Appl. Sci. Res. 24

A u g u s t 1971

ELECTROMAGNETIC DIFFRACTION B Y A T H I N CONDUCTING C I R C U L A R D I S K P L A C E D AT T H E P L A N E I N T E R F A C E OF TWO D I F F E R E N T M E D I A D. L. JAIN Dept. of IVfathematics Pennsylvania Stare University University Park, (Pa.), U.S.A. Abstract I n this paper, the p r o b l e m of diffraction of t i m e harmonic, electromagnetic w a v e s b y a t h i n ideally conducting disk lying at the plane interface of two different media is considered. I n this analysis, t h e incident w a r e is a plane w a r e travelling in a direction perpendicular to t h e plane interface of the two media. A H e r t z vector formulation is applied to reduce our electromagnetie diffraction p r o b l e m to a system of two scalar problems which are solved b y t h e kelp of two pairs of F r e d h o l m integral e q u a t i o n s of t h e second kind. L o w f r e q u e n e y a p p r o x i m a t i o n s to the t a n g e n t i a l c o m p o n e n t s of the m a g n e t i c i n t e n s i t y associated w i t h t h e diffracted field at t h e surfaces of t h e disk, t h e induced surface eurrent density on the disk and the scattering cross section are obtained. Nomenclature

A1, A2 B2 c C E /(±~, g(±) H H(1 ) I½

J~ k

radius of the circular disk far field amplitudes of M1 alld M2 function of 0 and k defined in (40) v e l o c i t y of p r o p a g a t i o n of the electromaglletic w a v e an unknown constant electric intensity of the total field functions of r defined in (16) and (24) magnetie intensity of the total field Hankel function of the first kind of order ½ Modified Bessel function of order ½ components of the current density Bessel function of order ½ wave number --

3 7 9

--

380

D.L. JAIN

/t'0, K1 #(M) l(:~) L, L(I), M, M M1, M2

kernels of integral equations (17) and (25) defined in (18) max(k(+), k(-)) functions of r defined in (22) MO) kernels of integral equations (27), (A 6), (20) and (A 1) Hertz vector functions of r and z defined in (8) p, r', v, w, 2 d u m m i e s of i n t e g r a t i o n space co-ordinates (r, % z) time t integrals in (37) Tl, T2 U(±), W(±) functions of v defined in (21) and (28) o~ 1

V 8

0 #o O)

--

oc

f u n c t i o n of p and k defined in (A 4) dielectric c o n s t a n t a z i m u t h a l angle in spherical polar co-ordinates magnetic permeability m a g n e t i e p e r m e a b i l i t y of t h e free spaee wave Irequency

superscripts

(+) (-) (o) (1) (2) (3)

refers reIers refers refers refers refers

to to to to to to

the the the the the the

half-space z > 0 half-space z < 0 incident w a v e field refracted w a v e field reflected w a r e field diffracted w a r e field

subscripts r ~0 z

r-component ~0-component z-component

§ 1. Introduction

The problem of diffraction of time harmonic, electromagnetic waves b y a thin conducting disk has been considered by many authors (Bouwkamp [1] ; Lur'e [2]; Lebedev and Skal'skaya [3]; Benkard [4] ; Williams [5] ; Inawashiro [6] ; Boersma [7]). They have assumed that a single medium occupies the region exterior to the disk. In a recent paper (Thomas [8]), the dual of our problem i.e. electromagnetic diffraction by a circular aperture in a plane conducting screen between different media has been solved using Hertz vector formulation and Williams's technique [9, 10] which reduces the problem to the solution of a pair of simultaneous Fredholm integral equa-

ELECTROMAGNETIC D I F F R A C T I O N BY A T H I N D I S K

381

tions of the second kind. Thomas's formulation of the two media circular aperture problem is exact but his solntion is approximate and is useful for low frequency waves only. Solutions useful at high frequencies have yet to be obtained. It was pointed out b y Thomas E8~ that the results of the electromagnetic diffraction problem that we present here cannot be deduced from his results by the application of Babinet's Principle because the media on the opposite sides of the screen are different and therefore our problem which compliments the two media aperture problem (Thomas E81) requires a separate treatment. We present here an approximate lowfrequency solution of the problem of electromagnetic diffraction by a thin conducting circular disk placed at the plane interface of two different media. In this analysis, the incident wave is a plane wave travelling in a direction perpendicular to the plane interface of the two media. Introducing a Hertz vector formulation as used b y Lebedev and Skal'skaya E31, Lur'e E21, Williams ES~, and Inawashiro E61, the electromagnetic diffraction problem is reduced to a system of two scalar problems. Applying Williams's technique E9, 101, solutions of the two scalar problems are governed by two pairs of Fredholm integral equations of the second kind involving an unknown constant. Approximate solutions of the two pairs of integral equations are obtained by direct iterations in the particular case when the two media have the same magnetic permeability but different dielectric constants and k(±)a ~ 1. An approximation to the value of unknown constant occurring in these solutions is obtained by using the edge condition that the radial component of the total induced snrface current density at the disk is of O ( ( a 2 - r2) ~') as r--> a. Approximate expressions for the tangential components of the magnetic intensity associated with the difIracted field at the surfaces of the disk, the induced surface current density on the disk and the scattering cross section are obtained. When the media are identical, all our results agree with the known results of the single medium problem (Bouwkamp Elf). An alternative formulation of the governing two pairs of Fredholm integral equations of the second kind is derived in the Appendix.

382

D.L. JAIN

§ 2. Mathematical formulation A cylindrical polar system of coordinates (r, ~, z) is chosen so that the conducting disk occupies the region z = 0, 0 < r < a. The half spaces z <>0 are occupied by two different media eharacterised b y dielectric constants e(±) and magnetic permeabilities #(±). Throughout this analysis, rational±sed M.K.S. units are used and a time dependence exp(--icot) is understood. The velocities of propagation of electromagnetic waves in the two media are c(i) = (e(±)#(±))-~ and the ware numbers are k(±) = c~/c(±l. Maxwell's equations in sourcefree regions of the halfspaees z X 0 are given b y

and

curlE--ico#(±)H=0,

divE=0

curlHq-ime(±)E=0,

divH=0.

] J

(1)

In this study, the incident ware is a plane ware travelling along the positive direction of the z-axis and is due to sources in the halfspace z < 0. Let E(0), H(0) be the time independent parts of the electric and magnetic intensities of the incident ware, then in z < 0, E(O)

=

(E~0),

E~(0),E z(0)) =

(cosg, --sing, 0) exp(ik(-)z),

)-~

u(o) = (H~.0, H(O), H(0,) = \~~2~-_) (~(-) ] × (sin V, cos V, 0) exp(ik(-)z).

(2)

The time independent parts of the electric and magnetic intensities of the total field E, H are given b y

(1) + E(3), z > 0, E=[E(0)+E(2)+E(a), z<0,

(3)

(1) + H(Z), z > 0 , H = ( H(°)+H(2) +H(a),

(4)

~E

~H

z<0,

where E(1), H(1); E(2), H(2) correspond to the refracted ware in the halfspace z .'> 0 and the reflected wave in the halfspace z < 0 respectively when the disk is not present at the plane interface of two media. E(a), H(a) correspond to the diffracted field due to disk. Values of E(1), H(1); E(2), H(2) are given by E(1) =(E(~l)'E(~l)'E(~l))=-°@°sq)'--sin~'O)exp(ik(+)z)

I

H(l)--_(H(rl),H(1),H(zl))=c~(@+)~~.(sing~,eosq~,O)exp(ik(+)z)['z>O' J \# / (5)

ELECTROMAGNETIC DIFFRACTION BY A THIN DISK

E(2)

3~3

/~(2) ~(2> E(2))=fi(_cosg, sin ~, 0) exp(--ik(-)z)

u+-~w

~> H +

--\

r ,

cp ,

<~>)-~( ~<->~+. - - \~~ß-__)B

, zoo,

• (sin ~, cos % 0) exp(--ik(-)z)

(6)

where 2(#(+)s(-))~"

(~(-)s(+))+ + (g+)g->)+ ' (g-)g+))+ _ (g+>~<-))+ B = (#(-)g+))+ + (#(+)g_))+

(7) and

c~ + fi = 1.

Introducing a magnetic Hertz vector M for the representation of the diffracted field, we have E(a) = im#(±) curl M

1

H(a) = grad(div M) + k(±)2M.,

z ~ 0,

(8)

where V2M q- k(±)2M = 0, z ~ 0, M = (Mr, M+, Ma) = (Ml(r, z) sin qJ, Ml(r, z) cos ~o, M2(r, z) sin ~), a2M1 1 äM1 ä2M1 ar2 + r Vr + &2 +k(±)2M1

] 0 ,

ê2M 2

- -ar~

1

0~~//-2

+ .r

. ~r

M2

~2M2

. r 2 .--

. 2 @ k(~:)2M2 ~Z

z~O,

/

0

J and M1, M2 satisfy radiation conditions at infinity. So we have

aMl~

E~3) = ico#(±) { M2r E(.3) = io9#C~) { 3MI&

~M2 .~ sin 9, ~r J

e~ ~ ) = i ~ g ± > ( ~~M1 - ~ ~ ~ ~ cos ~, ~2Mi ~9'M2 } H(a) = [ ~r 2 q- ~Tr~z q-k(~)2Ml~sin~o, H(3)={1

r

~M1 1 ~M2 } ~~ + r & -7 k(±)2M1 cos~,

H(a)={~~'M1 "Vr&

~2M2 ôr 2

1 aM2 M2} r & + r-g- sing.

(9)

384

D.L. JAIN

This reduces our electromagnetic diffraction problem to a system of two scalar problems. B o u n d a r y conditions : (i) Tangential components of E vanish at the disk. (ii) The diffracted field E(~), H(3) is due to induced surface current distribution which is confined only at the two faces of the conducting disk and therefore H~.3) and H(~3) are identically zero on the region r>a, z--OB. (iii) Tangential components of E a r t continuous across the region of the interface of twü media lying outside the disk. These b o u n d a r y conditions lead to ,u(+) vqM1 = ôz z=o+

[Ml]z=o+ =

(-)

: ôz-Jz=o-

[Ml]z=o- = O,

[~(+)M2]z=o+ = [~(-)M2]z=o- = Cr,

C - - - - iEx

,

O
(10)

r~a,

(11)

r~a,

i2)

O
13)

(11

14)

[#(+)M2]z=o+ = E#(-)M2]z=o-,

[~M21

- ~ - - z j~=o+

= [-~M2-]

L-FZj~=o

- = o,

r > ~,

15)

where C is an u n k n o w n constant. Also M1, M2 satisfy radiation conditions at infinity and are such t h a t the radial component of the total induced surface current density at the disk is of O{(a2--r2) ~} as r -~ a. We are justified to take Hertz vector in the above mentioned form as it ensures t h a t all the b o u n d a r y conditions are saftsfied. Introducing the functions /(±) (r) defined by

/(:~)(r) ---- [Ml(r, Z)]z=o±, 0 < r < a,

(16)

and applying Green's Theorem we get a suitable integral representation of Ml(r, z), which is given b y a

_]l/Ii(~,,z) ~~- -T ~-~-~-If ~//(4-)(~/) Ko(f , ~"; z, k(+)) df'l, 0

z X O,

(17)

ELECTROMAGNETIC DIFFRACTION BY A THIN DISK

38~

where 2r~

f exp(ik(r 2 4:- r '~ -- 2rr' cos ¢ + z2)~) Kn(r, r'; z, k) = 2~(r 2 4- r "2 -- 2rr' cos ¢ + z2)~ cos ne de. o Now boundary conditions (10), (11) imply

,u(±)l
= ~-

(18)

'le±)(r') Ko(r, r'; k(~-))dr' 0

0 < r < a.

(19)

Using Williams's technique [9, 10i (see also [111, this pair of integral equations leads to a pair of Fredholm integral equations of the second kind (sinh k(±)v) { ~} U(~-)(v)-- ± (k(±)ff(±)) C + C~

+ I M ( w , v" k(±)) U(~)(w) dw,

0 < v < a,

(20)

tl

0

where a

f l(±)(t) cos(k(:~)(tz -- v2)~) dt u(±~(v)

=

v

(t~ -

v~)~

'

(21)

l(±)(r) --

2 d f U(±)(v) cosh(k(1)(vä -- r2) ~) 7: dr (v~ -- r2) ~ dv, d

z~±~(r) = ~ - r (l~±>(r)),

/<±~(a) = o,

a 1(±)(r) --

(22)

2 f U(±)(v) cosh(k(~)(v 2 -- r2) ~) r: (v2 _ rS)½ dv, k

M(w, v; k) -- i(wv)} f 2I}(~v) I}(Zw) d;~ -- 2ikavw 3 ~ +O(kS)" (23) 0

Art alternative formulation of equations (20) to (23) is given in the Appendix.

386

D.L. JAIX

Similarly introducing the functions g(±)(r) defined by

g(~)(r) ~- [ . ~M2(r' z) .1 ôZ

,

O <

r <

a,

(24)

z=04-

and applying Green's Theorem we ger a suitable integral representation of M2(r, z), which is given by a

M2(r, z) ~-- T ~ r'g(±)(r') Kl(r, r'; z, k(±)) dr', z~0.

(25)

o

Now boundary conditions (13), (14) imply a

Cr -- + _ { f r' g(~:)(r') Kl(r, r '", z, k(±))dr'} z=0' 0 < r < a. #(~:)

(26)

o

Using Williams's technique [9, 101, this pair of integral equations leads to a pair of Fredholm integral equations of the second kind a

(2C sinh k(±)v) w(±)(v) = -7#(±)k(:L)

- fL(~,

v; k,±,) w,±,(~) d~,

o

O
(27)

where gg

W(±)(v) = v f g(±)(t) c°s(k(±)(t2( tz _ v2)~ - v2)½) dt,

(28)

a

g(J:)(r) --

2 d f W(~:)(v) cosh(k(±)(v 2 -- rg')-~) ~ dr (vz _ rZ)~ dv, oo

L(w, v, k) = k~(wv)~ f (4)-1 ]~(zv) J~(,~w) dZ + o

+

k

i(wv)~~f(k~-2 ,~z) I½(~tv)I[(kw) dZ = I k~v + 4ivwk3 37: + O(kS), w > v, 4ivwk a k2w + 3re + O(kS), v > w.

(29)

387

ELECTROMAGNETIC D I F F R A C T I O N BY A T H I N D I S K

An alternative formulation of equations (27) to (29) is given in the Appendix. So we have derived two pairs of integral equations of the second kind ((20) and (27)) which govern the system of two scalar problems. But these two pairs of integral equations still involve one arbitrary constant. This constant is determined by applying the well known edge conditions that at the edge of the disk,

jr(r) = 0 { ( 4 2 - r s ) :~} as r - + a , le(r) = 0 { ( 4 2 - r 2 ) -~} as r - + a , it(r), le(r) are the components of the total

where current density at the disk and are given by

]r(r) =

(30) (31) induced surface

[H(3)]z=o-- [H(~3)?z=o+=

g(+)(r)]+

{ 1 [ d/(-)(r) dl(+) (r) d r + g(-)(r) - " dr

=

(32)

+ k(-)2/(-) (r) -- k(+)2fl+) (r)} cos 9,

],(r) = l-H~3)l~=o+- [H~3)Iz=o_

:

d [ d/C+>(~) { = ~ L dr + g(+)(r)

k(-)2](-)(r)}

@ k(+)2/(+)(r) --

a/<->(~) dr

g~->(r)1 + (33)

sin %

where a

fl~)(r)

2 f U(+)(v) cosh(k(~:)(v 2 -- r2)½) --

~

(v2 _

r2)~

dv,

c~

g(~:)(r) --

re2 d~d f W(±)(v) cosh(k(~:)--(V2(vr2)~ 2 -- r2)~) dv.

After some manipulations, we get i~(r) -

2 ~os ~ [ u<->(4) + w(->(4) 7:

u(+)(4) -

W(+~(4)

(4 2 - - r2)~

+ O{(4 2 - r2)~}]

+

as

--+4)

388

D.L. JAIN 2r 2 sin ~ [ U(-)(a) -r W(-)(a) -- U(+)(a) -- W(+)(a) --re L ( az - r•)« +

]¢(r) --

+

O{(a2 ~,2)-½}] as -

r

-+

~.

Therefore both of the edge conditions (30) and (31) are satisfied provided U(-) (a) + W(-)(a) -- U(+) (a) -- W(+) (a) = O.

(34)

This condition is used to determine the u n k n o w n constant C as explained in the last section. § 3. S c a t t e r i n g c r o s s s e c t i o n

To find the scattering eross section, we first determine the far field amplitudes of M1 and M2 [9] which are given b y a

AI(O, k (±)) = £:i

--

w½1½(k(±)w cos O) U(±)(w) dw, (35)

7"7 0 ¢z

A2(O, Æ(±)) = ~=i

r¢ eos 0

-

cos 0) W(±)(w) dw,

sin 0 0

(36)

where 0 is the azimuthal angle in spherical polar coordinates. Scattering cross section = =

X

(92

+ (~<->u-,)',(t,~<--' ~(-> '7,s r~},

+

(37)

where rcl2

[[A1(0, k(+)) [2 + cos 2 0 [AI(Õ, k (+)) + Ä2(O, k(+))12] sin 0 dO, 0

T2 =

(38) i"~ [IAI(O, k(-))] 2 + cos 2 0 IA1(0, k(-)) + Ä2(O, k(-))l 2] sin 0 dO,

rd2

(39)

ELECTROMAGNETIC D I F F R A C T I O N BY A T H I N D I S K

and

389

(40)

Ä2(0, k) = --ran 0 Aa(O, k). § 4. An approximation solution

Now we find an approximate solution whert k(+)a and k(-)a are both small. It is reasonable to take each medium to be either paramagnetie or diamagnetie as suggested b y Thomas [8]. In this case we cart set (to withirt a good approximation) #(+)

=

#(-)

:- #o,

where #0 is the permeability of the free space [121. Then ~(+)2

e(+)

k(-) 2

e(-)

and

~=

2(k(-)) --

k(+) + k(-)

We now give the procedure to be followed for determining art approximate solution of our problem. Using the value of kernel given by (23) in (20), we find the first few terms in the expansions of U(±)(v) in terms of the unknown constant C and powers of k(+)a, k(-)a by direct iterations. Similarly using the value of kernel given by (29) in (27), we find the first few terms in the expansions of W(±)(v) in terms of the unknown constant C and powers of k(+)a, k(-)a. Thert an approximation to C is obtained by using the edge condition (34). We calculate approximate expressions for the tangential components of the magnetic intensity associated with the diffracted field at the disk, the components of the induced surface current density on the disk and the scattering cross section. We shall omit the details of the straight forward calculations which are given b y Jain El3]. Our results are given below. At the surfaces of the disk,

H(~3}(r, cp, 0 + )

=

8iak(+)2k(-) sin ~

×

3~#oco(k(+) + k(-)) X

[ ( r~)-~{l(kC÷)2-k(-)2) --

1

a2

Æ(+)2

@

390

n. L, JAIN

+(1_TQ

7k(+)2a2 10

¼(k(+)2 -- k(-)z) a~} + k(+)za2 (41)

H~3)(r,~,O - )

--

=

x

8ia(k(-)) 3 sin 9 x + k(-))

3~o~o(h(+)

1-- a T j

"

]~(-)2

" +

i (k(+)a--k (-)3) 1 (k(+)4--k (-)4) } + ~ \k(-)~ a - 4 6 " k(-)2 a2 + +

~ ~-~1 7k(-)2~ + l(k{+)2 - k(-)2) a2} + 1-TQ ( + ~o

r2 ~½~1

k(-)2a2}

+ ~-7]( + T + +(1--a--~]r2~ ~{.&(-)2~23~} .+O(aak(~)a)], H~)(~, % o+) =

16iak(+)2k(-) cos

3=#o~o(k(+) + k(-)) __

r2 "~-~

~ 1

i

+ -~=

X k(*) 2 -- k(-) 2

×[-(1 ~-/~T(

k(+'~

k(+)~k(+)2k (-)3

-~-

2k(+) 2aZ

5 +(1-

(42)

a-~'] r~'~ ~ {~k<+na2}+

)+

k(+)2

a2 @

~(k(+)~' -- k(-)~) a2} +

o(~3k(~)3) 1,

(43)

ELECTROMAGNETIC

DIFFRACTION

BY

A

THIN

DISK

391

16iak(-) 3 cos 9 H(~3)( r, 9, 0 - - ) = --

5 ~ o ~ ( k ( + ) + k<-)) ×

×[(1--rU~-~-~

+ ~

1 (k(+)2--k(-)2

k(-)a

a - 8o-

)÷

k(-) 2

a~ +

r2)~{ 2k(-)~a2 } + ( 1 -- a--~" 1 -~- ~ -~- ~(k(+)2 -- k(-)2) t~2 ~+

1 =

]

-~-] {~A~k(-)2a2} + 0(a3k(M)3) ,

(44)

it(', 9, 0) = H~)(~, 9, 0--) -- H~)(~, 9, 0+) = ,6ik(-) cos 9 :-

[(

3a~#o~o(k(+) + k(-))

r~ ~{(k(÷)2 + k(_)~) a~ + 1--

a~ j

-~- ~(k(+) 4 ~- k(-) 4) 44 - - 1(k(+)2 - - k(-)2)2 a 4} -~-

+

1

~

{i(k(÷)~ + k(-)4) 44} + 0(45k(~)5)

,

(45)

i~(~, 9, o) = H~?)(~, 9, 0+) -- H~'~)(~,9, 0--) = 3ik(-) sin 9 =

3a~:~oo~(k(+) + k(-))

r2~-i((k(+)~+k(-)2) a~'+

[(

1 - -j]

+ ~o(k(+)4 + k(-)4) 44 - ¼(k(+)2 _ k(-)2)~ 44} +

+

r9' \~ 1 -- ~ - ) ((k (+)2 -~- k(-)2) a 2 -[- -~(k(+)4 -~- ]~(-)4) 44} +

+(1-- -r2 ~ ) \~ {~(k(+)~ + k(-)9 ~*) + 0(a5k(~)5)], 256k(-)~'a4 [ k(+)5 S.C.S. = 27~.(k(+) + k(-))z Lk-k~--) ×

22k(+)2az x

1+

25

a z (k(+)z 2

--

k(-)z) +

(46)

392

I). L. JAIN

}

{ 22k(-)2a2 a 2 (k(+)2 -- k(-)2) 25 + -2-+~_ O(a4Æ(M)4) _~ Æ(-)4 1 + + O(a4k(M)4)}l,

(47)

where k(z~) ~ max(k(+), k(-)), and S.C.S. stands for scattering cross section, In the special case when the media are identical, expressions (41) to (47) reduce to the known results [1].

APPENDIX We give here alternative derivations of the governing integral equations (20) and (27) as suggested by Williams [9]. a

v -- fM(1)(w, v; k(±)) U(±)(w) dw,

O~v
o

(A 1)

where (g

U(:~)(v) = v

f

a

l(~)(t) dt (t2 - v2)~ ' l(±)(r) -d l(-)(r) = dr-(/(±)(r)),

2 d f U(~)(v) dv ~: dr (v 2 _ r2) ~ , (A2)

/(a)(a) = O,

(A 3) /(i)(r) --

2 f U(~)(v)dv (v2 -

r2)~ '

r

M(1)(w, v; ~) = (v~)~ Bo (~, -- p) J.~.(p~) ]~(pv) dp,

(A 4)

0

and _ f-i(k~ ~' I.(p2 -

- p~)~, k~)-',

k > p, p > k.

A simplified form of M(1) useful for k ~ I, is obtained by using a complex integration method of Noble [14] and is given by

ELECTROMAGNETIC

DIFFRACTION

BY A THIN

DISK

393

k

--i(wv)~ ~ (k2

-

-

p-)~ ~H (1> ~: (p w) J~(pv) dp,

W~V,

o

M ( 1)(w, v; k) =

k

__i(wv).~ ~ (k2 __ f12)~J{(»w) H~l)(pv) dp,

V~W,

0

k2v

2ivwk3

kSw

2ivwk a

2

3~

2

s~

k4

+ ~ß- (v3 + 3vw 2) + O(kó), w > v,

(A 5)

k4

+ -4g-(w3 + 3wv~) + o(ks), V~w,

employing the usual notations [15~ for Bessel and Hankel functions. C~

2Cv #(~:)

#,

tL(1)(w, v k(±)) W(±)(w) dw,

.

o

O<_v
(A 6)

where g(*-)(t) dt

W(±)(v)=v

(12 - - v~)~" g(±)(r)V

2

d

~

d~"

W(:~)(v)dv (~2 _

,~)~

(A7) '

r

(A 8) o

Similarly using complex integration method of Noble El4], the simplified form of L(1), which is useful when k ~ 1, becomes k

i(vw)'~ L(~)(zv, v; k) =

(k2 _ p2)~ H~-1)(wp) JdvP) dp,

w > v,

i(vw) 1 f (kl f12p2)~. J~(wp) H~l)(vp) dp,

v> w

f

P~

o

Ic _

o

394

n.L. JAIN

k2v

~-

4iwvk3 +

k4

3~

16 (3w2v + v3) + O(kS), W~V,.

k2w ~-

4iwvk3 +

k4

3r~

(A 9)

16 (3v2w @ w3) @ O(k5)' v~W,

In these alternative derivations, the edge condition (34) remains the same and the corresponding far field amplitudes of M1 and M2 are given by

Al(O,k(±)) = ~ i

~sin0 C~ t~

•cosOjw~J~(k(±)wsinO) U(±)(w)dw,

(A 10)

tt

o

A2(O,k(a))=±i( 2k(~)_r~_sin 0y. a

lw~J½(k(~)w sin 0) W(±)(w) dw.

(A 11)

t/

0

We have also solved the problem by using alternative integral equations given here and we get the same approximations as given before. Also this formulation simplifies algebraic calculations [131. Acknowledgements

I am very grateful to Professor R. P. Kanwal for his guidance in the preparation of this paper. This research was sponsored by the NSF Grant No. GP-7968. Received 6 J u l y 1970 In final form 5 J a n u a r y 1971

ELECTROMAGNETIC DIFFRACTION BY A THIN

DISK

898

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