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SIGNAL COSES ENGINEERING' LABORATORIES ivms SIGNAL LABORATORY . BELHÄR* NEff JEBSS2.

Title of lepoj't ,,.,.jSgfegäa&jfesii of Objects.. -----,.-' -

-Srograss Report Hö9 x^_u_JMj,|fc.8B . covering period _L-:._;/-' ;

Contract Mo. . W-56~Q39-.sc-888§4- ,

JjLä§2L*t§S~

Nam© of Contractor , fhe .Q1I3,Q State DttivegBlfey,, ColUBtfrus., Ohio

fötal Sumber of Copies Received? ^f16g_i_„_

pigTjffpaTioN,E4SL. Mfcft s>f "QffBba.

Sontract Advisor 1 Chiefg Radar Bran^si add File in Room 5007, SSL 2 , Director of Engineering» HOB. •> SCIL . "3 Contract File, Bcoa 3005, %t» 4 Chief* Enginsering and Technical feLrtsiojEL, SIGGB-2, Office Chief Signal 8

"'Office^, S?ashln|tpa -2$, B«.C0 ACTS» HrG Homas Abbott, fecnnieai Reports library e £§& ' 6 Director.. äaval Research Laboratory, Washington 20, D.O. ATTRs Code 2020. 7 Cests&nding General, A¥C, Wright-Patterson Air Force Bad©, Bay-ton, Ohio» 8

"AMBl MCBEEÖ-2» - " .Gossanding- Officer,. 3151st Electronics Station, Watson Laboratories, AMC, 9

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Laboratoryj, 250 Albany Street, Casbrldge 59, Mass* ATTNi ERGRB. Circulation, Radar Branch, ESL 11 AFf Liaison Officer, E& 12,13,14 C^manding General, Hqs., M&? Wright-Patterson Air ?orce Base, Dayton. 15

Ohio. Bireetor of Research,, Hqs.,. SCEL. ATTN« Mr. V.L «Friedrich. 16 ehlefg Engineering and Technical Division, SIGEi-S, Office of the Chief 17,18

Signal Officer* Washington 2S5 D.C, Applied Physics laboratory of John Hopkins Snivsrsity, 8321 Georgia Ave., 19

Silver fringe, Maryland. ATTNt Dr. D.E^Gray. British Supply Office, 1785 Massachusetts Avenue, H.W.T flashiagtoa so, D.O. 20 Itesal Electronics laboratory, Sen Diego 52, Calif. ATAMS H,R.Bryan» Code 21

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Ohio, ATT8: HCRBON-9. --" .. |)r. G.R« Haupt^ Head Radio-Radar Division, 0.S, Savy Electronic Lab,, 24

San Diego 52, Calif,

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ATOs WLCiL, *r. Bay C9 Spencer« Mr* ^&s Httys, Ofai«f, Spsc. & Br&fUag Braaeh, &Sb »all & fieeords, SSI» Ofeiefg of ^aval S«S9srch, e/o Science and technology Froject, Library

of Googrsss, «asMogtoia 25, D.C« SOS ingles Ordaanc® Biatriet, 55 Rorth Raymond Arenue, ?äsade«aÄ 1 Csüf. Can&di&a Aiay Staff» 1700 äassaehueetts Äroan«, H»W., Washington SS, D.O.

ACT» Major fred ** Ranga,'* SossBandar, »aval ^roving Ground-, ö«8. Ha^al Proving Ground, Dablgrs»,

firginia» Ut» 'Böodyard, %dar Branch, $SL (Seatral Air Boettstats Office, 1ri$rfc-Patt9r8oa Air Fore© Basea Baytoag

Ohio» MfMa GABÖ-S»

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| f&la eoatsaai is sopervised by Radar Bs^sli, Svaas Signal X^abörätoxf, Signal ßorps I H^aserisg laboratories. For further t®ebsi«al inforaatioa ooata@t %r, #.f.Svers, 1 $gäs? "Squipamt Sgetian. Evsns sigaal Laboratory, Betaas-, H„j. S®3.®sbons5 4ebury Paric

g-7449^ 1st» 523« Goatraeting Officer's Beprcpcntatives Mr. &.L„?teweger and %, L.J, | ?«lp9 **adar %&aeh„ grass Signal L&boratosgr»

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PROJECT REPORT 302-35

COPY NO. _ 36

REPORT

by ~~'~

THE OHIO STATE UNIVERSITY RESEARCH FOUNDATION

COLUMBUS 10, OHIO

Cooperator

Contract

Report Subject

Submitted by

Date

Evans Signal Laboratory Belmar, Hew Jersey Department of the Army Project: 3-99-05-022 Signal Corps Project: 1228-0

W 36-039 sc33634

The Scattering of piahe Electromagnetic Waves By a Perfectly Conducting Hemisphere Or Hemispherical Shell

Edward M. Kennaugh, Antenna Laboratory Department of Electrical Engineering

May !5, 1950

•*.*= ^R>» JCJ <i*^ V'ftV^sy&t'iiC fflfiüEjSPWJf TW*nS^j91*

-•ssäwr^-i-

THE SCATTERING OF PLANE ELEGTROMAÜNET*C WAVES

BY A PERFECTLY CONDUCTING HEMISPHERE OR HEM! SPHERE/SHELL

INTRODUCTION

EXPANSION OF PLANE WAVE

THE HEMISPHERICAL SHELL

THE SOLID HEMISPHERE

CALCULATI ON OF ÖOtFFTCIENTS

GLOSSARY OF SYMBOLS

.8.

13

16

17

i i

:.«. = . rJ»*yo •; '-->'«-^T-*,.i%fSP,..S?*^#f* T^_ftt***g<V,>^\* .i-.-^V^.^J-T*^

.-. -.^s&t **'-•

»sg,,

Page i of 18

_ " . THE;SCATTER IRQ OP PLANE ELECTROMAGNETIC WAVES

BY A PERFECtlY CONDÜCTIKG. HEMISPHERE OR HEMISPHERICAL SHELL

IHTRODÜCflÖH

The. exact form of the diffraction pattern produced by electromagnetic waves

falling upon a body öf finite size- is of interest in the calibration, of radar equip-

ment. At the present time, to the author's knowledge, the only finite object for

which this pattern is known for all angles of incidence is the sphere. Solutions have

been obtained for other finite bodies which are va3j.d for certain directions and

polarizations of the incident wave, as well as many approximations when the ratio of

wavelength to physical dimensions is very large or very small. This investigation

deals with the scattering by a hemispherical shell or solid hemisphere for a plane

wave of arbitrary polarization and angle of incidence, where the ratio of wavelength

to physical size is unrestricted. A formal solution is obtained for the general cose,

ia the fcaai of ant infinite set of linear equations in the unknosn field coefficients.

EXPANSION OF PLÄHE WAVE

Maxwell's equations in free space may be written:

a* 2 * £ + A = 0 b. V • £ « 0 (1)

Since £, and.^j. are divergence-less, either may be expressed as the curl of a vector or

of the time-derivative of a vector. Choosing JL, let

ß = Me 2* JL •

Substituting in a,

2 * (£ + t* S) » o ," or & - 2§~ V* Ö. .- where f is an arbitrary scalar. Substituting in c:

5Zx£*n-2§ + M££.=o .

(2)

(3)

(4)

<5>

"'\*f7.e., > .yy

Taking the divergence of eq. (5), the divergence condition d is seen to be satisfied.

IfH is a radial vector in spherical coordinates, then JI = it Ü, where ix is a

unit radial vsctor and D is a scalar. Eq. (1) may then b<_ written as three separate

equations:

) „BIT , - (sin 8 ) +

2 sin 0\B0 B 0 sin 8

1 B2 U\ B $

?)* B <p* J B r + M^ n = 0 (6)

B2H •1= 0

Lr Br B9 r B 6>J C7)

1 B'2n B<$'

[_r sin 0 B r B (£ r sin 8 B <£_

BD

(8)

If the arbitrary scalar <£ be chosen so that $ = —— , then two of the equations are B >"

satisfied, and the remaining one becanes:

32n B r2 r2 sin 8 B 0

B Bn -isme —

i B2n r sin 8 "d 4>

fj£ n (9)

If II = '"" , this is simply the scalar wave equation in sphericai coordinates.

To return to Maxwell's equations, another independent vector solution may be

constructed by expressing D as the curl of a time-derivative of a vector. Thus, if

j> = - p£ V xjj ,

then

where $ is an arbitrary scalar. Substituting in a,

B

(10)

(11)

3 t (VXVXQ -v$ + fj£n ! = o

v x 2 x~n* - y $* + ^ ft* « o (12)

13 satisfies the same vector equation as D. , but since the vector operations performed

on each one to obtain the field components are not. the same, they are independently

related to the field.

• '.*. «*•. *w>. :~-^a^*-m?,-/>:s*; --: "^^s-sr- ^r^^^w_KWrT.~ZT? ;*3^JS«?!'?????r?^?B^

-stjtesi-

Vft^^^: - •

If D. is a radial vector in spherical coordinates, JX_ = i- II , eq. (12) bacanes. •

thre» scalar equations similar to those obtained from eq. (S). Choosing: $ = f' •» , ' . . • - - or,...

two equations are satisfied, and the remaining one becomes the scalar jmxe equation, in.

spherical coordinates, where H = rv .

If the time variation of El is expressed in the usual manner by use of the relation "iwt , . '. - -

.;n_.=-.U-.e ., where.Il is, independent of-.time,. then...the .seals' way#Le.quafei.oh_becpaies-

where

V2!* + k 2« = 0- ,

I! = ru, k =.-£- . mr

Solutions of the scalar wave equation in spherical coordinates are of the- form:

Un* a Ka h (kr) + 6.-.s K (krfi K ^CpS 6) ^C0S- *$ + % sin **> '

n a na

To relate the components of the D_-derived field to the scalar II, recall that

J>V$-JU£Ü

(.14)

andj}.= ;i£ V xJl

-.2 3- = T

n 2 -

r "• bi

** 1

r

B2

Br n B0

p 1 B2 R

'* r sin 0 Br B<£ '

ß„ - 0

(IS).

K = -it >n

B.

* cr sin 6 3 0

ife 3 n

er B£

(16)

Since the radial component of the magnetic Held is everywhere zero, this field is

termed transverse magnetic-

If the time-variation of D is specified in the same manner as that of D, and

n = El e , where IT is independent of time, the conponents of the Z_ -derived field

>~-^^.Q^-ÄT?.H^^ :scv

are related to, X'he scalar Tl by ft'= j?$ "• At£ Ü.

ft

* r 3r35

$ ~ r sin Ö 3r 3d)

(17)

andJ); = - Aj£ V*W

Dr = 0

D = ik

^e cr , sin 8 B0

D* =

—ik

cr

olf 3 0

(Iff)

Since the radial component of electric field is everywhere zero, this field is tented

transverse electric.

For the problem at hand, the field components of the incident plane wave are known,

and therefore can be separated into independent transverse electric and transverse

magnetic components, and the corresponding II and U vectors determined. The total

field, incident plus scattered, can be considered in two parts, one of a transverse

magnetic type excited by the transverse magnetic component of the incident plane wave

alone, and the other o£ a transverse electric type, excited by the transverse electric

component of the incident field alone. The advantage of this method of approach lies

in the fact that a single complex field problem is broken doom into tso simpler field

problems, and each treated separately.

Consider a linearly polarized plane electromagnetic wave traveling toward the

origin of a spherical coordinate system along a radial line in the plane ^ * 0 making

an angle a with the polar axis, as iii Fig. 1. The electric vector of the wave lies

in the plane 4> u 0. In rectangular coordinates, located as shcsm in Fig. 1,

-iir co* y _ _ E. = - £. e cos a H « ö

- ik' cot -v Ey . 0 Sy - H0 * (19)

- ikr cos y Ez = £0 e 'sin a Hi » 0 ,

the time factor c bei?ig neglected» where E anJ // are the magnitudes of the elect.rir

n>

--••^--^'.^.^^^^^^i^rr^s^^^^^^eui ~>.«r:<;£Si^*.»''T'?'£r7^^

. _ ,—saiir?- - • *.":"••"•*.•* ^gS^5^" •*»* -^Si'-SfSi .-5

'*..!

fc-~

z

•<*".>'

Pia. I. Coordinate System.

= •—•-*» • •. f. •/• f3«ö5gi^ yj^'j^p^wmiarr'TSj'f^nafSpV. ^S?*•

and magnetic components of the plane wave, and y is. related to 6, 4>, and (% by

cos y = sin a sin 8 cos <p + cos a cos 6 . (20)

Hie transformation of the field components from rectangular to spherical coordi-

nates is given by:

Er = Ex cos 4> sin 8 + E sin cj> sin 8 + Ez cos 0

Hr = Hx cos 4> sin 8 + H sin.cj>sin8 + Hz cos 8 .

For the plane «rave, these become:

•ikr cot y Er =* Eg [sin a cos 8 — cos a cos <j> sin 8) e

-ikr cos y H. =» H sin <t> sin 8 e ,

T O T

which may be written more compactly as derivatives of the exponential;

E sJLjL(e-ikr eos y)

r ikr B a \ /

(21)

(22)

(23)

/? 3 /-ikr cot o ' a y r ikr sin a B 4>

H. > - = —- le

If the plane wave is to be represented as the sum of a field of transverse

magnetic type and one of transverse electric type,

r B r2

(24)

Hr - 0

for the transverse magnetic field, and

Er = 0

B'rf k2 . (25)

for the transverse electric field. Therefore, the scalar quantities IT and'O will be

givjjn by:

B" n .5 £ 3 f -ikr cot y\ T-r^n.f-« ;! (26) d r ikr d a ^ '

B I! , i „» W„ B f -ihr cet y\ 1.2

ä r lir sui IJ ?. qo

.""TET."

• •Ts.csr-.'ST.T .--_• v -.'-V";.***»i. .^-^.T «T T* •#•"•• Is?,'-,'' •~-,*'>T'.' T».'5»«*r^,S^3SJ."^*SiS»i V? ?^TP'*r*;**"^«"!Sr=JSr:",-'^?B

>*^"tVt-Uti. ULX U~J i ffffflff* ^^i>at«»ai»wtfKa^»j«^&^^^-a^yv-.-r

.^*jsuitfisr,i&;;.itig_a!a

, ^fe* £ - •> .starts --*

*

The expansion i»-of /"'"****•* »•**••

•it»».* £0,5 I (.-::)* (:2ft + D 4 » n=o <

R .(.cos a); Pn 0?°s y)-

+f| ÖP; (COS «) P" i«? -*>cos«<p

.. _ .."!i^" . i" ! 'i L/_r_-_ gffi 3

- J

-Upon differentiation, r '

3f' ifer-n=o

. (* --«)! ^P..,U?'») pB (cos e, cos «<M

+ 2 £ -—^IZ ~^Z n --- -J u=i (n + ») '• - °a

(25)

#J 1 r 3^ ifer sin ft n=»i

( »_-_•'1, 21-a (2« + i) v(fer)t

-p" (cos.a) P" (cos 8) sir. « (,l ~ »-)'1 nB ;„„* «> P" (cos 9). sin *<£(• C (30)

i=i. (/» + *)

But since Hand If are related to solution, of the scalar wave elation. »__ '**?* *

""H-i- rS.S.U-

if =:^%»

performing, the indicated operations

B2 n \

s. oh 13 and'T^' yielvk'.'

.+ ^H=^££uiiB(n.) tnU)

Bra- iSv. (n).(-fi.^1)

tm

(32)

^herefor^-, If:and^are:giv^by: _ _____ __ _ ___ - _, __:______ _"_._"-;.__„

• - . 3 p° jjcoit. a-5 -,* h « fn^«)-I (2n _ 1.) .,_ ." - -tfcrj p" kcoslß)^-%—-~

G3)

. -Ä.f"»' " ~ -"" V _-- *** . •- -artfr. * *•» • .2. . - -^7-;:-.-- "i

i«. ^Pfl - w -oo ,-n (n - 'Bf!' (2" + 1") •-=-•-•• » tf. >;-- . °' S -2»««.«* 2 .(-i). -f -4T --.-. r fcLW £„ '(=«» *l P„ (cos <x)

•ife-Äin-a «=i " n=» (n+ »)•!' nfn •+_!?• " n n

^ '.;"""..- --^=- ----:-*---- ; w .. fe

where Hä ^/'— ;E0 . r

The incident, wave haying .been Separated' into two excitations, one consisting of

transverse electric spherical waves, the other of transverse magnetic spherica-l WSTC'S,

the scattering of such a wave by a perfectly conducting hemisphere or hemispherical ..__

shell will no» be treated by-considering, the total field as composed of two parts, one

of transverse magnetic-type, excited by the transverse magnetic component of the plane

wave, and the other of transverse electric type, excited by the transverse electric

component of the plane wave. The »scatterer is pictured in Fig. 1, .with its symmetry

axis along the polar axis.

THE HEMISPHERICAL SHELL

Space exterior to-the conductor will be divided into'.two regions:

: in-region 1, r < a .;_.!"

. ~ ..__ in region 2, r 2. e ,

and; the II and IT vectors, as-well as their associate field components, will beaF

subscripts to -indicate' ;6he .region, in which they are valid' _^

— 'Boundary-conditions-, require that: .;---'•'— -^"_-..:-- - -

I. Tangential components- of electric .field mr/st vanish on conducting sucf&ces,.

- II. . Tangential ^components of electrj;c-and magnetic fields must match across

cemmön böündarLyjj'fregions i and 2, ' "V

III.. Total field. :must reduce to".that of-exciting field as distance from origin,

approaches infinity., . ""...- —-

_"for the ^transverse, magnetic field., these-require:

-•'" I

-~m) "V . 'r 'S4H^6' ^if "dW:: - ~ r'sin-G 3r

_i~irj*».>L..'«^". »'•i-"w* <sfjmm UP"'•». :^.A~n*"W^<^Na«ili!!Ht^Jv%!iWJ^^

, ---at- •?•: •

'•"-. - .." ,„ .„. ; i^SS*.-.-.- •Hff *- «-~.

'it

ÄS

'" " 'S,,

(, I-

**• V^«* '*»****. 1 T = a

B?nt _«a_ ill , Br^tf • B-r W" *r. 3*" :** ^ B

l3:v #"-*;§.$ '' ?-* "?*-7-:

036> -----

Uli. -

L« n2, --^ I (2 - s,j cos ** ^rrm **» + H l -n- "--.:-*tt

p—*'B>- - - , . —

r?0 a

-fij and ITa; öMst be öf the form:.

PL » 'r % £ a BBö n°s an

COS *^J^;^N;^-*C« fl)

151)

(38)

IL.r ! 1 c. cos n<t> J". -t*ir,) P* (cos 0) + r £ ? ^ cos ra=0 h (kr) P® (

'""'"' - - •-•- -- ...:. f39>.

where FI^ contains iiö Henkel functions* since these" become infinite &Z t-ho origin, and

the «£.variation- is restricted tö cos »<£ since'the incident ^excitation- is of £hi3 type.

Tfc~ are JO chosen that the first summation in the formula for 11, is equal to

the II csSiponent of the plans wave, then condition IIX is satisfied, since the remaining

part, involving Hahkel functions, represents- an outgoing wave which becomes vanishingly

small- as. distance from the origin approaches infinity. Then, ~~-

cos5#)

c „*= en. "tf, ijn S- ^ i^_ — ———: \ *-J -v _

(40)

Edition I will *"^fied Ur tor —^* »^^^."^ and. for-3 H^ r "and Ml^. r.. ^ associated, aerie, « « be set equal torero

•therange^^ö^^Änd^ä, Thus, iox every«. , . -_;

-as»

"r « J' -1'fcaV PB (cos'^j « 0' • - - ----- ----- - ~* -,»o - .....

¥ ^cr -i^K^-^B^^ft^*«»-^-*^:

--CD :r-» a (41)

(42)

Gsaditioai II will be .satisfied if, föt1 eech- value; <r a -in the :psir of -pexifes for

ft., ahdll', , as weH as for ihe pair BTTj/ä r andJ^B ELj/B-J?, ihe associated; .pair of seri««

^5

=' A»-^SB»,*«» ,V-T: ..*3^a9!SP"°- ^W0-"

M"'.r~

•?- i.) >

--"• ex *-^p

'"'% .55

•;-K-"

oh n be set equal over the fange Ö $.8 Zri/z, and r ? a. Thus, for every.»• ,

I '%n k ^°> Pn <COi Ö> ' ? K* £ '(*«) + <*snX l**$ P" (*p* ft):]. neu n~o - I

00'

Z s vj»(M;P*(cos_?):=.1.^ v^a» + 4,A;-F*«fl-*;(;dös-*g .'•0^0 STT/2.

444)=

fSoüriäary conditieris will tnus be satisfied for the transverse magnetic component o? the

total field- i-f- for every » i

'I [can ^n ^ä> + d«n #nfeo>3 ?" Jco? ^-5 - 0, V2 k 0' £ 7T C4Ö--

nag' Rsa __ _ _ "

'-" (47)

For the transverse electric fieldj boundary conditions require:

3 n* 2T^\ Iv Ea - 0 .

3 l£ o Ha £, = 0

#• _. .'30 ' 3 Ö /'

•r i a

7V^ <(?i.7T (48)

II. £a :E,i. B nt 31£

j- . ; ~— > ; V

- ' -ont *£

#-.". **; B2n;. a'ig.

9i «a T -"B> 3.fl B;r efc£ .1 --Ö.<0 $7^2.

- **_:-* 3?'it-^.l£

<*-* *3 '.'- a;? .3.^_3v~ BIS

ia, - " . . • •- '-:. """". • r " - - ..-.-• X-i*l£ »•.;,.".V.?'- Z 2? «;ift»<3& Z 4#;Hl?-*---w, (_tl J« -l**! A 5cos öj pn icos a). r -*or/ •ik iio.o. «51 n«a ('" + «')'! n(n -^lr)

(50

10"

•.T«air«asaf"Ji"3!3JS * "^f1!1«^*? .,*WHST' .

"J;"S'Oj. -.__-_ —: - - - * * ' "TSi"""!S^ -- " -,taS?"~--? '^•- -

j*s*»i*i?-*"- j"*».'wr»

But l£ and l£ must be« of the form:.

<s .» ft. = r X. _ a* . sin »0 £ j (fer:)~ Pj* V0* '&)•

l£ —r 1 c*- sin. «0 I ./. (:fer) P* (cos 6) •+ r I d«- sin »0 S/'hR (fer). p" (-cos,^):,-

for. the-same reasons as given in the transverse magnetic case, and condition HI will"

be satisfied i"f '-,."""

c' „ _^_ tjn"y El£U ÄlÜ^i^ (-co, a).. (53) /_ »n ik sin a. (s + B)1 n(n + 1) "

Condition-I will be. satisfied, if, .&r each value of.« in the double series on n

and «for l£ and lf2 ,. the associated series on n is set equal to zero over the range

v/2 £.# SL7£jand.r => a. Thus,, for each choice of a ,.

* a*- j {ka) PB {cos-^ey u 0~ . (S4)

- ... V .- '

£<4 •>» <fea> + £B V t*«l3- *>" "leovar - röT-: "- - - ; ^ . £550;

(sedition. II will be satisfied if, for each value of * in. the pair of double -_._

series- on -» and n for r£-.and t£, as wfH ms for the pair for 3 r£/9 r and 3 fia/o r»

the associated pair of series_-fnv:n-is- set equal over the range 0 1 6 £ V2; and f" * *•""

-thus,, for each-choics of •* y'"--. _ ^ -. -—-"^=-—^=—_rjLv_-L "-

'-£ a:n Jn(!fca)^P*.(-cpS.e.) - "1 _•[£, jTrfco) •+- d^ *B =(ka):]:7P" ico« ft^V . <5&);

r .-;

"_-"' --' '-.-: '-"— - "V --_.-; "" _-• _ - > Q < & <v/2_^

t a*sn \ (,feg): Pn (-cos ö). . 1 [<^„. < (=M) +^„%;vfe=» f^/{«*-5)y - 3S7) : V

boundary conditions, will thus ;be satisfied for tha transverse electric component of th*

feotai field ifj "for every,»-,. •-- •- ..-

2 [c*' > . (fed) + d*w >- ".fta)l P; (cos 0) - 0 , V2 Ü 6 <ir «9)

! a*-.- J" (fca) P" (.cos ÖS -- § tc;n J„ IM + C K. l*^ € l»' Ö}., 0 i ^ < ^,

-• ?fti =-•"--. !^_ . - "-• =- - ---V"-- mi

n -

r --.-•- -.--_._•_- -.1...-;^_^~...^^M^st-j^TgigHBÄ»» ewpwis«»^^^»^^^*?!»''«^"^»»s^MBi^^^S^

The equations satisfied .-day- the unstarred- coefficients ^sspc-iated-wxth; the-transverse- —

magnetic field are sTe'eh tö bs the sasne ias those for the starred, coefficients associated

With the; transverse electric field, with- the ordinary spherical Bessel and Hankel

•functions' interchanged' with the. corresponding- deE'i±a.tiyes:;of- the- spherical Bessel 'and

Hankel. functions* The set of equations for the .unstarred coefficients, will be solved,

/for the coefficients d in terms of the c r- ... and the solution' modified- by- the above -

rule to-obtain the sSärjed. coefficients a in, terms of the c. . •- - _

Multiplying eq~. (-46)'by,P^ («cos 8) and integratinE with respect to cos 6 over -

77/2 < 6 2 rr, a reiatipfirbetweeh the esrfficients is obtained:

^it^c^ifeQ):M^^{kar) - P=h) *»* Kp ?{ka) + ^Ä* m~'~ ("610

where

n .p" (cos 0} P"' (cos &•) d (cos 6-)z

P ' \ • -- -\-

,. Yl-P-i';/ 2 - ri.3.5,.. .1 r I + m»:3-[-l--3.-5--((P t^UJL

and' I is restricted to-be even,, and p -odd? The-double parenthesis (()). enclosing a "

quantity denotes-the largest integer (5o.f the .proper parity) less than., or equal to, tht

- number enclosed-. " ------- ~- .- - -.---_

-- '- Substituting from eq.. C-451)' into- eq-. (-470- •.t9'e"Mnui>a".te;-äiinj then. mü'l-.tipllYing.'by- -

P;| -(=cps^-0^-Emd-J.ntegr.ai.ing._with respect to cos 6 over Q £."'£> < TT/2, another relation"

between coefficients is-obtained: ----._.._ ...--••"

ft. ffcä:j-jn.-jffegj

:|l' (fea]

—-*"3 ,.*-. (£<?>(

T?-P ;U.p>p ^5' M.p-

*^; .4/''.

gySS^. -,, T.

''.pJL^*''"' ii^rttPr? f^P

•i'ifc«T

m. -HP.,

C|fc?VU.

iing^en^s: (dl> and t63), * «y*te« of equations -Elimis»^ng

-W3* -

slaving- the «f,.p to the

:y?*Js*'.

1ftSJ«9«S«3S*»fIJ"'«« 's »4*I<ftM , •* l~ ',fcü*£j*' ̂ S^^^^^TtSKffSr^^

" — H^r " **«?3> <* '-ata . ' <&* "*?•*.

~ . ~JfSii&^; v." •^sr "T -*r

C.ü £<>r each value of «5 is obtained:

V. (J.^1. _-,.

_•_ ; .JW** •" K '*-+ J''"'*L-V|-w.r'' dfc For the coefficients rf - -j merely interchange I and p .. the corresponding system of

equations involving the" d and cKn is then: .

(T-«j* 21 + 1 P°((*)) 'lp ~ap"p ^Tp % ip <*a> =

.1*

P" ft»//. r-n-/, ^-p

For each, choice of a,, this yields- one infinite set of Knear equations for the dB ^, one

for the d. -, arid: ä similar pair for the starred' coefficients d'mj and «Q- . _ . _

"THE SPLIDHEMISPHERE --_

For the scattering of a plane wave-by. a solid hemisphere, the same procedure is

followed, as. for. the shell,. Space exterior to, the conductor is divided- into two regions.

-- in region 1-, r <> a. , 0 $ 6 <, TT/2 ____-__ . _ ..

- -.--. _.-,". ia.-region 2; r < a .

Boundary conditions I, II, and HI require that for the transverse inagnetic field:

re

?fC

I, .-^M: = £* = 0 • _•- «a, - —jP.a-.

ÄS-rr - , 3* ft .iizr_%-; * :r 3718'. r sin 0 'd'fr 3 <£

-= ;0-

;(• r a a.-

75/2 ^ 0 < 7T C66).

=-- i

1

. .. N - . -

; r-i ". *t-.-'- -.

- 0 • 6» = ir/2

mr

nc '$QrV"v- Ö2' .- <pj "<pj. 'fil tf

#2

•^" --J»3;

\ r~s~a

^ Pa. 3? TL Bsn. 91' ••»'1

C6»)

B

•i',*

^•^.v'v'V.V^

- ! '•• j>:. •"•""' ^ •

rumwky.^jawJ»«awwg^tWft.iH.,%.^-TO^1" ^VtfaT7*?iW»^r-*»i„'?^B?^i'vT r - ^^^^gS^W.g^iV^'^1" ywyg^f 3^jresjf?tB«B^5i!l^s«

-~ .IG. P •'"..». , . -,-, •" -„, ,s> ,o » (n — mV! (2ti *• i.) n . » ?j •/>•" (cos «) Lvm EL =-—2- I ,{-2 - S-) cos, m cp S- i—=•^ "--.---•- -i) /• t*r) P. (:cos 0) *=-L*_ll2LS£

•Hj and IIj are chosen to "be öf the form

Hi = r- f £ a-CQsm<p /(M :P' (.cos-/9) .'."" (7,0:).

__ .Iij. = r X 2 c cos m <jfr. j t'/jr)-Pr (cos ft^ + -J?_ JL -Ü ß-:,cosa:<p' fr. (ifer) p" (cos £)

•aii

•Condition: III wili be satisfied if c. _ are so chosen tlrat

- Eo-, - " -°-,";(* ~ «)"': 2« +"• 1 , . n. B P„ (cos., a) c =-=-2- (.2 -8 ^4 — =(-£•} " n —^-=- (7oi,

The portion of condition-. I that requires £ = £, =-0 or it = " ' - = 0 when- ri #1 *-5r30 ' '

ft,=.77/2. will be satisfied if "the coefficients aan in the series for Tlj are set equal

to zero except when- m and rs are of opposite parity, since the Legendre polynomials n PB {cos ft) vanish at 6 = rr/2-, if « and n are of opposite parity. The remaining

conditions then require, for each m

- - - 2 [Cah. J„ (*•)• + JS ^ -r*Oj] P" (COS 0). * 0, 7T/2, <i 0 < 7T 073)

£ - «a* Xr (**> Fl lCOi> &) = S [aBn j (fed) + Ö fr ('fea)ij p" (cos 0), Q <. & £ TT/2

^ ~a«* -Vf*^ ;P» Ms-e) = £- [cBn J^ IkBJ. + & tin ('fea'Ö P" (cos ft), 0 -<^< 7T/S

where s .and «are of opposite parity. .-.."-"-"."

For the> transverse-electric fie*®! boündery c.ofidixiöns .req.uire-ithati, , _.-

±. E' = ,fc"-- 0 ^

a-ig a&.c

B ft '? c> f - ^ /' -I TT/2'•'•$ ft iV

•» —-_ n- -\ 'S»!- - -

B *ir/2 : ; . (IT)

..., •;%i^/

., V •'!M*-^«»w«Äy-Ps^ V.v •ygy*-"*a^-Ja

14

fc-^ t... "S"*,'-"-.

*i-, ,;:* >^V5:si«p--_--._- - '-

* *»^pÄTS*<!K3ai--is*?.wi • t*"ri»"*)^ "£*_-Hy

-' "" 3£ß .>ti£. B':H*/ö^ f f£ Ba'i& ?2Ht VIC ^(ß'löy W& 3 0 3'.r?'M:r3rfi'?T3^B'r?0

r = * 0 < £ <77/2-V.

(7ff)

r

HI.

[cos- tf);

17ÄX- _

i5-^. *=1-Jl=i?

'et Öl **• «n 9 - -. -, '9', -»»•"'2»•+ L .in -?.«•)!'- •> a •=*•» ife «in a-*-i n=» n(n *~ö (Jn:'+ f)•! J« . ' " ' "J >

I^.'ändfiLg are .chosen tä'be .of ,*he form-

Ä^sina0 ;'„. (*rO.-pV(.cor-ö|-_- (80)

Cg.= r f 'S c* sin't&ifi j -(*r). PJ (cos 6) + r 'S 2' /?".. sin, st <± A_ (Air) p'\.£6s 8). «= j .«=•_•" n "• ' «=x „=». »" " _ »

" V ;'" ,-,' .. ' (81) • -

Condition III wil'l be satis-fieä; if c^ are chosen in the same way as for the. shel.l:

.(82) C- . i •_£~ =2» '-I'J = - - - P- {COS <*) •" ifeiiw-g -- ,,n n>- + 1=) (-71=+:.äf-! "

The portion of condition I thsfc requires £,. = 0 or-~~i- = 0^ when # = 75^2 will

he satisfied ill the coefficients in the series, a _ are set equal) to zero except when a "." ""' - ,: . «n i - v 3Pn .{.cos 8)

» and n are of the sasre" parity,, since the derivatives of the Legehdre polynomials

vanish at 8 ~ ir/2, if ni.ahd: » are of the same parity.^ The remaining, conditions require

that for each st-r - . —-- -"-_"_/

se

S falt. J' (:fea). P" (cos flfrj,'.|;;!^^1fej.MI, #(•**$ P" (cos, 0) „ 0 5'^ «'S -'

'~»--^5H;^~*= ••",- 3-

9 £ KJ ;' IJta-} P Icps^R .S'/toJi#^--i4in:'!B •(**)•] P„i'te*l ^1,.X) 4 ö.«s^'2 ,

,r ^^Jfft^^I;_ -" """'" '''^^r^',' '^-(|5)> ' ;* •."•'- '' where t and a arc of the saifse .paritys.. JIhese are the flap» öjuotiorts ^atisfaed by the . .- .

ujSsta.rr«d'coefficients, äiüi s and t interchanged',, and the ordinary^spherical JBcsg.?)-. ., ...

andSHacSel functions interchanged, »ith the corcespondii.g derivatives of the,.spherical

Bessel and- Hankel functions. Tue eqs. 73 tp T5 fxir tha ünstärre<l^co«ffieien.ta^.w.i3D[ be . ;*•—-;.

solved to. obtain tl>e coefficients /S in terms of c.-. , and the solufcicfl^irocHfvid.jby \ ^

-this rjile ;to obtain the star-red: coefficients ,T. in ternss o£ the -c- ".. /Fofjowing ihe .,-.-.-

procedure' cuitlihed for the sheji, t«ro systems °£ equations are .fJe|-^fedL"£©J the ceri*i;aniis, ...

-.-/-

-if

1

*t

0 ... one system involving, the c*^£ficien!ts /ft\ ;"• where s. and rc are of dif£feri?rit par.'ityv%,

and .the other system involving the constants.^ r, where t and »are of the same-parity:1" - " -f

-irr^Ffl^-K - .1,*" "- * «•> *^g 4^. **.*£rf -;-; "V ß B Cßff) >V«r=—!:^~* •^%r* -.- i s

,-"

ffl" B ^ tSl- U. fMJ' 'S. '«

cO'-, <";,';.

and- ... i^W£ .^ - - '

_Ji—-, - ',— .^-'*t ;,. .- ;, _ f -y c .,' . (CfcaV r-A X7- - - -A' •- • -—:H'-&- / A C .1-»., *U' i • ' -

For the starred coefi'icieÄts /Sin». these become

(s +"'»)'!' c

.(;?, — B)-!: 2>

•i• s'~ AT" •." . ^ ^a) - -a -^ .. -*< ,. *^—-.,Ai--j •b-'5. y . c.' j- (*ö) t -A/ -',-" • ' 2 7... Ci» J* I**0')

p. .. , ^. 't*«') A--. . , - (ft. [ka\ ?X t-fcq)'- £ .-v . - 5* - .ft- -(.fee:). * .X7---.: ; £ y - «*, ;h,, ;fea').: T^rr1 " ••••' -,-.•;•

and-

,_i>«-±*v!...fwt. (re. - ja)/?. 2f + i 'fci h~. :(*S) «=«+ j,'

.my»*:-

* -'* • ..L i

^-,^.•.-•..1—.•• --^.^-^.V .2 y„ßh If**), T 2T y„fJ,"|wv

GÄ!SeUtMj;p» 0:lE C.QEFFtCLEN*S: * .;

the eqiiftticoB invbisiiig. the scattered field coaf f-icienta;"för the shell and. the solid

'boiiB^isre are similar. In each case, four sets of liftäar equations, in infinitely nsaiiy

.variables are defiiied for erery vaioe of «• , Ibese eqmtiohs (-g4 tö 65 snd -86 to J39-) "

may b« wriltin in-.B«trix form as:

i#Ni^i u<ii:v i\ffn.> Ui£fci ii^it ra-= riCiJi 11411^ iic« i ii%5

fM5 -,-

liMj iii l&L-M.- H'AIBI! •bd'-liipifi ^re' iflfiAite square iratric«? cbiKp<ised GX feowb"

"IS"

*. a.irt^ff?tfWE*aM«*^limgt^iW)lgI«Mi|^ ^Si^a^

•^ri-S* ***** •> .'"•

:#

elements which, de-.end' only upon the radii»1 of the hemispl?ere. '|=IC-.||i |JCD|U J jOj j |{,.

r!£p IV ?i"a infinite column iroatrices1 sith> known element» depending upon the direction

öSthe incident wave and: the, radius of the scatterer. H<^li:, Irl^J],. It«'; H »ndl (-{'•£, I

a*-e infinlCe: cplslhnitniatribes',. She cpeppöherits. of which determine the unknown field

•coefficient;!»."" These, unknown coefficients :tnay be readily expressed for arbitrary angle

of incidence ayi

Solving finite subsets' b"if increasing order; the approximate solutions fqt the cplumn

matrices N«£'|-|;» J-I^.llv H<,*|J- &$lT6'i°II converge, to the exact solutions.. 3he

•number of terms that :rnus,t be included to give sufficient accuracy .depends üpeh. ths-

rediua of the hemisphere;; äs well' 6a the angle at which the plane wave is incident.

Most öf thö cpmputaticr« is involved in the evaluation of the matrices IJdj | j;

•i?|Sp.jt"|'» r-M'j._;|'K and. |;j;fi'-|j| | • and the calculation of the approximate" inverses. Once

these have been obtained', the scattered field coefficients for various angles of

incidence; are given byseq.. (91:)-.

GLOSSARY OF SYMBOLS:

in (fcrj •!•"'Spj&sricai iBaase'l,, function of'order-n and argisssnt kr.

hn :£&r) -_ Sp^rical ffsnkei .function of -order «- and argranent ir{ first kind)..

" : • %^'^m^r ^ «*?# •• :... /, ."

PK ieps &Y -r Associated Legendfe polynewi'si-l of order n and degree n .

= 1.t-fi!BÄTOra«.i»j»5^E3-^^ws-Bii:j-- -r?o«ra*:..-,iT7 ^^.im^^ff^w^F^m^^^^Tf^^^^^Wn^rW'm^^g^

•"- .TÄSoHKvl /J»"f" *'ÖMt!p-~ — -"«tot-.

«•-..* .JW-.Ä^ fv^S. • **!«>»-!.>*-•< "

V* ts-

fiRQdECT DEPORT 302^35

COH^RÄGf "<f 36*039 $03363*

Mffir I5f i960

KOTEt jn submitting this report it js understood that all provisions' of the -eesträct between the Foundation and the Cööperator and peiv taming tp publicity of subject matter will be rigidly observed.

ünvastlgaior. .T&. ..; i. :.r... i... ...,r?r. ...; f.TTkr......... Dt.te:. .T.... / '.

inV6Gxl QAXOTi t • • t t< e i t » i t « * * • > 9 • * * ' ' * «*•-• • • o • o • e • »•-•'• *"* * * *'*"• * UÄtöc • i

invest! gatör. s,.. ;..................... ._.-.=. .. Date, i

investigator........ o.......... -. ,......... Date..,

• • Or* • ••* • <

Investigator................u * Date.

Supervisor:.,.^............... J. ...... ...--..-• ..£.............»..> DatOy»..«... >.»y."". • ••"•••=-!- ->-*--s ;

FOR THE ÖHJQ STATE ÜNIVERSTiY RESEARCH F0USDATJÖ8

Executive D]'r*e.tör.i <? .£..$ D.a;te..../...

^gs, c-

ia

fJ»g. eraiJUBjjL~j.; ssBHMBaraBa-TaBfs