llllll

1

 

RADIA‘HON OF A POIN? 536905.23 LOCATED
AT THE T}? Q? A PROLATE SPHERGH)

Thesis for me Degree of M. S.
MSCREGAN S?ATE CGiLEGE

Eugene C. Hamhes 32'.

    

lIIlIIIlIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII I

1293 917014790 _ g

This is to certify that the

thesis entitled
Panama“ ofa Pom'f DI o/e iot‘qkol
CH" We WPO a]? a .Pf‘qulla Sf‘nptmmg
presented by

cc. C. Hdaimflt,

has been accepted towards fulfillment
of the requirements for

_M'_\<j'_ degree in M

Mud “@43an

fiajor professor

Date W53

 

 

124/

PLACE IN RETURN BOX to remove this checkout from your record.
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1/96 chIRCJDaIeDuepS$pJ4

 

 

RADIATION OF A POINT DIDDLE LOCATED AT

THE TIP OF A QRJLATE SQHEEDID

by

Eugene C. Hatcher Jr.

A;Thesis
Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Physics and Astronomy

1955

ACKNOWLEDGEJUNT

The author wishes to exoress his sincere thanas to
Dr. A. Leitner, under whose inspiration this investigation
was undertaken and to whom the results are herewith dedi-
cated.

He is also greatly indebted to Dr. R. D. Spence for the
use of unpublished tables of the prolate spheroidal wave
functions.

Grateful acknowledgement is also due Mrs. M. Muller for

her assistance with illustrations.

817510

II.
III.

IV.

VI.
VII.
VIII.
IX.

Introduction. . . . . . . . . . . . . . . .1
Definition of Coordinates. . . . . . . . . 2
Maxwell's Equations in a Homogeneous

Medium. . . . . . . . . . . . . . . . . . .4
Solution of the Partial Differential
Equation and Determination of the Green's
Function. . . . . . . . . . . . . . . . . .6
Use of Green's Theorem to Calculate the
Magnetic Field at any Point for an Elec-
tric Dipole on the Axis of Symmetry. . . .11
Energy Radiated. . . . . . . . . . . . . .14
Results. . . . . . . . . . . . . . . . . .16
Apgendix. . . . . . . . . . . . . . . . . 30

Bibliography. . . . . . . . . . . . . . . 46

Introduction

The purpose of this thesis was to find the radiation
pattern of an electromagnetic noint dipole of unit strength
located along the axis of symmetry of a perfectly con-
ducting orolate spheroid. Maxwell's equations are solved
in terms of the wave functions which have been discussed
before by Stratton et.al.,5 Page2 and Spence.5 Green's
theorem is used to calculate the field. Our calculations
were restricted to the limiting case where the dipole is
imbedded in the surface of the spheroid. In that case,
however, there is no radiation when the point dipole is
of the magnetic type, as is easily proved.*

It is believed that these results will be of use in
antenna theory and may shed some light on the mode of rad-

iation by shells or other missiles.

 

*See Appendix I.

II. Definition of Coordinates

The prolate spheroidal coordinates are defined by

the transformations

X=afi§§7a~74j case? (I)
Y=¢J7§‘-n(:-71) 51w (.1)

Z: l§7

 

 

where §Z|) —lé7§b 05?:532“. (3)

and (1,13 the length of the semiamajor axis. The metri-

cal coefficients are defined as

‘k: a. 2.22:. ('1)
k7: a I: : (S)
5‘? -.-.-.. aJGE'UU-f'). U”)

The practical utility of soheroidal coordinates may be

 

surmised that as the eccentricity approaches unity the
prolate spheroids become rodeshaped, which is approxi-

mately the shape of an antenna, see Fig. l.

III. Maxwell's Equations in a Homogeneous iedium

VX§+H9;%:O (7)
vxg-e%—%=O (3)

where g and H are the electromagnetic vectors and
[1.6 are the permeability and dielectric constants,
respectively, of the medium.

We assumed a harmonic variation with time and define

a ~c' w
Then equations (77 and.(37 become
VxB:-iecu§ U0)
We assumed that g andflare symmetric, i.e., they
are independent of the azimuthal angle? .
Expanded in suheroidal coordinates Maxwell's

equations become

37557 Hstiewafgi-Z-F 5., (H)
371W Hw' ““W’h "'7"

Mam-r); 5-)— 3:9: Es
*Q’Gatgvt) 371/? I

“gfflgE? :31“qu

 

 

(l3

 

gffifi’g‘strwa figs-7‘) H1 (NI)

gig? 13., -.-. -1pwaL§‘-7’) HS (15*)

B’i-év’é-i-w-sJi—Eéiws w

A. (557»

 

 

Z". ~££co Egg

These six equations have been so written as to
emphasize that they separate into two independent
grougs which correspond to the two types of dipole
excitation, electric--equations eleven through
thirteen and magnetic--equations fourteen throuph
sixteen. Upon combining we obtain the following

two differential equations

5%(5- 0%“;“3, (' 7/9" “Ht/"[17 117]”? (‘7)
nd : — a} (31- gt)“:

4 d
a§G;I)%fiE%+‘a§1§-(k773%7?£q "L‘J—f- +§t11t4€ (18‘
- avt
where 6". z. ‘11::- yéZ/w‘ij f 2. Elf/A

C, being the velocity of light, A the wavelength.

A solution) of these equations is ( l7]

f4k? -‘ Sig” R115)

The funct ons S ‘Wand R‘giare the ar;,;jular and

radial prolate spheroidal wa ve functions of order one,

IV.

reSpectively, and are defined by btratton et.al.D

We shall discuss them further in Appendix.III.

Solution of the Partial Differential Equation and

Determination of Green's Function

We restrict ourselves to the case of an electric

dipole located on the Z-axis. Let us write equation

(I?) as
L H. = 0 («20)

where

L156-7) 1% 75; “6:? iii-[6235 1']

£411 1?};
L is the fprolate soheroioa +§operatoz 8231an to

 

functions of order one. On the perfectly conducting

spheroidf': 30 , ”we must in addition satisfy

iggprflx) (2!)

while at large distances from the origin HY must
behave liae an outgoing wave function (radiation con-
dition).

The Green's f1mct:i{:cn6’(§37l g, 7;)", satisfies
LG{§,7j§;7'}=" 1’? A? I»)?

and the sa;=ne boundary condition as Pk? , namely

i%(h.e)=o)€=§a (13>
and the radiation condition at infinity.

Once Green's function is determ ined ”Ifcg 7) is

found from Green's theorem

H660): ,E‘v(§17)3:£a7§"“?33”')]‘°i W)

in.terms of the values of the integrand on the boundary.
1

The surface 0': 0? +0; +03 , where

2:: boundary spheroid g.

=.a small surface surrounding the point dipole

03. ““1- ”fi-

=the great sphere, fifiwo
The significance of the similarity in the boundary condi~

tions (1|) and (23) is ta image the integrand in (‘1‘!)

I

vanish. Warn 0'20; , anda‘z 6; . Equation (2") is
so written that. 394977 signifies normal derivatives
into the field space.

The central problem of this thesis consisted of
constructing this Green's function, from which it is
possible to calculate the total magnetic field £1 at
any point in space in terms of the current distribution
on the antenna. The Green's function was so constructed
that it satisfies the same boundary conditions on the

spheroid as the electric and magnetic fields E; and

Li . Because of the symmetry of the problem, the mag—

-8-

netic field consists of the single azimuthal component
It? and therefore the Green's function has a simple
scalar form.

We found the Green's function by the use of eigen-
functions as described by Morse and Feshbach.‘L We

assume Gig/7} §;7’) to be of the form .
66,7167? 2 ; (Luggage) My) (.25)

I
and that it is symmetric in (6,71'517". Substituting
(25)in (22., , multiplying both sides by S'flC‘7yand inte-
grating with respect to 7 and (P we get

1 , , lo - l
(§r|)"a')0. +[Au +1504 6 Jam- :- AQEJ (24,)
a 6-:
where N‘lis the norm of the angular function.
By referring to equation (59) of Appendix III we
see that 0118,?) satisfies the radial Spheroidal
- I
differential equation at all points §# § . The
differential equation (2(9) asserts that the left
hand side is infinite when g: g, . This means that
dug/£1) and/or its derivatives are infinite. I
If we integrate (26) with respect to g over

’

any range including the point -- ' we get
’5 Z 2 z i _._:_L_ L2")
4 #‘Q‘E +All+tJ§-§-2#'} 0‘1! df" 2114M,~ )

The result is clearly independent of the range of g .

We may therefore Shrink it arbitrarily close to the

I
point 6:? . In that case the derivative term in the
integral outweighs all others because of the singu—

larities discussed above. We obtain

(1 (1.16,?) g _ -' L“)

d§ gL. 11W» Nu. (€14)

l‘ 1+
whereg ,§ are two points arbitrarily close to

 

l
3 at either side.
In additionauG'g )must be symmetric in (g, g ) and
must satisfy Sommerfeld's radiation condition when So

0
§_7oo. Therefore

. IRIO-‘é’aknl‘bvg' (2m
a,,(g,§)—. @Rutg BOWEN/g“ 3'

where ”R ”(UR +l$ i the radial function with
outgoing-livave behavior,* m‘ancg R‘L is a linear combi-
nation of the fundamental set of radial wave functions
(”Rugg) Bu“ ('3‘ , such that the condition (.13) be
satisfied at the bouand ry spheroid go . T hus e vrite

u) L0 e)
To solve for , 3’» , two simultaneous equitions are

required. As the first we have the boundary condition
L23) , which can be written in the form

 

*See Appendix III.

-10..

(1) (I) (I)
7,117; ($03.30 a... ufimsfiz‘r Rug) (3.)

As the second equation determining 61’ ’3L we have

available equation QwL * Thus one obtains the result

L _ __ ( 2.
Egan} L‘s-) airlift» ALIS.) wk mg) 3 )

where Q.) / l (l) /
milk) = WTLQERtLC‘) "W‘gfktégj (3’)

The calculation of Green's function for our

 

problem is now complete. The exyansion (28) takes

the final form

96'7ng 7)

:31 S l S L I a“) Bk|é§b§<§ (gq)

N Kim‘s) define (30,57?

 

fIt can be easily 3 lOWIl from the radial differential
equation (5’9), Appendix III that the Wronskian of its
fundamental set of solutions is

(I) (J (2 (’1’ /
Ru. R11“ “kw IA: Ed 3- -l)

i. e|.’~ , essentially the ri ht hand side of (.23)

 

V.

-11-

Use of Green's Theorem to Calculate the Magnetic Field

at any Point for an Electric Dipole on the Axis of

Symmetry

In accordance with the remarks on page seven,

Green's theorem reduces to

I
i g a I I
Hawfiwws—sawmills
1
where 0;. is a small surface surrounding the dipole,

and where H? (3317') , the field in the immediate

neighborhood of an electric dipole, is defined4 as

l
L'k R
’ 36
He' wees—see "
he )7 ‘HF R1
where E: is the distance from the dipole to the field
point.
I
The surface q: over which we are integrating is
infinitesimally small and approximates a circular
cylinder, see Fig. Two. The dipole is located at D.
I
The side of the cylinder )4: surrounding D is
constructed from the coordinate hyperbolohd :7 equals
l
constant and nearly unity. The bases of q are
cut out by this hyperboloid from two coordinate
+ —-
Spheroids g, and 3. , symmetrically above and

below the point D. Substituting (3‘!) and [36) in

(5‘) and integrating we find that the singularities

-12..

 

 

 

 

 

 

 

 

 

 

 

 

ll 2(7 :1)
i I
/ \\§;§f
{R “is—era ““““
_"l"/‘"" D(§,,1) \§=§,
. /" \6’=€.‘ I
x z
534%. i
I // 721-5 !
! ” ' f
E /// Iii/é *
/ W ,
/
f // /)£=€. j

 

 

C I
FIG. 2. P(;)7) is the variable point of integration;
J is an infinitesimally small positive number.

-13-

are integrated out, with the final result* .
__ é'é 8’ (g ' (‘3 7

where

B _____ fitCLl L” (58)
X Nu. U :15»)

As gl—y §bthis coefficient simplifies to

I!
__ g c. . (a?)
B. -- Nae-aw

which applies when our dipole is imbedded in the

 

spheroid. This coefficient is singular whengo‘" .
Since the Spheroid in the limit§o=l becomes a rod

of zero thickness and length 3a. , this result means that
the system would radiate an infinite amount in this
limiting case” At large distance fflufaeaG) be—

comes (-iffle‘hflag and fire? becomesfifi, . Thus,

for the radiation field we find

in.

HQGIV): 11.". LP‘ _e—fl:_ (“4x BX Suc7) 0-H)

 

*Details of this calculation are described in Appendix II

VI.

- 14 -

Energy Radiated

The radiation intensity is defined as the real
part f the complex Poynting vector
sfllg x1“)? (4))
1 —

%
where L} , is the complex conjugate of }+ . When

we substitute (#0) in Q0) we find that

EFWEH.
ES: 0

Thus we obtain the Poynting flux

to"): $1913"; 7'; pews.) H 34

2T Ed.

when 3’ is large (7'1)

where w . ‘ 2
fie/£473.) :. (get)! B, swam (7‘3)

Here 6 is the polar angle measured from the positive
z-axis. The coordinate 7 becomes 6036 at large
distance as the hyperboloids apvroach the cones of
constant £3 .

The functioni represents the variation with

angle of the radiated energy, and depends in addition

on the parameters fiat-377% and g .

- 15 _

AZ

 

 

 

 

 

 

 

 

Proportions of spheroids

3.

FIG.

VII.

Results

The radiation pattern for a number of values
of $0 and Q& are plotted against angle in figures
four to fifteen.

The values chosen for 30 ‘were 1.005, 1.020, 1.044
and 1.077. These represent prolate Spheroidal radi-
ators whose eccentricity decreases in the order given,
and their proportions are illustrated by figure three.

The parameter the was given the values one, two
and three. Since “=51?de and 25L is roughly the
length of the radiator, these values of fie. correSpond
to wavelengths equal to 1T ,IT/A and "/3 . In other
words, in no case considered is the wavelength
shorter than the spheroid, being roughly equal to it
in the shortest case, “~23 .

We realize that it would be of interest to com-
pute radiation patterns for wavelengths shorter than
the spheroids. However, the results are given as
infinite series representations in terms of Spheroida
wave functions, cf. equation (‘{3), and the convergence
deteriorates as *& increases. We limited ourselves
to cases where the computation was accurate while
using no more than four termsin the series (”3) , i.e.,
fl:O/|I3)3 . As it was, this required an extension

of the tables of spheroidal wave functions over the

_ 17 -

existing ones of Stratton et.al.,5 and of Spence.3
Some of these results are tabulated in Appendix III.

As regards to the radiation patterns of Figs.
4-15, the spheroids have the effect of radiating into
directions where 9>70°. Vihereas for the dipole alone,
the radiation is symmetric about the z-axis. This
type of radiation oattern becomes more pronounced as
thL increases.

A greater amount of energy is radiated as the
soheroids approach the limiting case, where ‘7— , as
was predicted by equation (3") .

 

 

 

 

 

 

 

 

 

 

FIG.

T

-24-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10

FIG.

-26-

T
_
q
_
r
_
w

i ’

 

a \

a

 

80.8

\\

oood.

 

 

 

 

/

\\

 

k
\

 

 

 

 

nun;
80.. u aw

 

 

12

FIG.

I27-

 

 

 

 

 

 

 

n «.9.

80.. EM.

 

 

15

FIG.

 

 

 

 

 

 

 

 

FFFFFF

APPENDIX

I.

II.
III.

TABLE OF CONTENTS

A Magnetic Dipole Near the Tip of the
Spheroid. . . . . . . . . . . . . . . . . 51
The Integration in Equation (55). . . . . 55
Formulas Useful in Computation With

Prolate Spheroidal Wave Functions. . . . .36

I.

- 31 -

A Magnetic Dipole Near the Tip of the Spheroid

The electromagnetic field can in this case be de-
duced from the component £5q> alone. It must satisfy

the boundary condition

E'f : 0 when g: go (17(5)

The calculation for Green's function yields

G67; 57') Utmé Gk
2% W (‘5) (3) 3‘? Q1“)
2 mfiaue.) (”RZGWRmu)? >§’

where now

. I :2)
“Li? (3) 3R; s”) R. (9-9 Ruej'RL LS) an)

in order that Gfi7jg'l7)satisfy the same bounds 1y

condition as Ehe .
The calculation of En? yields an expression
analogous to equation (37) , but now: the coefficients

of this reqresentation have the form

81: taC. fkm (SM/{ET
NUF’R (3.) (“m
u

where g , is the coordinate of the dipole at any point
I

of the pos tive z-axis such that €12 go . Now as
§,->§°,L u , and therefore 811. as well as the
entire field must vanish.

A simple physical argument can be advanced to
account for this fact. When a magnetic dipole is
approached to a perfectly conducting surface, with its
magnetic moment perpendicular to it, there will arise
surface currents producing an opposing magnetic dipole
field vanish, as required by general theory. In the
limit of zero distance the effect is complete cancel-

lation.

II.

The Integration in Equation (35)

We substitute from equations (34) and (35) into
the integral (35). The integration variables are the
I
primed coordinates g , 7’, and all quantities are re—

presented in terms of these. Thus one finds, cf. Fig. 5

and equations 0) , L1) and C5) , that

sin e: f/R H?)
where

R= a J(§'-U0-7")+(§'-§.)‘ (‘9’

f z: aJ(§”-’-l)(|“7'T) (5‘!)
also z! =a§‘ , and $2 fig. since 7:: I“; (5 2’)

where J- is an arbitrarily small positive number.

 

a

 

In substituting from (5‘!) it will be convenient to
write down here just a typical term of the infinite
series, and furthermore only that part of the term

which varies during the integration, i.e.,

, ( :
512(7) 11(3) (53)

This clearly is the function to be used in finding the
field at points (3,7) more distant from the origin

I
than the dipole D itself (§?§) . (If the near field,

such as the current on the spheroid, were desired,

Lu’RU. would be reilaced by (”R [L ).

The integral breaks Up into three parts, the top,
bottom and sleeve of the cylinder. Since the cylinder
is infinitesimal in size, those terms with the highest
negative power in R predominate, all others can be

neglected. Thus on'esobtains
_ we ,
~3‘fkw 153-??— 5‘77‘9'“ r‘ 7

9R “Ur-3...:— (3 6" 1R 5,1977”? 'hf' d7 ' (5”
+4 S,,(I~5)f r7K’é ”fig (3' )RSA p M

The choice of prOportions of the cylinder is

 

arbitrary, and if we assume it to be long and slender
such that

/°<< «(3"; §,)
then it is easily snown that the first two integrals

cancel each other, while in the tJde

'11; Z: "t— r*' ‘II -+ .1 éyi
R a‘[u-7'v(§'—I)+<3—§.)j

can be expanded by the binominal theorem.

 

I
Since the value of the integral over 1‘ must

be independent of its size so long as it enclosed the
dipole, we may now shrink it arbitrarily close to the

point D.and obtain the result that only the first term

- 55 -

of the binomial exoansion will contribute an amount
waich is not arbitrarily small. The calculation

yields the result

_.’2Lw 311("'$) m (57)

a “4”); Jacki,”

 

One finds that the bracKeted expression becomes the
IQ
coefficient Q , defined in (B7) .

Remembering that the actual term of the ex-)

. (3 (,5
pansmn (3"!) is not (53) , but “(7 )WRMW\
times (5.5) , and imposing the summation appearing in

(3%), one Obtains the final result embodied by

equations (37) and (58)

(A!

III. Formulas Useful in Computation with Prolate Spheroidal

have Functions

The partial differential equations (/7) and ([8)
upon separation of the variables yield two ordinary
differential equations

(7LI)%SIA +[Afl :2 O (58)

Isl

           

31¢.

(59)

 

        

J- 1 J l
--( 4.8.
«1? g )ds' 1i” §*—

The solution of (5:9 is given by Stratton et.al.5

‘5

0’3 May)" - i 0/: H (7) “0)

The prime on this summation and those which follow
indicate that the sum contains only terms even.ir1 *7
if 2(- is even and odd in VI if! is odd.

The norm of the angular functions is given by

' , z 00 I l :L I
N";[ [7%)] ‘17 2 ggfihiifii (a!)

 

-37-

The functions SUE?) possess singularities at
7:1. and therefore must be eliminated as a possible
solution in problems where the 7~ axis lies in free
space.

The radial equation (57) has two fundarinextal

(1)

solutions, )1? “(3) and I“) . The radial function

of the first kind is given by
0" IO. K-l-l/A
Q __ A. I 1—1 (62)

where the factor I or g are used when l is even

or odd, respectively and V'here

___l_' 1.... (as)
x" Iii-41k (1:1)!"

x ‘Efl.’ toil w»
111 :Jfi-ch #4} (£11).

11

The coefficients CK are found from the

recursion formulas

3K(2g+2)CL1+{UK-Hulk}:+1151 +2 4%?! 01:] (a) 5)

+£zatCk-1JKGVC’?

- 38 _

4K(2K+2)C.l+ §(:K‘H)2K +A|Q +fiz¢ 1}C‘Q "l ((0‘)
Mi? 1 C11 on

(L7)

 

”I a l
and (:;:l::: 1:5: ‘4” (7Tf4?
m9: 271,

We did not use the expansion of the radial

function of the second kind as defined by Spencea

for two significant reasons, namely

(1) his eXpansion is a power series in the
variable f-lz' (é‘fUG'U. If one wishes
to compute the value of this series in
the range 3 just beyond-bl the use of
the variable 3-] alone is clearly of
advantage;

(ii) the coefficients in the expansion given by
Spence are determined as infinite series,
which do not always converge raoidly,
whereas the coefficients of an expansion
in the powers ofg'l are detafmined in
closed form from the set of (fé previous—

ly defined.
Consequently, we found the following expansion
for( useful:
( .— g)~ +6; afiurf"fifi§:+dzz(gb+dlg +1 (ng
(9%} 0k}
1+4
where

{=§-l

cf: I/ei
G": -— as /e.,3

61 = ( 3811“ 2 £061 )/Q7

.1

G; = -- Hels- L625, €1+1e:93)/ef

and where

60 :: 2C?
:1
e, : (fl-MK,“ 0R. 3C6.“ 4-qu
, Q IL
e12 aqfi. 3c,“ on cg‘L-z-sc,’ +9c,

. . 2
e, ‘-’.- c,“+:.1c,"+mg" on. 5c, ll+2acf~+m€§

where the first alternative corresoonds to evenl ,

the second to odd ,2 . Finally, the constant K1 is
defined in terms of infinite series over the coeffi-
cients A), which, however, ravidly converge in the

range of our calculation, viz.

 

«40—w

K = 3:“. $114: 22.." «1+! 6"
‘ 2&6"? (0&3)?ij .2 °
l "’ Q
«.1. :§[Jf(fl+l)(n+a)2%]~l'w Z (.1 "1.94:- i};- L Qfi)

ital) (:F‘WF‘Lif ' Keven
Aim

 

('2 i 1.3+! C'1
K‘ 2%. 3"" +1(C.‘9’ :otQ“)‘.‘ ‘ta °

Z[J.(w)tn+z)§"'] 'ZJ (Mani; (7o)

.31" (MM-1)} ’- 0H
”:5 =

Other radial functions arise in the solution
of boundary value yroblems involving Hrol te

(I)
spheroids. They are linear combinations of RIO.

and (”Rm given) in L51) and (b!) .
Among these 13(3) RIF—(S)

0“ (S): ORR ,(E) +‘LdRmtv (7’)

which for large g takes the form

(‘43
(s __ _ 14-16 (71)
i”(3) I ) £03

In conjunction uith the assumed time dependence

 

-41-

this function satisfies the Sommerfeld radiation con~
dition at infinity.

The extensions of the tables of prolate soheroidal
constants and functions by Stratton et.al.5 and by

Spenceg are given in the following tables.

 

 

A15 A14
at»: 2 -21. 94014372 -51. 96275891
3 ~24.408312175 -54.45404458

4 -27.91179

CN

(04101

-42-

II
10

ka

ka-5

 

 

 

0003me

10

0.000004 f)
-0.000055 6
-0.0008115
0.0058659 6
0.086649
0.97072
-0.055648
0.0005902

-0. 000002

-0.000001l e

-0. 000016829 Q
0.00050497
0.0051979
0.068002»
1.0158

~0.052729

0.0004799
-0.000052

~0.00005 r
0.0004587 6
-0.0026501
-0.019055
0.042681 6
0.18268
0.95056
-0.07719
0.00288
-0.00002

0.000010 9
-0.0000€364 e
-0.00045567 6
0.0056564
0.015012
0.15410
1.0246
-0.074684
0.00246
-0.00006

223,5. ka
0.0000000
0.795116
1.508551
2.07587
2.45897
2.561776
2.452795
2.0675668
1.507590
0.8179867
0.080981
-0.6142452
-1.18185
-1.55095
-1.67654
-1.545167
—1.1805748
—0.658058

0.0000000

- 45 -

= 2
0

3:5,}:{82'8

 

0.000000000

0.716985
1.57558
1.913126
2.28670
2.46921
2.40488
2.12959
1.65492
1.02998
0.52501
—O.57405
—0.990095
-1.59907
-1.58209
-l.49921
-l.16551
-0.65606
0.0000000

L41 Rag-15

 

0.000000000

1.22076
2.27047
2.99461
5.50957
5.15275
2.55889
1.62687
0.514655
-0.58565
-1.47125
—1.98655
-2.05822
-1.62865
-0.85874
0.9015 .
0.99475
1.64097
1.87500

The table appearing on this page was computed by

H. Myers and is reproduced by kind permission.

 

 

 

If
C0 11:: 5
ka - 2 9.2697
5 6.6629
)nfl.
ka - 2. 999.69
6 190.06
ka = 2 4.2142
5 5.9656
c)
Ru ,2 =5, ka=2
g .1.005 .00094696
1.020 .0020092
1.044 .0052755
1.077 .004901,4
(11R '
In
5 :1.005 .096646
1.020 .058550
1.044 .049859
1.077 .049565

-44..

‘X . 4

 

14.720
14.524

12,946
1751.5

5.6501

9. :5, Ka=3

K :4, ka=2

 

.004461
.0094066
.015115
.022155

.46222
.26741
.22060
.21015

.00011757

.012558

 

.00085277

.0018652
.0051594
.0049581

.090415
.057522
.052962

.056820

(#93

I!

Ԥ;1.005
1.020
1.044
1.077

(1) l
R
l!-
3:1.005
1.020
1.044

1.077

I :3, ka=2

-45...

 

 

 

 

-262.46
-l52.07
—l46.5

~2.x 102

52858.

1745.2.
-556.5
-1 x 103

:6, ka:5 1:4, ka=2 i=4, 149:5
~51.977 -1792-1 -166.62
-10.59 «60.65
-2.26 -28.

2.8
4.1 x 103 2.6609 x 105 21525.
591. 2560.
2.1 x 102 7 x102
1.2 x 102

- 4e -

BIBLIOGRAPHY

Morse, P. M. and H. Feshbach. Methods of Theoretical
Physics. Cambridge, Massachusetts: Technology Press.
Massachusetts Institute of Technology, 1946.

Page, L. Physics Review 65. pp. 98-110, 1944.

Spence, R. D. The Scattering of S und From Prolate
Spheroids. Final Report. Office of Naval Research.
NONE—02400, 1951.

Stratton, J. A. Electromagnetic Theory. New York:
McGraw-Hill BOOK Company, p. 456, 1941.

Stratton J. A., P. M. Morse, L. J. Chu and R. A.
Hunter. Elliatic Cylinder and Spheroidal Wave
Functions. New York: John Wiley and Sons, 1941.

MT“

-

"tynygfigyy ~11an 1m: my“

 

470