aé’n}+ —9,——QFA{°}+ M10} n=1 ”+11 I" (I) (x) m - i I +2W+2£L+zsf +—-9—N+—n-N 11 00 Na 0° N ( 8) + Z ( )Z + Z n n=1 H n n=1 P’ for b >5) TH ES) XI. The Infinite System of'Equations for A{:n)3 If, in equation (117) we let p = -m, where m may be any positive integer, we obtain the infinite system of equations 00 oo oo 00 P A -n} QnAfi—n} Mn _ 112 E n - m + E (n+m)z - E n+m E n+m n=1 n=1 n=1 nfil n*n (119) +1 0 A10} MLOL (-1)m 9 -m ieU‘Zflicos H13C°31l(11'§)A£J-1}/ for b < r0 in which it must be remembered that Mn contains a linear combination of Ai-n} and A'{ -n}. Similarly, 0') CD CD (I) 23 P AL-n _ z onAi-ng + 2 Mn + 2 [Sn m=n (m + n) .msl! n m-n n=1 n=1 n=1 n=1 mtm n#m oo oo Nn .N' o Ato} Mfof ’n’flmztfl—n—Mm ' Jar-t m _ _ (120) m N N' (-1) a .. _.02._ _..o. = ______... _ m + m 2 11 a p [E "Pg-#1008 p (Boos pCfi-fl) A683 [- 2(-1)m a + ‘i'? 571} FIT-81°05 P- 4’0 C03 1”("4”I-tii-grOfl-m for b > rc1 31 THESI XII. The Infinite System of Equations for Bi -mZ Following an analysis similar to that used for A i p}, we have 9m: 2.12.11 {-cm-ii sinpfi mm emits 4 . . + inp Kpigro}s1np 4’0 s1an1T-8)} (121) or, substituting from equation (84) _ TY (B E (.13 - Bf-p} )sinpp sinpc 17-8) DZP}_ ZsiinT sinp‘fl’ 4 . . + inp Kpfigro} s1np (1)0 smp C 11- 8 )3 (122) The application of the Wiener-Hopf technique follows exactly the same lines as that for Ai p}, except that there are no poles at zero. In fact, it can easily be seen that Bf -m]obeys equations (117), (118), (119), and (120) with the provision that, in each case, the cosines are replaced with sines and the following changes are introduc ed: 00-9 0 14120} -> 0 N —> 0 N), —> 0 (123) 32 TH ES) XIII. The Series Expansion of the Solution According to our original representation US = IL (F1£p}COSp¢ -+ FZIPI sinpcp) KngrJ pdp (11) in Region I Changing p to -p , and changing the direction of integration, we get Us= - IL*(F1{'H} €0wa - in'tllsmtl‘i’) Kplgr} pdp (124) But relation (42) shows that the integrand of equation (124) is analytic in the strip -1 < Re p < l at least. Then if the original contour lies within that strip, we may now deform L* into L, and add equations (11) and (124). 2US = [Limlinl - Flt-p3) c05p¢ + (Fats) + F2 L-ubsinno} ' Kpigr} dd}. (125) Using equations (73) and (84), 2U: ILfiAt-ni - Aim sent...“ 8 - 8) p 8:1an +(Bf'H1‘l' Bil-*1) sing¢sinp_(1T-(3) } Kpfgrhidp (126) u sinpn' In the following, it will be assumed that L lies in the strip -1 < Re p < 1. Let us split the integral and consider, for the moment, only the terms involving Ai-p}and Bi—p} . 33 TH ES) 34 Changing the variable to -}1 in these expressions, and changing the direction of integration, we have A131} cospocosLCw -p) 1L H 8mm KningH 9P A p co§p¢>c03p 1r -]L Kpigr}I-L d” (127) - -IL* p siinr sigisinwsinim ~13) 1L H 8mm K {grim dH - - f >:< EiLl‘Sinpcbsianuj) L _ P 8mm Kpfgrj n dp (128) We will close these integrals on the left, so we must deform the * path of integration of the other two to conform with L . However, the integrand of the first has a simple pole at zero, so we must include the appropriate re sidue . B£p351np¢slanTT -£3 [L H Sing,” KH‘I 81‘} #911 >I< Bip} sinp¢sinliC1r - (3.) J.L . p 8111pr Kpfigr} p dp (129) but A{ pjco§i¢c05pCN - 8) IL p sinpn‘ KPi g1?“ d“ (130) =1 ‘i‘filj‘S’flW" ‘J’ K {gr} ids +21A101K {gr} Therefore 7 US = - iA{0}KOLgi-3 - [L A[ n}°°§SE1:Cp:S”C" ‘3 K {gr} du SIJLOSIDJLTT- -8) -IL Bip 1 8mm, Kpigr} du (131) I1 TH ES) 35 It can be seen that these integrals converge for I¢I < 8, which is exactly the range of validity of the representation (11). It can also be seen, from equations (79) and (80), that these integrals can be closed on the left if r > b. In that case CD US = -iA{O} Kofgr] - 21 E (-1)nA{-n} cos n(1r -8) cos n ¢angr3 n=1 oo - 2i 2 (-1)n Bi—n} sin n(11'-8) sin no an gr} n=1 for r>b and lo) < 8 (132) Expanding cos n(1T - 8) and sinn (Tr - (3) 00 US = -iA{OI Koigr} - 2i 2 Af-n}cosn 8cos noangr} n=1 oo 4- 21 Z) Bfi-n} sinnB sinn¢ Knigr} (133) n=1 for r >b It can be easily shown that equation (133) is valid for all values of (1:. TH ES) XIV. The Scattering Cross-Section for the Symmetric Case In expression (133), let ¢o = 0. Then pr} = 0. Returning to real wave numbers 00 TI’ US = —-2— A{0} H;{ er+ W E (i)nAi-n}cos n8 cos n (I) Hhi kr} n=1 (134) But, for large _I_:, H1 {kr}—-> 2 exp i(kr— {Zn-1451)} (135) n 1T kr 4 So, as r-> oo / i{kr- 1] 00 US —> 211:!- e 4 A{O}+ 2 Z A{-nIcosn8cosn¢ n=1 (136) and, since U = Ez 1 EU H = '1—‘—— 137 (I) 1 00 [.10 a r ( ) T k' -1- R E H’”) h th t ' a mg 2 e i z , we ave e ou gomg energy __ . oo ExH 2—17—— AIOI+ 2 ZA(.-n]cosn8cosnI2 r 401 For n=1 (138) and the outgoing power 00 3—1:, s: 2:: {IA‘LO}|Z + 223 IAf-ndz coszn8} (1391 0 n=1 36 37 The cross-section is this expression divided by the incident energy, which for the symmetric case is (140) 2 b ° " arcsin ( fl ) WHO r0 So, for the symmetric case, the cross-section is 11-1 {lAi0112 + 2 2:1 IAi-n} Izcoszn8) b sin 8 (141) 11' . T [arcs1n ( I'o TH E81 A XV. Special Values of $7] If, in equation (88), we let p approach any positive integer 31, the function Cip} on the left is bounded, so the same must be true of the function on the right. Moreover, since I-m {gr} = 1711ng3 (142) inserting this relation into equation (78) shows that Ai-m} = A {m} (143) Using this information, and applying L'Hospital's rule to equation (88), we have A'fm} + A'{ -m‘g: ii Kmigrog :2: 2 go (_1)m (144) and, in particular, Aiio} = 13 Koi grog (145) 38 TH ES1 XVI. The Case of the Strip A. The System If we examine the special case 8 —§ 321 , 430 = 0 the wedge becomes an infinite strip of width 26. Since the arrangement is symmetric, we can disregard the B{ p}. The infinite system reduces to g -€-2nf62n} Ai-Zn} + c; €2n Ai-Zn} n=1 2n - 2m n=1 4ff 2n+13 (n+m)z n=m oo .3.— [ E H Af'p} | 00 2n n _ Z 841 rgiwi 2n _ 2: {26 (-1) Kznigrs} n=1 2(n+m) n=11,['£2n+l](n+m) A 0 1 z - gangeem pgzmiiismi-6-2mri2miiizm1Ai-zmi -2m 9 Ai 3 + 6 PmeI 3 PH ~2m (146) The solution of this infinite system is not expected to be materially simpler than for the unsymmetric case, since we still have a system involving both the unknown A [p] and its derivative. We could further simplify this system by allowing ro to approach infinity. In that event we use the source function l 111' Ui = expi-gr cos ‘1’} = IL cos pd) Kpigr} dp (147) 39 THES' 40 instead of the Hankel function [ equation (1) ] . Then wherever we 1 have the MacDonald function of r0, we may replace it by -2- . l ‘ Kn {groi a E (148) which does not ameliorate our difficulties to any significant extent. We can, however, separate the dependence of the unknown on e from its dependence on p, which is done in the next section. B. Expansion in Powers of € b Aini= —2—1-— 10(4181- At-ai) this}? <78) But, by symmetry, Aifii= -A{-B} (149) Treatment of the local field near the origin shows that -AT{E} may be expanded in even powers of _1_'_. The radius of convergence of this series extends to the singularity of A nearest the origin. This can hardly be nearer than the edge of the strip, and in fact, the analysis related to equations (54) shows that A is bounded there. Therefore, the series must converge at least up to r = b. Explicitly: 1 CD '1? 4101: 2 “2n nm (150) n=0 where, for convenience, we are expanding in terms of .. 5.1“. ’7‘ 2 (151) Then, using the series expansion for the Bessel function, % 41 i i '2 16 m m 00 nsz-p < ) A p = . 23 u n E . . dn 152 1 fig 0 n=1 Zn j=0 J: {3+1-p} Interchanging the order of integration and summation and integrating . oo 00 mi” 23' + 1 -}1 21 u 6 Ai = 2 ‘2: .m. . 153 “3 1T g n=0 j=0 jlfifil - p] (2n+ 23 + 1 - p) ( ) where we must remember that the uzn are unknown functions of 6. Inserting this expression into the infinite system (146), we have: (- 2° u :5 r: -P n20 Zn 3:0 8:1 2(5 -m)]:' (j + 1)_|:1(j + 25 +1) (Zj + 2n +25 +1) 1-51m €43+2j+zn+1 (I) 1 +5317 1T(25+1)[7(j+1)ff(zs +j+1)(2j+2s+2n+1)(s+m)' .(4?(23+1+j)+"7[/(25+1)-21og€ + 2(S+m) +2j+2~$ +2n+1)) ng a“ (4)5 K35 {gig} '1' 2 5:1 (25 +1) (3+m) CD CD €2j+3n+1 + 5 ¥ _ - :j uzn :0 [511+1)] [Zj + 2n+ 1] 2m [2m 2 10g E . 1 +‘4’(1)=+')U(J+1)+ 2j+2n+1 l €2j + ZD‘FI P(Zm) “110+ 1)f1(j+1+2m) [2j+2n+2m+ 1] ' . 1 ”4sz YIZm+J+l)' 2j+2n+2m+l [[2 0 an infinite system of equations for the 112n- TH E51 42 The system (154) is an improvement over (146) insofar as there is only one set of unknowns--the uzn. However, all attempts to solve this system by iteration for small values of 6 have been thwarted by the apparent logarithmic behaviour of the um. We can, nevertheless,” obtain reasonable agreement with Bouwkamp [2] by means of reasonable assumptions about the behaviour of the “Zn as 6 goes to zero, as shown in the next section. C. Approximate Treatment of the uzn Formally differentiating equations (153), and letting p go to ZCI'O _ - 00 2n+2j+1 _ . _ 1 A7i03 = “21 2: “Zn '5 (“gee 1"“ 1) 2j+ 2n+ 1) g n=0 jzfgj+13 (2j+2n+ 1) (155) But, using equation (145), we have _ - CI) u. 2n+2j+l _ . _ 1 EZ'Kofgrol = 1.21 2 2“ flog“ NJ“) 23+ 2n+1) gn=0 jtfzj+1} (2j+2n+1) (156) which must be identically satisfied in 6. Since the left hand side does not contain 6, the right side must also be a constant. Therefore, for small e, um is bounded by n,m <7 idn (157) o where the dn are independent of p and 6 , and some or all may be ZGI‘O. TH ES1 43 Inserting this expression into the formula for the cross-section, we obtain n72 4 g2 dn 2 - . < — L — X section 2b [ TIC gz IlOge€ '7 ] :0 2n + 1 I (158) for the symmetric-plane-wave case. 1 - ) shows But equation (156), for the plane wave case (K0(gro) —-> 2 that 1T = T7.— (159) $0: . 1T2 kr '2 X-sect10n<_ 7 (logE -2-) (160) \where we have returned to real k and ignored the constant added to the logarithm. XVII. The Case of the Wedgg The expression for a wedge corresponding to equation (153) for the strip is easily obtained. For the case of the source located on the axis, it follows from (78) and (149) that i b dr A111}: "T,— 10 Aifillpfgr} ‘1.— (161) But our treatment of the singularities at an edge shows that A {8} may be expressed as A[8]= :70 {an 02 n=1 11' 1111' + 6n n2("'m} (162) where n is as defined in (151). Inserting this relation into equation :3 .03 (161) and integrating term by term 2' + “17 Aid-12°? anew “3 — T)‘ 'I ' _ - _ n11 =1 j:0 J-fil’” ”3121 ”+2-8- n11 (163) 2' - +--——— + bL€ J [L 2("-B) . . . n It J1P{J+1'P}(ZJ'H+WI Differentiating this expression, and setting p = 0, it follows from: (145) that xn In an: n11 ’ and b“: n‘n' , (164) 62B logee €Z(1T-(3) logee where the xn and yn are unknown real numbers. 44 TH ES' 45 Therefore A60}: 21 W (165) loge 6 where we have used equation (145) as before. It will be noticed that this expression is the same as that for the strip, so to this degree of approximation, THE RADIATED ENERGY IS THE SAME FOR ANY 8 . The cross-section will be different, but only because the intercepted energy is different. A more detailed analysis suggests that A{ O} can be written _ - K0 {gro} A{0}_ 21 logee + x (166) where X depends on 8 and possibly on 6 . Comparing this with the results of Bouwkamp [2], we see that X = - loge 2 (for the strip). Unfortunately, we are not able to obtain this number even by approximation. It appears to be necessary to solve the infinite systems (119) or (154) in order to improve upon the solution (160). We have attempted to solve (154) by iteration, but the presence of the logarithm in the denominator casts considerable doubt on the validity of the procedure. We have also tried to extract more information from the definition (78), using equation (144), without success. THES) REFERENCES [1]Kontorovich, M. J. and Lebedev, N. N. J. Physics, Moscow [2] Bouwkamp, C. J. Diffraction Theory. Reports on Progress in Physics, Vol. XVII, p. 35 (1954), equation 8.3. [3] Oberhettinger, F. Comm. Pure and Applied Math. , l, 551 (1954). [4] Oberhettinger, F. "On the Diffraction of Waves and Pulses by Wedges and Corners, " Journal of Research of the Nat. Bureau of Standards. Vol. 61, no. 5 (Nov., 1958), Research paper 2906, equation 10. [5] Meixner, J. "Die Kantenbedingung der Theorie der Beugung elektromagnetischer Wellen an vollkommen leitenden ebenen Schirmen, " Annalen der Physik, Ser. 6, 6 p. 2 (1949). [6] Erdélyi, Magnus, Oberhettinger, Tricomi. Higher Transcendental Functions (Vol. 2, 7. 5.5; McGraw-Hill Book Co. , Inc. , 1953). 46 1.74 THESI .I a: ‘1‘... (1)17... «. MM [:8 "‘1111111111111)7