DIFFRACTION OF ELECTROMAGNETIC WAVES ON CERTAIN BODIES OF ROTATION

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP81-01043R001700010009-9
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• Original Classification: U
• Document Page Count: 110
• Document Creation Date: December 23, 2016
• Document Release Date: February 14, 2013
• Sequence Number: 9
• Content Type: REPORT

.,
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"~' AIR TECHNICAL INTELLIGENCE
TRANSLATION
STAT
(TITLE IRJCLASSIFlF_D) ,
DIFFRACTION OF ELECTROMAGNETIC
WAVES ON CERTAIN BODIES OF ROTATION
(Diffraktsiya Elektromagnitnykh Voln
Na Nekotorykh Telalch Vrashcheniya~~
by
Various Authors
Published by
Sovetskoye Radio, Moscow 1957
?AIR TECHNICAL INTELLIGENCE CENTEf~
WRIGHT-PATTERSON AIR? FORCE BASE
? OHtO
STAT
STAT
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TABLE-OF CONTENTS
GES
. ." ? : ? 3:
2. From the Editor~?s Office : : : .- : :: .? . . .
~3. The-Theory of Diffraction of a Rotational Paraboloid:
4.? Radiation Characteristics of Spherical Surface Autennaa~
by M~ G. Balking and L, A. Yaynshteyn.. : .. .' 63
STAT
;_,,,_
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IFOREWORDj
The collection contains articles devoted to tha strict theory of
diffraction of electromagnetic waves on conductive bodies.
The V. A. Fok report explains the mathematical apparatus allowing
9 '
boundary problems of electro-dynamics in the system of coordinates of a
rotational paraboloid to be solved. As an example the author offers a
detailed analysis of~the electromagnetic field excited within a rotational
paraboloid by means of a radiator (emitter) oriented in its focus. ?
The M. G. Belkina - Z. A. Vaynshteyn report .discusses the radiation
characteristics of vibrators and slots oriented along a conductive sphere.
The report offers numerous characteristic graphs plotted according to con-
,
yentional diffraction series and in accordance with formulas obtained as a
result of asymptotic adding of these series.
The M. G. Belkina report offers a solution to the problem concerning
a dipole oriented on the axis of an elongated rotational ellipsoid or disk.
In this report the radiation characteristics are calculated (for a majority '
of cases) allowing the effect of an elongated body on afield of a radiator
situated near it to be estimated. Anew solution is also given to the prob-
lem concerning the diffraction of a plane wave on a disk and the numerical
results are compared with the approximated theory o.~ diffraction. The radia-
tion'characteri.stics:"of~an unilateral slot on a disk are described.
? The book?is intended for radio-physicists and radio engineers dealing
in supersonic frequencies.
STAT
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Fxom the Editors Office
' The theory of diffraction of electromagnetic waves on metal bodie s
is ac uirin eater practical importance in recent years especially in
q g ~'
connection with the development of the centimeter radiowave technique.
Radio a sneering has brought up a series of new problems differing from
,the classical diffraction problems in optics. These include problems of
diffraction of electromagnetic waves radiated'by various antennas on metal
bodies situated near by. The articles in this collection are devoted to
just such a type of problems.
The V. A. Fok article describes a new mathematical apparatus enabling
a
bound roblems of electrodynamics to be solved in the system of coordi-
~'Y P
hates of a rotational paraboloid. In the second part of this report an
i ortant problem is solved concerning a dipole oriented in the focus of
mp
the aboloid. A solution is given i.n the form of integrals and series
1~'
tion established between the obtained terms and theRlaws of
with a connec
eometric optics.
g ~
v ti ates the electro-
The 1`i. G. Belkina and L. A. Vaynshteyn report in es g
o and slots oriented on a conductive s here. For
magnetic field of vibrat rs P
lds as totic formulas obtained from ordinary diffraction series
the fie ~ ymp , ,,.
s ~ ,
b the'me'thod of improved asymptotic adding in accordance with the method
Y
ction were derived
introduced by V. A. Fok in his transactions on diffra
acteristics for-
The report contains numerous graphs for the radiation char ,
mulated in accordance with the diffraction series, as well as in accordance
frith the asymptotic formulas.
STAT -
re orts offer a solution to the problems concerning
The 14. G. Dellc~.na p
o 'entz d on the a:tis of an elongated rotational ellipsoid or
a dipoles ri
o calc~llated (fora majority of cases) the radiation char-~
disk. The auth r
an o inion to be forn~d on the e'ft'ect of an elongated
acterxstics allowed p
bed on a .field of a radiator situated near it.
Y
o~d re rt also offers a new solution to the, classic problem
The sec t Po
t'on of a lane wave on a disk or circular orifice in a flat
of diffrac i p
accurate results obtained in these cases are compared with the
screen. The
ed b the a roximat~ methods of physical optics. The report
results offer y pp
also describes the characteristics of slot radiation on a disk.
s ite of the fact that several years have already passed since
In p
e e orts have been ublished ands that during this time a greater
the s r p P
of re orts on this very same subject have appeared, especially re-
number p
diffraction on a rotation ellipsoid and disk, the articles of this
ports on
'o till bear a scientific interest both in a methodical respect
collects n s
and with respect to the numerical results as well.
STAT ~~~
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on of a Rotational Paraboloid,
The Theory of Diffracti
by
V. A.'Fok
Intro~du-ot, ion:
' of two rts. The first part describes the
This report consists Pa
o Par 1-3 and presents certain analysis
o abolic functi ns ( )
theory f par
of solutions
tions Par. Lt and 5). Next is given the theory
of these func
abolic coordinates (Par. 6-8). The most im-
of Maxwell equations in par
st art is the introduction of the parabolic
portant result of the fir p
dar conditions to be formulated
tentials P and Q allo~nng the boun y
Po
tions in the final differences which simplifies
without the aid of equa
roblems connected with the rotational paraboloid.
the solution of all p
auxilia functions (Par. 7) connected by simple
By introducing four r3'
it is ssible
th each other and with the P and Q potentials, Po
ratios wi
ification of terms (expressions) for the field; '
to attain a,simpl
' ~ tics/ a ~aratus necessary
Thus the first part contains the mathema PA
fraction roblems of a rotational paraboloid. ~ '
for solving dif P
~',
roblem of di le~?adiation in a
The second part is devoted to the p.. Po
. ~ ma . field ,
- flect~n rotational paraboloid. The pri ry ,
focus of an absolute re g ;
throu h the abolic potentials of the gen-
of the dipole is expressed g P~ .
ld tential``of a reflected wave is,~expressed
era/ theory (Par. 9). The~fie Po
r
the form of~series ,
' the form of integrals (Par. 10) and then in ,
first in
e e ress3ons
with different convergence Zones). Th ,xp
Par. 11) of .two types ( ,
( ,
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'v in 'the form 'of series. In
for the auxiliary functions are also to en
and 1 afield in the wave zone is discussed
?? the last two agraphs (12 3)
1~
o es ndin to the approximation of geometric
and lain expressions c rr po g
p ;
correction members are ,given for this field.
cs includi the first ) ?
opti ( nB
~he field am litude u n the
In rticular,,we obtain the dependence oft p po
Pa
Part 1. General Theory
Pax 1. Parabolic Coordinates.
The letters x, y, z will designate the rectangular coordinates and we
will write the rotary paraboloid equation in the form of
STAT
~9.th the letter k we designate the absolute value of the wave vector
(1.02)
where lambda is the wave length) and we shall introduce the following
( o
s in the role of variables: first, the angle phi bettreen the sur-
value
ou h the ven oint and a certain
face easing' through the axis z and thr g gi p ,
P
surface xOz
i throu h the ve same axis (e.g., the ),
fixed surface pass ng g ry
an 1e will be the same as in ordinary cylindrical coordinates; secondly,
this g
abolic coordinates u v are connected with the rectangular ones accord-
the par ,
ing to the following formula
u = k (R -{- z); v = h (R - i), (1.03)
1- .. ~ u-a ~ 1 u u 1.04
Tt 2k 27
r = x2 ~' R = ~/ x2 2 z2.
V +y . +y +
(1.05)
The rectan lar coordinates are expressed through a parabolic according
~ ,
to the formula
- t - ~ -v 1.06
- uvcos ?~ =-yuosinT; z= (u ). ( )
x= V ~P~y k 2k
k
z 'z._ -- a2 = 0. 1.01
x ~- y 2az , ( )
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distance from the axis. ,
d to include in our re rt as part three,
At first it was intends Po ~
o tion obtained b us in 191, '~ the problem of diffraction of ,a
the s lu , Y
eve falli on a aboloid from without under any given angle.
plane w ~ 1~'
s an a roximate summation of series and a de-
This solution also contain pp
ield on the surface of the paraboloid in the zone of
termination of the f
? mi-shadow which is of importance in connection with our established
se , ?
ield in the semi-shadow zone. However, in view of
principle of a local f
first two its of our re rt we found it
the considerable volume of the 1~ 1~
advisable not to include this third part. In the role of a?brief resume
? -
t o the distribution
of the content, of the third part we refer to our?repor n
'~ 'v bo of
of currents excited by a plane wave on the surface of a cohducti a dy,
a ' given form. (1).
ny
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const and v = const represent a system
The coordinate surfaces u
0 oral raboloids of rotation.' The equation of the
of mutually-orth g I~
iven 1.01 has the form of
araboloid g ( )
p
diroct substitution of (1.06) in (1.Q1).
as can be easily verified by
zones external with respect to the given paraboloid have corres-
The ~
values v v ~ the internal zone has corresponding values
ponding ~ o~
a fable u chap es within 0 L u infinity.
The v r g
re of the linear element in the parabolic coordinates has
The squa
the form of
durin the utilization of curvilinear orthogonal coordinates
As always g
enient tv distin sh between the projections of the vector in
it is corn ~?
to direction and the covariant components of the vector
a given coordina
derivatives of a certain scalar
(the?latter are cornerted as partial
The ro ections of
in accordance with coordinate parameters). p j
function
~O
nclosed in entheses e.g. (~ ) ( ~:)
the physical vector will bee per' 9 ? s ,K p ,
leavin the designation for the covariant components.
g ~ '~
.~ ~
Then it will be
and analogously for other vectors.
Par. 2. Parabolic Functions with Continuous Parameter.
The Laplace operator in parabolic coordinates has the form of
consequently, the equation of oscillations
o~+k2~-0
will be written in the form of
Assuming that
.(2;02)
(2,04)'
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and substituting this expression in (2.03) we become convinced that the
variables in a cation (2.03) become divided and~we obtain for the func-
q
bons U.and V equations
where t is the parameter which was formed during the division of the
variables. '
In order that solution (2.010 should be a syronymous function of
the `int in s ace it is necessary that s be a whole number (whereby in
po p
0 2 0 and 2.06 it can a arezrtly be considered that s~
equati ns (. ~ ) Pp ~
0).
The ameter t can be arbitrarily substantial or a complex number.
In
many cases, however it is convenient to introduce integral representations
of solutions~in which s lays the role of an integration variable. Keep-
P
o ider U and V as ana ical functions
ing`in mind these cases we will c ns 7-Yt
variables ~ '
of the ? .
th am t the
For the functions U and V equations of one and e s ?e ype ( y
differ from each other only by the sign at t) were obtained. The solutions
o these a cations are thorou hly discussed in literature (e.g,, in (2)
f q g
and ~ - ?
(3).
rst discuss the solution of equation (2.0~), finite at
VJe shall fi
0 Thus assuming the role of the solution the function will be
u ,
Breakin down the integral into power series we will also obtain
g
chi is the series compiled in accordance with the
where f (alpha, gamma, ) ,
.~STAT
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By changing the integral (2.07) z into 1,- z it can be easily shown
that at essential values u, s, t the value xi (u, s, t) will be sub-
stantial, It is evident from formula (2.09) that xi (u s~ t) will be
an integral transcendental function of t and s. ,
We will now analyze the second solution of equation (2.03), namely
the solution which at greater u has an asymptotic expression con~~aining
a multi le ei u Such a solution can be obtained rovided we car out
P ~? P rY
an integration in (2.07) according to z within the boundaries not from
0 to 1 but from 1 + i infinity to 1. The constant multiple factor in
front of the integral can, of curse, be selected in a different form
than in (2.0~). Converting the integral by substituting
-r
x s
l--r.-e
-u
;eand by selecting the constant multiple in a proper way the expression
R s+d r u W
? ~J-d 4 R -~+2 r2
Ci(dr~ s~ t)~e 'u 'e 'Fso~ 2,11)
(
may be used in the role of a second solution where
s-~-? ~
F _ - e-xz s x
so s _ _ ~ _ ~t ~ 1-}-! dx, (2.12)
~, f u
2 U
The expression (2,12) can be broken down iirto an asymptotic series accord.
1
ing to powers _ applicable during u --~ infinity, namely, if we write
u
then expression (2.12} will be equal
The function zeta ,(u, s, t) in contrast to xi (u, s, t) is also
1
complex during substantial u, s~ t., Among others, the function satisfies
equation (2.05) with substantial coefficients. It is evident, therefrom,
that function ?
obtainable from zetal (u, s, t) by changing i into - i will also serve as
a solution of this equation At any given u, s, t the functions zeta and
1
zeta2 will be linearly independent integrals of equation (2.05). Just as
the functio o t
ns f parame ers s and t the values zeta and zeta will bey
1 2
like xi, integral transcendental functions. By changing the signs we
will obtain
(2.16)
We will establish the relation between the functions xi zeta
1 and zeta2,
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(2.17). .
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Takin into consideration equation~(2.17) and an analogous equation with
g
a reverse si n at a and utilizing the ratios (2.16) it is possible to
g
express
Cu sf- t. ~
I(' ~ ~'-slnsn
The value zeta will be derived therefrom by changing the sign at i.
2 ~
If under xi we co rehend here the series (2.08) then for zeta (u, x~ t)
mP 1
formula (~.16~ offers a break down according to ascending powers u, When
s tends toward a'whole number the r'~ght side of (2.18) shows a tendency ,
toward the final boundary; but, after converting to the boundary the
series for zetal will contain logarithmic members. '
i u s t with rameters s differin b
? Between the functions x ( , , ) pa ~ g Y
+ 1, and with parameters t, differing?by ? 2i; there exist different re-
current ratios of which we will mention the following= - '
We will have
''rDe 12:20)
t
..-~r
s~-trE u, s, f ?-r) .~ c u s f-'-i = `irE(,r, s-l, tD~
z
It is sometimes convenient to introduce the function
into the calculations instead of xi (u, s, t) which will no longer be
an integral transcendental function of sand t. Recurrent ratios for
psi (u, s, t) can easily be obtained from (2.19)-(2.22). These ratios
however, such as psi (u, ?s, t) are also satisfied by function zeta
. 1
(u, s~ t). Therefore, we will have ,?
(2.2~})
'STAY
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The recurrent ratios for functions zeta2 (u, s, t) are obtained from
(2.210-(2.27) by changing i to -i.
In order to evaluate?the Different integrals and series having
parabolic functions, it is necessary to have asymptotic expressions for
t se functions a ro rrate at e ter values t The asym totic ex-
he PP P ~' a P
pression for the function xi (u, s, t) has the form of
' ~ ~ ' ~' J 'lur),
! u ,c, 1 -t. f ~~
(~ ) , ,
where j '3.s the Bessel function. This expression is appropriate at finite
s
all the wa u to u = 0' of conditions where t` ~ 1. For
and small u_( y p )~ ...
functions zetal and.zeta2 we will have t in .the upper simi-space
and in the lower semi-space
~2.2~t1
' t2,31))
(1) (2) ~ v s
Here Hs and are the first and second Henkel functions. The aloe:
H2
l+it . .,
1 - it and G~ ._.__..) can be'changed'by '
of the functions G ( ) ( 2
~- ,
their asyc~totic expressions.
hem a ~' "s' cia
In axial symmetry problems and in such which lead to,t ~,. 4,, pe,
role is played by functions, with the parameters equal to' aero;>_ `'
r ~
33
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(7,OG) ,
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Cs- - -~' l Ds-1=
d iu '
du ~ s - - (Fg -~- iGs) _
,~
a i
d 2 sl
d !u ~
d 2~ s ?
a r
(7.09) .
(7.10)
STAT
34
a u
?-~--- G
u - -~- S F
1 ~"
_aC
'D
1
7,11
( )
s
s
du 2
s-
au
~
s-
~
2
r ?
u a
a
.
_ _ C
- D
F
G
(7
12)
,-1~
r_1
=
u --}- s
- -
du :s 2 ~ du 2
.
.a _ ? - a ~
7.1
du ~ 2 ~ du 2 ,
a ~? . - a 1 C
7.14
a~ d 2 u..
The auxiliary functions introduced by us make it possible to express the
field through ordinary scalar and vector potentials and also through
,,
corresponding magnetic potentials. Tho fact is that the field with para-
bolic 'co onents ~~
mp
E = Ets~ cos s ~ E = E~'~ cos s ; E = Ets~sin s , 7.15 '
.~~ o 'P T (
u u o 4 4
if =.N~a?sin s ~ . H = ~s~ sin s FI = tits cos s ~ (7.16)
u a ~ ~~ a o ~ t 4
can be represented in the form of
E =?~kA - grad Ao; H = curl A, (7,17)
' divA=ikAo,
The functions Cs, D , I's, G satisfy the very same equation as P ~ Q ,
s s s s
'
d~
d
du
1
4
and two other ratios are obtainable from (7.10) by also changing i to ~i.
o ~ 1 c ' ma ' na arts we 5.1,1 obtain
B,~ separating (f really) the xea an 1 a. ga. ry p .
o d= d I
-- ~, -~- (s -~-1) - -~- - o F . (7.07)
~ du 4 s
It is in itself understood that the functions C _ , D satisfy
s l s-1
the equation of the (7.07)~type a.n which s is changed to s-1. The ratios
between functions Cs_l, Ds-1~ P - , Q _ can be written in the form of
sl sl
equalities
F i~= a ` d t
s ~" s - P
du 2 du ~' ( s-t ~' {Q _
a : ,)
=ud s_tu d t _
du 2
=_v a s rU a_i
u ~' + 2 ~ (ps-~ + ~Q _) (7.08)
d du l s ~
and in the form of analogous equalities obtainable from (7.08) by chap in
g g
i to -i (just as if the values P, Q, C, D would be substantial On the
).
other hand, from formulas (703) and (7.0!}) we obtain
u
,and an analogous equality obtainable from (709) by Chan in 'i into -i
g g ~
Two ratios' become evilent ty comparing (7.08) and (7.09
as well as in the. form of
(7,18)
- ~ - ae '7.1s
E = curl B, H - --- tkB -~- gra o, ( )
dIv B = IkBo,
STAT
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f .
r(n+S+1) _z _
u, v = (uu) E ~(u) E , (v) (8.01)
X?~ ( ) r n ~ ~ ~ .
, (+)
' ial cation 0
ha e. then on-the basis of the different ~q, (3. 3),
We V ~
These ratios will be satisfied by~function chi provided we change
ns
derstand
ono or both,xi functions to eta Furthers under chi we will un
ns ns ns ,
one of the four functions obtained in such a way.
have the form of
and also
whereby the electric potentials are equal
and the magnetic potentials
(7.22)
8 riea for Potentials and~for Auxilia Functions
hr. S~
Tf the tentials for P - , Q are known one can then easily de-
Po s 1 s-1
?rive analo ous series for the values C . ~ D _ ~ F , G from the series
g ~ sl sl s s
arranged according to parabolic functions.
_. of brevit we gill write
For the sake Y
'(8.06)
(8.01)
(8.0'9) .
(B, l0)
(8.11);
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The latter formula acquires a specific meaning only at p ~ iq
0 0
because tho value nChin~l s is specific only at n 0. '
Analogous formulas for tho auxiliary functions .are obtained if the
series for the tentials P are orie ac
. po s, Qs rated cording to functions in
co lsx con anon with Chi , ,
~ fig. s?
n
ri'it Will be
(S.12)
(8,13)
(8. (5)
(8.f6)
In conclusion we shall present formulas for the derivatives of the?
series arran ed accoxdin to Chi functions. If rand z are cylindrical
g g ns
coordinates then we have
If F is a series
(8.18)
(8,19)
then, generall~t speaking, the right part of (8.18) is a series of the
very same form
and the right part of (819) is a series arranged according to~'Chin s+1
functions which we write in the form
The clauses generally speaking is necessary because our statement
o ervatio s? but on if Chi has the form of 8 O1
is true with ut res n , , ly ( . ),
ns
i e i:f this value is co sed df xi functions However if Chi ? .
.., ~'0 ns ' ns
8 8
includes one or two eta functions then the. formulas (.~21).ana (.22)
ns
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In order to find the relation between the a and b we will set-up
n ~ n
a~ F
....._- ~...- 8 20 series and
an ex ression 2 u -,v ), where F xs the ( . ) ~
P (
~u ~v
we will utilize the recurrent formula (3.03); on the other handy we wall
multi the. 8.21) series by u + v and utilize formula (3.0~). By ad-
Plv (
'n both ex ressions for ?
dusts ~ P
we will obtain
In an anlo ous manner b means of recurrent formulas (3.~) and (3.09)
B Y
we will obtain
(8.24)
4
h =c ~- c , ~{-cons/. (8.25)
nil n nil
If Chi contains only xi functions then the series (see circle)
ns ~ ns
o d coincide atd we then have
sh u1
Analogous transforms can be applied also to integrals, but we decided
? o his matter
not to Waste any time, n t
,.Part 2. A Dipole in the Focus of a Rotation Paraboloid.
Parma 9-Primar~Field from the Dipole
We will consider a di le oriented in the beginning (base) of the
~ ~
a d havi mo t directed alon the axis x' r ndicular
coordinates n ng a men g (pe pe
to the axis of rotation
).
T e ma netic field from the dipole is expressed by~formulas
h g
through the vector- otential having only one component different from
b P
where~C is a certain constant.
The covariant parabolic components of the magnetic field are deter-
mind from formulas
and the com rents of~the electric field are derived from, these formulas
Po
? ~.
- ,
Maxwell a uations 6 02 . Further we~will require. both `?
b a 1 , ng the q ( . ) , _ . , .
Y PP Yl,
' mar. field. and .we shall
aocurate~as well as approximate terms for the?pri y . . ,.
therefore, write them out in full, We have
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and .are the functions not depending upon psi and
Tn this case P Q
u+v
- ! ~ u-v u=v
2uE =C)!uv a cas~p ,-~6i -}-
w (u t v) (u -)- v)a
.4f 8 12 (u-v)
vt-u v u u '
(ut) (t)' (-f-N
u+a
- r .~ u-v u-v~
2vE =CVuve cosrp - ,-6I -}-
a (u-I- u) (ut v~
4t 8 12(u-v)
-}' , - -}- v ~ '
(u+ v) (u+vls (u t )
~ u?a , I ,
- ~ 2!
E =CYuve sinrp u r, + ,-
t + (u-r- u) (ut uP
- ~ M','Y
1 " '21
2uN =C~uve ~ sink -+ ,
u u-I-v (u-f-v):
ru}a ,
2 ~- - s i 1 2!
rtN - C ~uv c s tl rP -- -- - , ,
a u+u (u-I-v)
r u+a .
H~ = u e ~ COS (u "' v) 2t (u - v)
C -~ v -
+ ~ (u-I-vp- (u+v)~
a to' th ri fi ld fram the di le as well a
It is necess ry express e p many a po ( s
the total field including the reflected gave) through tentials P Q.
~ ~
The dependence of the total field upon the angle psi~will be the same as
for the primary field. Assum).ng that in (6.20) and (6.21) s = 1 and
omitting the signs at P and Qiwe will have
I dQ I
l
I dQ
t
-
,
P
. dP' d
-
- rru(2 -~--- --= P - --.2 du cos 4?
2NE - V dude 1 2 du
w dv
4 ,
` ~ P I I I aQ i aQ 1
dP ~ ~ - cos
VE V u, - ~ dodo 4 2 du J
..,., l d I
- I dQ I ~Q d?P I
- : ?u ~ ?- -- - _ -{--P sin q,
E -- y Q 2 du + 2 ~ dv dodo 4,
. ,. . ,
' ~ ~ 1 I dP I dP
2 -- uv 2 --- +' 2 u 2 du
luNy dv dndur 4 d
~ ,.
?? i
s
I dP ~ P dP
~-~ ' ,~? _ ~'4 - ~ _ _ .-1 sin
2luN-- uu - 2 _.... .,- - Q -- 2 du 2 dv ~t
,dudv !
o du.. 1
~. i d p 1 JR. d!Q .' 1
- ~ cos
- Q ~
~ ._
,
/uu P --T + - -- + ~,
1N; ti 2 dt 2 du and 4 ~
(9,47)
(9,04)
J~P :' I I aQ I ~
Fr= ~~`'~ P-~ 2 Vii - 2 cv '
uuv ,
dfQ I
G, _ - -~-; ,4 ~Q
t~udv ,
I ap ~ r ~aP
dir+2 dc'
2 ,
4)ith these designations we ti~ill haver:
STAT
h can be ,written in
satisfying the scalax equation of oscillations, whic ,
the form of
- L=urn d !u.
(9.08)
~# ~
According to general formulas (622), the functions P , Q~,are
connected with P,,Q by the ratios
u~~tio also saris 'the scalar a uation of oscillations;
These f rc ns fy q
P? = LAP = - LAP
Q~ = LyQ '_= - LvQ'
(9.09)
(9, 10)
The formulas for the field will become simplified provided we'intro-
duce auxiliary functions (7.01) to (7.010. Tn this case we w~.ll Write
C~ = S; U~ =- T
(9.t1)
and we will retain the designations F and Gl. Thus we assume that`
1
.'P? - ~
S 2 Q?
,_
"luE = ~ rrv 2 d~ -- ~ cos ,
1 , dv ~ ~ p
2u'E - 'riu -=2`~S -F co's
- ;
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?1e 11 find the values P ~ P? Q = Q? which correspond to the
{ wi
field of the?free dipole (without reflected wave).
formulas (9.0~) that
It is evident from
as it should actual] be because the magnetic field from the dipole does
t m of the first two a easiona
not possess a radial component, Thus he su xpr
(9,07) for the primary field should be equal to zero, This condition can
? ^ '0 Re ardin the value P? it is not difficult
be met by writing Q g g
to rove'that formulas (9.07) will coincide with (9.03) or with (9,0~),
P
if we should write
(s.ts>
~wliere is the point characteristic discussed in paragraph If. The
auxiliary functions (9.12) to (9,15), corresponding to the found values
Pi0 Q?~ .are, equal
~: '. ..
(9.20)
(9.21)
(9.22)
(9.23).
ect calc a d accordin to formulas 21 ?
The ordinary el ric,potentials~ ul to g (7. )
rT~ . ~~
t a ? t
with onsiderati:on of designations (9,11) appear to be equal (c ngruen )
~ .
which is in conformit with 9 02
Y ( )
Par, 10. Expressions for the Field of a Reflected Wave
in the Form of Integrals __ __
We-shall find expressions for reflected wave,potentials~ in the form
of integrals. For the potential Pp of the primary wave we haves according
to (lt,12)~ an integral representation
~ u?o '~~
Pa ^ 2C 2 _!C df _ .
~~ v e 2 n t;~ (u, f) Gi (u, t). (10.01)
~ ~
_~ch 2
The overall field is written in the form of
(10,02)
~ d ~ cor o to ref a ed av It~is evident from
where P an Q, resp nd . the 1 ct W e. .
considerations of geometric optics that the reflected wave should have
phase multiple ,
-
.-
ik, ~ s
e ^e.
(10.03)
.But this multiple has. an asv=nptotic expression of the function
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(10,10)
where~~for the sake of brevity we will write
a a o ous ma ner--for zeta and zeta
and in n an l g n 2 2
The solution of equations (10.10} gives
(10.12)
(10.13)
~.
It should be~notieed that the numerator of the fraction for q(t) is~the'
tixenskiy determinant which is equal
(10.14)
. ._. ~;,
' i:n .formulas ~10 0 and
The substitution of the values p(t):~, q(t) found ( . ~)
~.
o P s reflected
(10,06 , . ff..r
Par: 11.Representation~of a Solution in the Form of 5eries'_~~
The reflected wave potentials ,can also be represented i.n~ the? form~o
series.' -Similar, to that which has been done, irk paragraph li,for the.
'~ ~ ,t .
' ~ ~ ~ ~.
ma 'disturbance these series cari be obtaa;ned e3:ther as~ a'. suin'o~ d
,z'Y ~. ~ ,. ..
uctio s in o ints t ~ - `2ri * l i of the .lower `semi-s ~~~ ce ;or;'as~~: ~a~~ 's
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t are sub'ect to determination from
own functions p(t)~ q() J
where the unkn
minas 6.26) and (6.27),
the eneral for
boundar conditions. According to g
Y
utel reflected paraboloid have the
bounds conditions for the absol Y
the ry
form of
(10.00
(10.08)
' essions (10.00 ; (10,06)
chap in into boundary conditions the'expr .
By g g
~ ~ ~~ assurian that the value v in them is v ~ vo
~dand 10.09)forP~Q,,Q and g
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sa to kee in mind that the general de-
the calculation it ss neces ry P
? a t has a simple root in the point
functions p(t) nd q( )
nominator of the .. ,
Thus the. 1e t = - i appears to be
t ? . i and has no other roots. , Po ,
s wi11 be single (simple}.
double whereas the other pole
deduction in the double pole t - -i.
We will first find the
` om formulas (2.11) to (2.15) that
It is not difficult to. conclude fr
near t ~ -i it will be
wherefrgm
' ~ t can be written as
o nator of functions p(t) and q( )
The common den mi
follows ~
?r' s ~ (~ - 2I a-tv~ /t j)X
-- 2 4 t i ~. s 1 ~' ,
C ~,) -1- (-E-) ~
(1 ~
? ~ t i
i,_ ,v _ + __._ .~.. , , , (11,04)
~ 2 vo 2 u
_.... ..
ators t ?and (t) at t ? - i are equal according
The valves of the numer, p() q
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where p and are constants, the values of which will not be de-
0o qoo
termined by us because they drop out from the expressions for the field.
b titutin 11 0 11.07 and (11.08) by integrals (10.05) and (10.06)
Sus g ( . 3), ( )
we obtain values of the deductions in the point t = -i, namely,
These expressions contain logarithmic terms and do nat remain finite
alon the axis of the aboloid. Hoyrever, the va]:ues S~ T~ F , G com-
g ~ 1~ 1
filed from these terms according to formulas (9.12) to (9.15) do not ,
P
the field comes ndin "
contain any logarithmic terms (members). Therefore po g
tha electric field, for examnle,..~we, nh~~aax+
r u_v ?. _ "
'~ _ c ~ r?' ~ ~uu cos
'lu (L??)o - c 'e } gyp,
va
which corresponds to a plane wave polarized in the direction?;'of axis x..
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of deductions in poles t = -(2n + 1)i, where
The calculation
with a difficulties. Employing for-
e = 1 2, .,, is not connected ny
, .
2 we obtain the following series for the refleoted
minas (3.01) and 3.o )
tentials.
wave po
. ~ _ '~ l i (I 1.14)
_21 1) ( ) 1 ~
p~ (
Ill ._ I ' . , +., lr,
9?
E - 4;n ^n - 4l (?n -I-~) lingo
.{11=_2J. nn_" ;- _, ~ (11.16)
Pn ~ g; ~ 4i (2n+I) E~;~
fn'- ~ -
III_ " 4+ ,~_ ~ _~ ~ (11.17)
n
a ue v a ars to be? the argument of functions xi , etan~
? where the, v 1 , o ppe n
resented
?o i t out that the denominator of value p , can be rep
We like t pop n ~~
-- ~ t
4Je will now calculate the potentials P , Q as a sum of deductions
semi.-s ce are
in points t n (2n + 1)i,. Since all the poles in the upper pa
single ('simple) then we wi11 have
(I 1.20)
(11.2'1)
(I 1.23)
(11,24)
(11.25)
/calculate the field corresponding tb the zero members of,the'
We yn.ll
~~
a -dow ll 19 and (1~,~20) i,e; ~to the potentials
bre k , p ( ), > >
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We have '
X111=~~ TI?I~O+
0 0
Cc' 1 v,+r~o
~'I- ~ e +
lu -"? uv
Cn ~ 1 0.+" ~ '
? al-_ j-e
eta - v
u
The parabolic components of the field are obtairred by substituting the
0 16 ?and 1 This field
values (11.28) to (11.30) by expressi ns (9. ) (9. 7)
corresponds to electric potentials with components
Alst_0~ AI'~~0
: r '
At u r~
These expressions represent an approach to the actual field under con-
v2 Tn the eneral case it is necessa to
ditions _~rhere u~ 1, uv o. g r5''
take into consideration further members of tt~ series.
. ,.. _ ?
The 'series for the auxiliary functions S, T, F ~~ G are derived from
. 1 1
~ ~ ~ ~'brou ht forth in ra a h 8.
the series^for P , Q by formulas ,, g pa gr p
lu. 1~1 _?i ~~- n= I, 'l , , .), (11;35)
bA -~(PA NA)( t
(I 1.36)
b~ --- e
lu_2i ~ i ~ (11.34)
ao (noa+ 1100)
nru=(n+ll(a~~-I-iu~l~ (n=(; 2, ..),~' (11,33)
STAT
We gill write
o.
~ '- 'C alts E u c v, (11.37)
S +iT -I A A() A()
n-0
S - iT' = iC blt~ c (li) E (u), (11.38)
A M A
W
' i~'= ~_ rt-}-I)al'tE (u)E (v), (11.39)
FI -{- t ~ ( A ni n!
f uv
n-~~
' _ G' . ~ rl I L't', E (u)E (u). (1.1.40)
F l -' ~ // ) nt
~ t l ~ n{1 nl
~uu ~
n-u
'----iC ara (II),, (t'). (11.43)
S -}-IT n ~n n
n-0
.0
+ - '. - iC blYl Y~ (It) r, (~) (I 1.441
S IT n n n
n_u
u-o
r -QCI'p Ins 1 7
F i0 - - c c -}-
i ~ 1 u~
.~
~' l111Oala '~nl (u)rn/ (U),
-(' ( -I, ny 1
uu
1rJe will then have
to functions eta (u) eta (v). We will write
n n
al?1= n ~zt _ i 1:1) a171= 0~
n (Pn ~' 9n + u
bn - ( f) (1 n 9n
Tn an analogous manner one can obtain series arranged according ?
n L blz1- a .~ (a),
( -f- I) n rnl () nl
Substitution of these expressions in (9.16
J
fleeted wave field.
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' ate the convergence of the series obtained.
It now remains to investig
task rovided one utilizes the asymptotic ex-
This is not a difficult p
ah
o s xi and eta described in paragr p 3.
pressions for functi n n
n
u .- J l -i V(2n-}-11u) (I(,47),
E (( )
( ), o
u =laN"' l -i ~(2n-(- t)u) (11.48) '
~( ~
n~ O a
ex ressions we will obtain, for the coefficients
By utilizing these p ,
di taut members of our series, approximate values
of s
'r~ - = i4+~e-2~+sra~~
ph -
(~~- 4a a 2~1J~c~
P~ ~ e
~,~ ` ^ i es~*srm
yn ,~ (2n -1- I
where for 'the sake of brevity, we wi11 write
'It is evident :therefrom that the
as other series arranged according to
conditions
.u+ uC2~'va, ~ (11,54)
y V
a d ' 1 20 aril other series arranged according to'
'and series (11.19) n (~,~ ) ,
co ition ' ?
eta u eta (v) functions coincide at the nd
()
n n
(1 l ,53) _
series 11,12):and (11~13)~ as well
(
' u xi v functions coincide at
xl(~ ()
n
54
STAT
The boundary of the convergence zone is the surface determinable by
R ~ = 2a. (i 1.56) .
equation
This is the surface of rotation, the cross-section of which in the
lane of s et is .the rabola with an axis r endicul.ar to the axis
P Y~ ry Pa Pe P
the intersection of the focal lane with?the paraboloid.
P
Par. 12,?The Field in the' Wave Zone
Tn the case when all the three numbers u, v, vo are great iri compari-
' s'o s for the'field corres ondi
son to one (1) (large in ratio to one) expres z n p ng,
' a oximation of eometric o tics ma ?~be obtained. In our case P~
to an ppr g p ~ y ~
nd ~ wi11 be re resented in the form of integrals (10.00 and (lO.Ob).
~ @ p
In the case investigated, it is important in these integrals that, part of
the integration corresponds to finite values t. But, at finite t we can
utilize the as totic ex ressions? (2.11) and (2.1~) for the functions '
ymP P
'?~ o n~
zeta and zeta2. Using these express~.ons we r~.,l btai .
1
...~, .~~~.o~,
he dotted runes desi ate the " mbar ~ of 'the oxder l ' With the
w re ~ 1~ , ~ ..._-?~ - .
'~
vory same degree,of accuracy we?will also have
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 ~a
~Z irr!! --21 1 { z~~,
g( +
x! -" z
(l-1-i)ciJ 2
_~
t ~ tlz
(l - i! I -ir ,
rf
ch-,
-a
1
rf- ~Z ! - (I 1..zil~
~ ch 2 .
?{? ao
-
1
(12.06)
(12.07)
(12.08)
oximate a ression for Q~
As a result, we arrive at the follow~.ng appr ~
Qr = Ce
u-r
1 0.~l~ 7
_ i air a~ (u - u ~{- 2uo)
v Ig I + uu'
. ? (uo+ uu~
(I 2.091
STAT
5E
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? Substituting these expressions in an integral
b
1 C I A q (1
'._?-- ) : u, t); u, -l)dt (12.00
Q .r rr I { 7 (
ciJ 2
we can yn'ite it as folloti~rs
u-a
1 n. h ~
r
~
_
,
Q-
Y )~uv
~. ~. _~~
~1 7
Qf n
x! ua
_ ~ ch ?.;
X
I- I - !! u - u I -{- !I .. , 12.05
(!-t?! 4 uu l o
' to ral can be calculated without further disregard with the
This in g
aid of ratios
PI. but here
In a similar way one can also calculate the integral ,
we can do ataa with the integration because in our approximation, accord-
Y
_ _ ~ 1 _ ~iF F~. S
in to 12.02 t - t (t) and consequently P? - 2Q , where Q ha
g { ), P{) q
a 0 Calculatin P~ b formulas
the v lue (10, 9). g Y ?
r__ r ~ 12.10
1 _ 2LuQ ~ . { }
we will obtain
I I I 1
The auxilia functions S T F , G are derived from (12.09) and
~ ~ ~ 1 1
(12.11) by differentiation, We will have
~ ?~
The difference F, - iG,; will be of a much higher order of smal7:ness,than
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? ~ uxiliar functions found in
itution of tl~ values of a Y
. -The subst .
- om onents of the reflected
~6 and (9.17) offers parabolic c p
~' mu].as 9.1 )
for ( .
' ~ ... ~ e ex xessions (9.010
n 'these components nth th p
wave field. By combzni g .
' taan the overall field. Since
the rimary fti.eld We w111 ob
and (9_.05) for . P _
? ve a com lex
c com onents~of the field ha P
formulas for the parabola P
the
5
onl the realization of tho
t list them but toil/ check y
form we .~ri.17. no .
1'a _ ?p? 0 that
? t from condition i ~
ar conditions. It xs eviden ps
bound y
' = v it should be
surface of the paraboloid v o
on the
o ormulas 9,20) to (9.23).
o imar field (f (
where T and F pertain to the pr y
1 0, ~
that G ~- 0 and T = Oo When
a took advantage of the fact 1
Here we alre dy
_ e have
v-vow
? .. ~ sions 9.21) and (9.22)
erence of expres {
Com riri (12,18) Frith the Jiff
Pa g .
' o~ ~ ed that the rrembers of the order
`^ ?~ F we will become convinc
for T and l ?
' ~,2 is of 12x16) coincide.
' t and left ar {
and 1 (u + v) in the high. P ,
1.uv ) .~ o ,
~( ? ?
' oancide, .
' ~' ~taon both` its of-(12,17)'also c
In this appro~.lma Pa
. , - tha the expressions derived for
fusion toe wish to point out t
? , In conc ,
? o . t ,. ~ t ~ t the values u v are
obtained in the asaumpt~.on that ? s
,?S~T~F~Glwere
. , , ;1
~? ~ ~ - ~ fated on1 for greater distances
conse uently~ they are substant Y .
,.;,,high;aand~.. q ~
ex ressions re resent holcmorphic functions
from the axis. But these p P .
vicinat of the axis. They cans
of the coordinates x,~y, z also in the y ,
" corrective members
efore be a lied in all these cases where the
then , PP
main ones including also the axis of tho
are small in comparison with the ~ ?
paraboloid.
lar Co onents of the Reflected Wave Field
Par. 13 Rectangu mp -
the rectan tar components of the field it
In order to calculate gu
ormulas 7.21) and (7.22} for the electric
' most convenient to use f
is
ate first the accurate formulas and
and magnetic potentials. We will wr
oximation discussed in the previous paragraph.
then take up the appr
rectan ular co rents of electric potentials
Accord~.ng to (7.21) g ~
are equal
orda to 7.22 the magnetic potentials are equal
0 the other hand acc ng ( )
n ~ ,
f3 =U; 13 .-.~; l3t= -kyFi;
a r
(13.03)
div6=ik13~=k~y~i? (13.01) .
~' ld in two forms namely,
Tt is therefore possible to represent the fie ,
' JkF dS
E _ -~ -~ kT = dy _ iii
?? Jx
'L = -' Ox
r ~y ,
(13.05) .
x ay ~ .
~ ~).-! . -lkS, 13.06)
N --1--+!
y dx~ dx ay
H -- ! - ! -{- Y, t ,
~ dy dx ,
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r as for the reflected wave we will
n~.n to the approximate fo_mul
Tur g
? . ? ? ected wave field will' therefore corres-
ve T = 0 F =? iG . The xefl
~ ~ 1. 1 i
? ntial with the single component A .2
roximately to the vector-po{,e
pond app
?? is a ual
' to 13,01) and (12,11)), Q
which according (
different from zero
?x ,
a 4~a:r,! _L 4la9z
q 2 8 T k! a! r7)8
(13.07)
? d r endicular to the axis of the para-
Th? components'of the f~.el , Pe p
~oid scan be represented a.n the form
bol ~
k a (r~F,) k aF,
~ ; y' =...,- -}-cos 2~Q? 2 r dr ~.
x y 2r ar
a aF,
y x 2 r
._ Ce z a !
~Ia ff f )+
k a!''. r! 4~ r
( ~ )
tv~exe
(13.08)
(13.09)
(13.10)
kr aF, ~ 1k (a?)?z) ark.. :~. ~ialr? (2r? - a! - 3az)
2? dr ( +X)
1 field in the wave zone along the axis of the
exam~.ne the overal
t E wi 1 be different from zero which is equal
Componen x ~
o the strai ht. line and the second?
where the first member corresponds t g ,
to tie axis can #~eca~,c~.ated Leith the
The components, paxa.].~.el ~
? ~ to e cessions '
approxima xp
S -- Ce ~ , ?ra !
k (a -I- )
r~-o. ps.t2~
In consequeria: we obtain ,
Bfrcas V ' ~0, (13.13)
s~
60
aid of
STAT ,
tration we gill write plain expressions
For~the purpose of demons
all confine ourselves to mayor
wave field and we sh .
for the reflected
ver hin through rectangular coordinates
rimlr,~) terms and express a yt g
(P
E~ ~ C a, ' x, .)- Y')s
( r
_ - 'a_,_, xy _ er" c?'~'-, (13.14)
Y ~
8ialx etk (a?~t) ~
~r-'-'~k a~ ' x'+Y'1!
( r 1
~? , r ,
Lions div E 0
The wra.tten out express~.ons strictly satis~`~r equa ~
uations
? roximatel satisfy the remaining Maxwell eq
div H Q and app y ,
the rime field) and the boundary conditions,
(together with p
leoted wav . Since z = R the amplitudes will add:~iu;
member to. the ref ~ _ ,?;
?
at a condition
' m is a~` wits ~~ ,al
-v - 2m ~ l ~ (rtt u ) ~ )
ka o ( f ) ?
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' 11 be subtracted when
and they wi
~, m+~. (13,19)
va- 2
stance between the focus
indicates that the di
elation (13.19)
The corn f
multi le of one quarter o
a e odd (uneven) P
~' a x ..- is equal to th
and Pe 2
the wave.
yiterature
cal sics~ 1945,
' ntal and Theoreti ~Y
A Journal of Experime
1. Fok, V. .~
Ed. 12 ~ pp: 693-702
Vol. 15~ ~,
cs Isningrad Regional
ci lea of Quantum Mechani ~
2 Fok V. A., The Erin p
Technical Institute, 1932
Scientific
' Main Publishing Office
Iii, her Mathematics, M.
Smi.rnav, V. T. ~ Caur~se ion g
3.
GTTI Vol. 3, 19h3
teri.etics of Spherical Surface Antennae
Radiation Charac
b3'
M. G, Belltina and L. A. Vaynshteyn
Introduction:
surface antennae in e71 those cases where
We come in contact with
'ion of radio waves takes place in the
the radiation (emission) or recept
' And so for example, an.
ei borhood of massive conductive bodies.
n gh
anted on a metal body together with that
ordin oscillator (dipole) mo
~9
tenna the radiation characteristics of
body forma a complex surface an
ferent from the we1]--/mown radiation charac~
which can be entirely dif
teristica of an oscillator in free apace.
to as the radiating slot is possible,only
Tn regarding slot an nn
ctive surface (walls of waveguide or cavity
in the presence of a condu
cut Conae uently~ any slot antenna ie
resonator) in'which this slat is q
e antenna and the form of the surface may
at the same time also a eurfac.
teristic
xert a 'strop effect on the radiation charac .
e g
aracteriatics of surface
The_ roblem concerning the radiation ch
~p
ti at~d
slot antennas, was theoretically invea g
antenna'a~ particularly .
,
~ductive surface;
salt of diffraction of electromagnetic waves on a can
re
s
,of surface an enn
o ~'d as ~ "
,e J field of a surface antenna is being f xme ,
the radiation
becaus
T e oalculation of radiation characteristics
onl in a verb low degree, h
Y .,
~roblems '~ .
is based on the solution of diffraction.p, ., _ ,
a
t
63
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.. ~ characteristics of
a stud of the radiation
This re rt presents y
. po
eo of electromagnetic
IIsin the classic th r9
al surface ,antennas. g .
spheric
zin the general reci-
as abasis and utili B
iffraction on a sphere
wave-d
' tion characteristics of
er formulas for the radia
rocit theorems we off .
P y
a a here or on the sphere
tic di lea situated near P
eleotric or magna po
. e case of a spherical slot
din we inolude also th
itaeli. So in o g
ere ie electrodynamically
etic dipole on the aph
antenna because the mega
The caloul.ations~ according
the slot out in the aphe~.
e ivalent to
~ e
aeible. In other instant s
as axe carr3.ed out whenever po
to these forn~ul ,
ant for the derivation of ap-
ana a ar to be the basic po
these expresai 1~ ~
ristica are calculated,
. hi,ch the radiation characte
roximate f ormu].ae by w
p,
? face antenna we have the
ais of ,a spherical our
ping the analy
t tine viewpoint, what effec
in from the quanta a
o rtunity o~ explain g,
Ppo
acteristic and we encounter
has on the radiation char
the surface curvature
e entire class of surface
wh ch are common for th
'ser a: o!~ phenomena ~
a ~,e
antennas.
' :of a lane Wave on a S here
,Para a l Diffracti _
' wave diffraction on a sphere
.the stud of electromagnetic
During y
-,' ~ u and V potentials which
troduce the so-called Debya
?it is customary .to in
' ? etic fields 3n a' spherical
electric and magn
~' ive the components of the
g ., ,
. ~ formulas (aee~ for example
rdinates in accordance with -
eystem of o00 .
~~ 1 or, (2) s -
` STAT
t wave number in a vacuum (time depen-
where k - ome - 2,~1,,,, is he
- a -
c lambda
i omega t The u and v functions should serve ae
n as e- ~ ?
dance was take
solutions for the wave equation
the fields in vacuum when the diffraction
ourselves to the study of
considered as ideally conductiveo Tn ,
occurs on the sphere which.can be
- ante oP a a ere
c as exclusively) the prey ph
this case (which we will die u?
u a'is considered as boundary conditions:'
of the. radi a
? 3
. _ - ~ 0. n''E~ I ~- a ~ )
Co-E~-N, P ~ , , , .
the diffraction of .a plane electromagne$ic.,wave
We will discuss ..
irection of the negative'axia x and having
.falling on a sphere in the d
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For each a case it~ie convenient to~utilize the P and Q tentiale
Po
Introduced Y 9 Fok ar a
by (3) p agr ph (6) which are connected with the
u and' r ? tential
po s by ratios .
aP aQ
u.=cosh ~~, u=-sin~p db ,
(5)
?The potentials Po and Q? determinable the for?:ul,s,
00
1?
.... ~- n ;I - j n'~ . kr P ,(C05,~1), (6)
rk'r J a I f- )
~ ~''1 ,
give?ue the f3sld of the incident wave (!~) by virtue o! formals
n
P~ (cos 8),
The overall diffraction field of an ideally conductive a >re has
P~
potentials
P=
(--I) ors (kr)- ~ .(kr P (cos 8
jkr nn ~ n ) n
.1. -f-) Kn (ka)
n,1
,~ ~ ~ Q=
0? ka
_ 1 2n 1 n ~'n( ?~ ?
.~_ _ kr -- P cos 8 ,
I ~n () C (kr n ( )
( ) a n )
.n (n + 1) Kn (k )
? nwt
wherefrom we obtain components
E = cos 'V~ ~, r), !~ _ - si n ~ V (a, r),
r ~ ( r
= os Ml'i b r H ?=-sin VI'I(a,r),
(81
= in Vhl 8 r ?H =--cos~W~l(8,r), STAT
E s ~ (+ )' 9
a~. ~ aN ,
k
V g~ r1= fir, 3u
d a'P ~-~ +
?~u ~ 1 r , -tkstn~ de
V (s+ r) ~ ~' ~~
I a aP'~ ?k a'Q
$ r - ~ dr a~ )
V (+) r sUl
as ,s r ~ ,
V it r- ~+~ as
(~) Jr
I aP J'4
~. J r +
~+P 1 a c1Q.
12) ~ r ~ . j ~~ .. i "~~ r slii >! Jr r ~i-8 '
.V (+) au
the functions (9) it is necessary to differen-
When determining
the variable theta in segments. The derive
tiate the aeries (7) by
' to the an a theta can
t e Le ndre of amisls F according ~?
tivea of h ge P ~ n
be beat calculated by formulas:
(1) to ther with
is the associated Le,geaadre Planation tabulated B'~ , .
where P
u~ - ?
ri _ l1 y2..... 20 we calculated these functions by
P (see ./~ )j for _
n .
The functions a and set are the.apherical`
recurrent fonmalas). p ~
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~' ~ nt caloulated with the aid of tables' 5
These functions are oonvenie ]y
The derivatives are calculated bf formulas
it ,-
Tn( )--~n-I (,) x n
ac 'the following, valueat .
,~ .. . e I d~Pn (cos D)
~? .) (~ a t
..lip _vlzl pia ^0, - (I,la)
b a = V 8, a) ( )
V(,) .(
. n I dPn (cos e) 116
I _ ~ .-~--- ~ ( )
dp (cos i)
't
~ I d~pn (cos ~)
-n ~ i
( ~ ( ~" n
7 I' n j, ? dP?(cos 0) llr
< .
' ~~ ~ ant en-
' ~ d ~ ~' i ' at a the
this case the letter alpha ea gn ~ ~'
I'u . , , .
,. ,
e when r - a the 'tlusotiana (9~
On the, surface of the aphere~ i? . ?~ .
'~ ~ ~ ~ nsionlesa rametsr:
crnmtered dime , . , pa
.. , . .: - ~ 2aa
u , , aka =-~-,
,.~ .
,;~~.. ~ ~ e u ~ on ,the' oircumfe~ence
eve wh~,ch .
ich'is:e usl to the number, of~ w a ~,. p., ~ ,,
hh.. . , 4 .... .. .. .. ..
. ~ e ~ni ~ scale the aeriea~ (11)
~be ?lar~ r ~ circle ~ of.~ the sphere. ~ In 'a d f to ?
..,...a Se. ~ , . .. ,,;. - ...
~,.,, ~,. I_.
'di ~bributio~? of the surface charge and current on~,an ideally co>y-
::,the . a ..
;,;ductive eplie:re;irr~diated by the plane wave.
.- -
STAT
(lla)
d~P? I dP" - - n (n -I-11
lim ~ =1im sin oo do -' 2
a--o d0 a.+o
"- at theta - 0 and theta . pi there are
t en the functions (9~
be we
h acteriatic of a 5~herical Antenna
paraarAnh 2 Radiation C ar
f the receding paragraph make it
rmulaa ) (7) and (11) o P
Fo (9 ~
the radiation characteristics of ,
aible to obtain rated formulas for
poa
di lea oriented near the surface of a
electric or magnetic elemsi~tary po
Y and to ether with
re or on th',very surface of the sphere g
conductive sphe ~
. e it is necessary '
formin a aurfacd anterina..For thin purpoe
that sphere g , ,
r element di s? We 'will
rocit ;theorem fo eaY P~
to util~.ze the recip y,
for the radiation field of a radial
formulate for examples an.expreeaion
here can be considered as
le The one wave (1~) near the ap
eleotrio d+po 1~ .,
? ~ ~axis z
cave radiated by the dipole situated on. ?
a of the spherical ~ .
ere, and '
co arisori with the radius of~the?aph .
at int B at a greater in mp . , .
Po -
' e arid' ~havin ~ a' moment ,
diets ce from the cantex of the apner ~ g
length of~wave) n , , .
~ ~ ~ . .
' ~ '~ ?that in irit"A ~ ',
see Fi . A .page 62). Assuming .po
irected along aria x ( g ~
px d ~ - ' ,.
:';
' ~ th
- ?theta exhibits a radial electric dipole,.w~. ..
with coordinates r~ psi - 0~ .
t out that thanks to the relations
Wa wish to poin
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STAT
We wish to ~int out that thanks to the relations
po
n (n -)- I )
1 lim
m? dUs - oao Sint} d0 ='" -- 2 ~
e--o
lim-=-lim --- ;~ )
e-.R d8~ a?x sln a da 2
d'p^ 1 dP?
1
t en the functions (9) at theta - 0 and theta = pi there are
be we
relations
V ()
Para a h 2 Radiation Characteristic of a S erical Antenna
rmulae 7 and (11) of the preceding paragraph make it
Fo ~9) ~ ~ ~
btain rated formulas for the radiation characteristics of
possible to 0
e t is or ma etia elementary dipoles oriented near the surface of a
elcr gn
re or on th ' ve surface of the sphere and together with
conducti.e aphe q rY ,
ere formin a surface antenna. For this~purpoae it is necessary
that aph g _ ,
ze the reai roait ;theorem for elementary dipoles, We will
to util~. p Y ,
field of a radial
fozmulate for examples an .expression for the radiation
~.
the here can be considered as
eleatria di 1e. The plane?wave (!~) near ap ,
,po ?,
r 'gave radiated the di le . situated on axis e ,
a of the sphe ical by Po
with the radius of the a ere and
ant B at a star in comparison Ph
at po 8r''e
e ter of the a here. atad,,havin .a .moment
length of wave) distance from th can p ,. g ,
_. f ,
?that in,~ irit~A
re to alon axis x (see Fig. A page ti2~. Assuming , po
P~ di a d g , .
s
th coaidinatea r psi,- 0 theta exhibits a radial electric dipole with
wi , -
~' tables' S .
are canvenient]~r celaulated with the aid of
These lbnctions
~'e derivati~rea are calcule~ted bf formulae
Th it
i ' e when r ~- a, the ~it~notione (9~
On the surface of the ephere~ .~ -
values: '
acquire the lollowiug
(12)
~~~ `~~" to the mrnber of waves wli~cfi` pile up ;on`,tbe' circumference
~hioh ie. equal .. .
. ~ ,. ..
Ba=V 8, a) ( )
v(,) ,( ,
^ I dP? (cos e) 116
dp cos S)
1 (8~ a) -' a sin b n (n --? U ~? ( )
`~ dip cos D)
,~ .{ .n I ?( 119)
.n ?{ -~ _ ~
n (n ?1? I) ~? () ~
? dsP (cos D)
^i ~~ - (, ~) -~_ dpi
this case the 'latter alpti~,i de.aignates. the? frequently en-
~. ,~, ?
, , ..
. ~ the rie ll ?
re In.~a definite?:scale ee e ~ }
~he:larger, airale,;of:the ephe ".., ., ..s ?, .
~'dis'tributiat~~?-of the surface oliarge ? and Current on , an iCleallp co?-
.,the ?,, .
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such amoment p - p that its primary field has a component in point B
r-
Aa result of diffraction on the a ere the field f t
ph o he plane
wave (1~), in point A according to'($) has a com Went '
Po
E?=sia8,
x
Sinoe the ane wave ha a c m t
p1 (k) a o ponen is point A
-~~' the reciprocity theorem g#.vea for the dipole momenta at points A and B
z ~ J
-r~re.s ?
px _ e pr.
According ~o the reciprocity theorem for overall fields the radiel
dipole, situated in point A, should produce a field in point B
where theta is the angle betweon the radii-vectors leading from the
center of the a ere toward iota A and B, If the axis of thF a rS-
Ph po ]~
cal system of coordinates should be drawn throu int A in which
gh po ,
the radial electric di le is situated with the moment then its
1~ Ps
primary field is the wave zone ie
where R ~ the distance from the observati A
on point to point and the
angle theta is read from the direction OA. As reau~lt of diffraction on
the sphere the overall field has the farm of
~rkR
Eo=H4=-kip -W(8,r), (146)
k
where "the Weakening Factor" W (theta, r) is connected with the funo-
tion V (theta r) see formulas 9) and 7 the ratio
( (} ~
W (0, r) = crk, ~~~~ o ~ ~~ r 14n
( ) ~ )
If the moment p of the electric dipole, oriented in point A is ,
. ~
perpendicular to the radius-vector OA then we guide axis x (beginning
,,,
of the reeding of angles psi} parallel top (so that p ,~ p )~ Then in
x
ane ai - 0 where the rimary wave field of~thia dipole is given by
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 Declassified in Part -Sanitized Copy Approved for Release 2013/02/14 :CIA-RDP81-010438001700010009-9
re ai ie i the primary field of this dipole has the
In the plane whe p .~
2
form of
and the ovorall field
ill now investi ate a case where, in poini~:A, an elementary
We w g
' a moment~m If the di le is directed
magnetic dipole. is situated with Po
o that 3n the absence of a sphera~ its radiation field is
radially a r ,
then the field of the surface antenna has'the form~of
However if the moment of the magnetic dipole is perpendicular
to the radius-vector OA then wa as before, select the direction oi'
~ ~
the axis x as coinciding with the moment m . m~ and in the plane psi
0 the primary field in the wave zone containing the dipole is preaentQd
by formula
and the diffraction field of the spherical antenna equals
(186)
. }, ,
In the ane si - i ndicular to the di~'le moment ,the,~~'
pl P ....~.~ ~x'Pe ,1~ ~ .. ..pr
2'
field d e n t de nd u n. the an 1e theta ? and a ual~ ~= . `,
radiation a?a a pe po g q_,
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tenna has the form of
on field of the spherical an
and the radian
tAk ., ,,
~,,-~~ c - ll u~ (11, r). (196)
s
ve the a ere for any arbi-
tio di 1e abo , Ph
'The field of the magna po
formulea:
the an le psi is given by
trary value of g
electric dipole. Formulas (20)
aimilax man~-er also for the
and iu a
Prom fore ($}?
are derived directly ,
e ant
' ation chaxacteristics of e1 m asy
- we wi]1 calculate the radi ,
Ne~t~
? ere i. e. at r - a, For the
tuated on the surface of a sph ~ ~
dipolep ai , - ,
e of brevity we will'write:
s~ ,
~V(~)~1V(,) ,
l4 (il) lV (~ )
Figure C'
2) .... ,.(1),
Tt.should be mentioned that the flinctiona W (theta) and W ;
' e iat n characteristic of an eleinent ~ dumb-~~~''
(theta) also offer th rad io . ary (,?.
F ~-the 'ere because h a slot as '
bell type) slot (see ig. B) cut in? sph suc ~ ,
he,e nt ~ etic+ di~ ~ le~ situated ~on the
is lwoun~ is equivalent tot lame ary magn po ,
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 Declassified in Part -Sanitized Copy Approved for Release 2013/02/14 :CIA-RDP81-010438001700010009-9
s here, The function W (thota) gives the radiation characteristic of an
P
annular attics/ slot (see Fig. C)~ provided its radius'~is small in
com arison with the wave length and the radius of the sphere; such a
P
'~ma etie ring" is equivalent to an elementary radial electrio dipole.
Threshold C sa e. Graphs of Radiation Characteristic
The calculation of complex functions (21) is of greatest interest
in~the theory of spherical surface antennas which ie represented in the
form of aeries (llb c d),
~ ,
If
(22)
then these aeries are approxS.mately reduced to their primary terms
arrd ate obtain
YI(~1 ~ = 3 CoS ~,
~ (?) 2
y
~=gyp=~,
W (~ z'
(23)
Thusi fin's specific eaae~of infinitoly long~waves~ the affects of the
ore n the radiation characteristic of an element' electric or
sph o ~`Y ~
netie dipole, which ,is situated on the surface of a aphere~ fie three
.. ~,,>
times equiva).ont to the' ine?ea'se, in the moment for the radial eledtrie
di le and one and one-half times for the~ma etic di le~(element
Po ~ 6`n ~ ~Y
., .. ..
slot) From the vie int of electro-and ma etostatica this inrease
~ ~
' STAT
' eaence of a sphere) of
connected with.the appearance (due to the pr
is
static "ima a" of the primary dipole.
In the case when ,
(24)
we actual/ deal with geometric optics.
y
d zone i. e.
x~ait of ometric optics and irradiate ~ ~
In the pro y ge
have formulas
theta i ~ ,
at 0
2
~I (t-) = 2 sin a,
W~'~ 8 =2cos0,
()
he shaded zone at .,~ rtheta pi
and in t i
2
(25a)
W 0 . W~'~ (0) = W~~~ (a) .~- 0. (256)
( ).
theta --~ eonaequently~ 'appears to be the goo-
The equatorial plane _ , ,
z
the a here doubles-the
tric'baund of light and shade above which , p, ?
me ~9
arance~of the "optioe]." image~of the ,primary.
.dipole momenta due to the apps _ ,
. , ~,~
field of di le
the s ere c lately shades the primary f _ po
source. Below it ph ~p , . , ; ,, ., ,~, ,
f static ,and o tic images takes place, according
radiation, The formation o p., -.
,~ _ .
.. 2 std ~ 25a)'
coefficients in formulas,: (3).., .,:(.
to different laws; therefora~ the , . _ ,
are also different, ".
,77
76
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`STAT,~
 Declassified in Part -Sanitized Copy Approved for Release 2013/02/14 :CIA-RDP81-010438001700010009-9
e into consideration the diffraction
Farcau7.as (25) do not tak
the absence of a distinct boundary
ch a era first of a11~ in
whi ppe ~ ? ,
the aence of a
ated and shaded zonea~ i. e.~ in Pre
between the i]lumin
utilize the series (11) for the
.. eded zone. It is impossible to
semi sh i
? tion becauae~ in
undor condi (24)~
ation of a diffraction chart .
deriv _
' fowl However, the approximation
thin case the aeries converge very a y.
rSved a g the common.dif-
n 21 can be de b'9' 1~?Y~
a for functio a ( )
forma]. a
onvex conductive bodieai developed
fraction theory of a plane waves on c
A Fok 6 7 These formulae are
by V. ~
.u
~:
M and xi are the dimensionless parameters
where .
~ i
kn ,% ~ ~ r p
~~:
(~7)
x v ble t is
where theacontour r (G) in the plane of the ample aria
d Fi D and w t is the complex Fkire lunation (see
re resents in g. ~ ()
p 1
r The formulas (26) Big a continuo~is ahnnge from
6 0 7 ).
li t to shadow and in the illuminated zocae at greater negative xi
6h
the convert into foxmoulas (25a) and in?the ahade~ at greater positive
y j
the are raotic zero i. e.~ thep comrert into formulas
value a xi y p e11Y ~
he sib olute values of functions F(xi)~ and G(xi) de-
(25b).- Since t s
in the ter xi the change
orease maaotonoualy during an increase parame i
d the increaae~of the ter
in. the radiation oharaoteriatiae wring P~'~
al - ka fron sero to infinity may be assumed as taking place aocord-
P~ -
in to formulae 26) in the following manners the radiation in. the lower
g
emi-a a theta .~ weakens monotanoualy, and in the upper semi- .
a pac
2
a ce theta ~ the characteristics pass monotanouall over fraa~
Pa
2 I
2 to t'unations (25a) the semi-shaded sons having an angu-
funationa (3) ; e
tar width of~ the order .~. ~ deoreasea. ~?
. M
in radiation characteristics is more `
The feat is that the change
cam ex ae is indicated by the direct caleulation according to the~dif-.
PS
fraction. aeries paragraph 1. '
e at the end ' of ? the ?re rt) the 'resat
In the drawings 1 to 5 (ae Po,
ctions determinable by means of integrals
and F(xi) and G(xi) are the fun
. ~i `~ 3r
I ~ c~
yr. ~~ (~)
r
:,
I ';s ~~;r
Yn ~r, tr-
I'
STAT
e for al" `-1 .'. ` .are vea
. of calculatin the functions W (th ta) pha _ .3.. _.9., . , ~? , ,., .
B
;~,;
and 1 to'23 ' rea ctive the::'~aulte~~of, cnlct~=.:.
and in Fig. 10 to 14 9 i. Pe ~,Y~ ,..
' ~
~
~ '
:
.
(1) (2).
. 'the functions W. (theta) and W. (theta),,, a~ro,~ given for .these.
/sting ., .
al The ~casculationa'_ Were'`,carrie
very same_ value s of the parameter pha. r r : ?:: _ -. :' ;
out accordin to formulas (11) and ;(21)`;' where+iPon~'~it? xae `necees
g ;
;
,
.f ~,
? le a than 2 ~el ~~ ~? members "rind=~ eo ~,v at,
the aeries (11) to take no a ph . ~,,_ ? ,
,
... _
q
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1ve and:at al ha - 10 we took twenty members of the aeries
we took toe P
and a a result ~obtaining~an~error of not more than two unite of the
~ s ,
f h ai war) after the comma.
Dort gn (po
W theta' -the am litude characteristic the phase arc W (theta)
( ) P ~ ,
the ass characteristic of the given antenna. Of greatest~intereat,
ph
certainly, is the amplitude characteristic often called simply, the ra-
c e i ti
diation haract r a c.
The a s mentioned above show the characteristic features of the
~~
am litude radiation characteriaticas
P ,
the a 1 arance of new~ma~dmums the total number of`~which
1) P1~
increasos with the increase 'in alpha;
2) the oscillations coimected with it which are i'oularly
P~
intensive in the zone of the ahadaw and where, due to these oscillationa~
the radiation characteristics acquire a multilobe form;. and
3 icularl stron oscillations in the vicinit of the
pew Y g Y
dole theta : pi, where the amplitude oharaoteriatic asaumeq relatively
Figure D
The graphs show the absolute values and phaoe.s (in degrees) of the
..,, _ ~ ~ 2
(1) ~ .( )
~o lax functiona.,W (theta) W (theta) and W ~ (theta) in relation
to the an theta, (0?? :, theta 180?) at a van value of the '
.~ ~3 . ~ ~
. \ ~ ~
parameter al.
For example, the aomplex function W (theta) should, of,eaurae~ be
called tie complex radiation characteristic and its absolute value
STAT
greater values.
~Aa to the phase characteristic, it experiences rapid changes near
the 'minimums of the amplitude characteristic;
We shall analyze morn thoroughly 'the individual seriea'of graphs.'
e 5 " '
adiat c eri 4l
The r ion haract stic ~hete,) of a radial electric di le, at
( ~ ,
alpha~_.l,ia different from the Characteristio nt alpha's 0 by a small
.,,-
di~ 'cement of the maximum .into 'the' z e of the shade; ~ ~ When el~ a is
~_ Qc ,l~
2 this maximum was ahead broken'down into~two " rotubarances~!" '`Duran
y P , B
. ...
.. ;
further increase; in the ammeter al anew maxima and.minima'e . ~sr
P ph PTA ~
E.
and the numt-er of oscillations increases monotonou~ly~eccuppiug;e t~iroxi-
.. pp
- ~ - ;.
mately -an 3r~torv 60? theta 180.?.
$1
so
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(1) etic di le in
aracteristie W (theta) of the magn Po
The complex ch
(2)
d criteria are possessed by the function W
The above mentione
it form - the complex radiation characteristic
(theta) in the most distinc
? ~ lane ndicular to the
f the ma etic dipole are in the? mei idian, p P81`Pe
o ~
ceable ovex,the.entire
le Here the oscillations are quite noti ?
dipo
? ~ the fact that
T e oscill.ato nature of 'this 'function leads to ~
diagram. h .rY ,
e t0 Calculate
order to, ot,a graph it is necessary, in many cas s,
. in , , P1
? o (z)
the ar ont theta by 2 . The function W
this.funotion according to g~
taut and - e ual to :~. for all values of
(theta) at alpha is Oi is cons q .
STAT
t e di le has at the very same values
the meridionel plane containing h Po ~
ler form. At alpha ,~ 0 it consists
of the ameter alpha, a Bch situp
Pte' ?
? .. the u r and lorrer - separated by
of two larger lobes of uniform size ppe
ri tic at theta - 90?. With the
the zero value of the amplitude charade s -
:e rear "shaded") lobe, always remaining
al ha aramoter th ~
increase in the p p
e aduall narrows and weakens. With
atrietly isolated from the upper on gr y
a ar noticeable especially in the inter-
thia weak oscillation begin to ppe ~
e u r and i.ower lobes. Because of the
mediate range of angles between th ppe
a],u?a al ha 5 for the lower lob:, should,
e cillations v P ~'
presence of thea os
at theta -180?) in the amplitude characteristic.
maxdmum ,
aximum of the oscillations (having a
in essence, be considered as the m
the increase in alpha it acquires a wave-like
the angle, theta. During ,
the oscillation am litude
and at a chan in theta from 0 to 180 , P ~
form , ~ .
in generals increases.,
the value of the parameter alpha ? 10 is
~t can be concluded ,
1 hi in order to allow a olear formulation of the
still insufficient y gh
d " aded" zones with the monotonous transition between
illuminated an sh
10 the radiation characteristics
them. In""other woxds, when alpha
~ ~ `~.
tativel transmitted by the approximation formulae
are not even quali y
ct utilization of the diffraction sexiea is already
(26) and the dire ~ ,
enco of vacillations from the physical viewpoint
impossible. The Ares
remains incomprehensible.
i n with the statements made above we ha`re two problems
In connect o
closely connected with each other.
T e lain the on and physical sense of the oscillations
1) o xp g~
at on chaxacteriatica of a spherical surface antenna and;
in tha radi i
2 To derive such a proximation formulas for the radiation
P
tic which would ive satisfactory results at alpha 10.
characteris a g /
- lotion to these roblems will be given below, in paragraphs
Tha so P
!~ and 5. " ?
Pa?~a a h ' Def3nitian of the Theo of Diffraction on a S ere
at Greater Values'of the Parameter Alpha
P formula (7) of the electromagnetic field
The potentials and ~ .
~be written ea follows:
of a lane wave being diffraoted on a sphere, can
P
~ .. _. '-'y _ ~>> ?, P , cos II ~ ,
_ ~ ?:
Y
~ t.,,~ .
-~~ '~' ?~) J' i
=---
83
STAT
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is carried out in accordance with the nu values
The auaffiation here
obtained
_~ y~l r.
i ~ ~~~ )
,kp y- 2
slue al a is determined the ratio (12).
and the v ph by
It oan easily be shown that the potentials
=i
(30)
i -,,-Y__, --- ,- .
~~ v2 - 4 cos (av) 2
rom
q-i
Y2--' COS'('RV) 2 ~ J
Ci ~}
where the contciur C.'compriaes the right-half of the material axis .
? .1?
tentials ive fields ($} - (9) such as the
(see Fig, E} and the po B
' p d ~ differ from P and re- ?
tentiala (29}. ?The fact is that an Q Qi~
po .
-- -- - - .. ,. . ~ 1
but the`fielda of these items as can easily be proven are obtained
a ual to zero.
4
$4
STAT
' Phi tra and Pai (nu)
p ( )
Talon advantage of the fact that - ~ .
g .. nu ~
2
exam e 8 Fig. 68)
f nu we~convert (seep for p1 ~
are the even functions o ,
the material axis and above it
e aurae C into C which?ie para11s1 to
th
1
II the ratio
(see Fig. E}. sing
~ ~y- ~-
fl -?i cos (r.?.) e a G
_P ,(" cue )-_~~,.~~~~~
i~
~ p 'i (cos ~)
Y- ~~
* lamed in formula (2.12)
2 e resaion G was exp
(see 9 par. , ~ nu
A A
ok we se ate P and Q into components
this bo ) p~
?, -1 ~~ .,
r ~ ~' C _-
p- = ~
V~i~, .,_
~{ '
) ~~ :gyp , (cos 8) dY,
ti ~1
(33)
of
and. Q" are expressed b'Y such integrals in which
and the components Q
nu 'is substituted Psi (nu}.~
Phi ( ) b9'
('r) Cj vG~ (~'~,
a
85
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to a].s aro based on the following
Further transformations of in gr
aracteriatics of aubintegral functions.
ch
function at Re nu 0~ having the
1) G* ie a holomorphic
nu
e ne ative art of the material axis;
poles on th g p
zeta ~ are integral functions nu,
2} Zeta ~ ~ l
nu_ 1
2 2
t havin zeros in the fourth quadrant;
no B
functions of nu;
and air are integral
3) Pai ~ 1 p. nu,..+, 1
. rru ... 2
' n Phi nu) and Pei (nu) am
C e entl the fiulctia s {
4} ons qu Y,
" ~ the first.
the~fcurth uadrant and meromorphic in ,
holomorphic in q
~ ~ e c n'tour C can be eub-
4lhen stud in the into~.ala P ,and Q th o
y g
r materials real) axis in
ch inter.,ects tha {
atituted with contour C whi
+r
bottom see Fif. E). On this contour the basic
the int na from top to { 6
~ o
toaratian r. gars to~ba th?,vicinity of tho point
part of in n pP~
nu .~ a].paa
o -
where at condition
ail
..
one can a 1 the a totic a cessions com 9
Pp Y ~p xA ( I~ Pte'
~ _ r ~P) = V Mrn u' (rir~ -- Ott J),
y 2
- ~ . u~ nrt - nry),
V nf~l~
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variable nu b~ the relation
le ? t is connected with the
and the variab
ae followa
~_ x _~ `_ Alt
~ _~ at'= 2 ~ `~l~~rr1%\
`'' ~~~ to -ul --rrry) ,
~ ~~ ~~~
r _t r.
l (~)-kr~
e hits oriented at such diatancoa from
If we are interested in th po
which are small in comparison with the
the surface of the sphere r s ay
thon in forn,ulaa (!+1) it may be written a6
radius of the spheY~e
? ? m-l
emitterl~)
as th? rimaiy radiators
In tho case of surface. antenn , p
' on can be
' e the surface and the latter conditi
ara usually sztuatsd n ~r ,
considered as fulfilled ?
' * is a ied the asymptotic representation (see
and the function G pPl >'
nu
then P~-and Q~ wil]. be represented'ea followa
and zi designates
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. ~ rote ela (~) ~ are
bona V and V ~~detexminable by Br
The~f~nc 2'
l
f the universal fl~nction P
exce tional cases) o ~,
artial cages ( P ,
P
ee b formula (1~~22) ~lY, '
xi' introduced by V.~A. Fok s
( ~ Ys 4)
o cation w t) - ~ and t~ are the roots
the roots of the eq . ( - ., s
where t are
a
. ~ .t - 0 see ? 9 Par. 7).
of the a cation w ()
. 4
? elds
formulas ive tentiela end fi ~n
Under condition (35) ~heee g 1~
' these should be supplemented in such a manner as
the~zane of semi-shade and
' m~ the
r fields in ~ given point separated fro
to obtain expressions fo ~ .
ce which is small in comparison to the
surface of the sphere by a distan ,
radius of the sphere.' '
' ~ ~ moat an-
ti ate the field in a deep shadow it is
. In order to roves g
~,a 32 where the contour. C was aub-
-t be in with the integr ( ) ~ l
venient_ . o g . ,
,., ~ _
e tion The lea of
in in o site dir c ). Po
tituted with contour C (rune ,g ppo
s
ctiona
coons Phi nu) and Psi (nu) are zeros of fun
he aubinte al fun
t 6'r
' STAT
al ha which we designate.by
~ d zeta ( P )
eta (alpha} an ~ . 1 2
z ~
nu 1/2
~ alculated by formulas
? ctivel . These zeros can be c
nu and nu ~ respe Y
s s .
this ine uality~ the functions
i i of the poles satisfying q
In the vic n ty
b formulas (Ltl). Since one can
d Psi rnl) can be calculated y
Phi. nu) an (
(
dre.unctions near such poles
as totic formula for the Isgen
apply the y~
r.-b ~
P i -cosf- .P _! cos (r.-6)~= -Jo(~~(,.- 11~?
~_ _ ( ~ ~ ( sin d
1
= i then at an additional
table all the way up to theta p ~
which is sui
condition,
~ and Q
orm of deduction series for P
obtain expressions in the f
we
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At a condition
( )
tituted b its own asymptotic expression
the Bessel function can be subs Y
will be brought to a linear comtiina-
hich each of the series (~2)
after w
i.e. to functions V and V , namely,
tion of series of the type (47 ), ~ 1 ~ 2
where
.However if we fulfill the condition ,
;; - a < 1, ~ (5i) ?
M( )< .
then we can write
'ons be and the sign of summation,
sustain these functi Y
after which
and ~ ac uire the form of
for ~ q
the ex ressions (~2)
p
_~~ s T=a~
kp= .. z 2 ~/;. c sln as
~~
~ r. 1
X Jo ('~i (~~ ~ ))
r.. 3
~ - -~ ~ ~~ rib
-. - _ ___. 2 V ;, e. X11- n ti i,
kQ. fa1 sl
R
' ~f i ~
. :. ~~(,(
2 it was assumed that conditions
' ation of formulas (~ )
During the deriv
(k9) ana (51) are
ntial material) members of the
' ed for all esse {
fulfill
ctuall take place because in the
ies obtained. These conditions a Y
ser
or which t ~' 1)
of the first members (f s
series obtained several
the boundary with
and the number M is high. Only on
are usually essential,
- V xi and V {xi,y) be-
' s for functions ( ,Y) 2 ,
the illuminated zone do the serie l
But in this case, it becomes necessary
gin showing a poor convergence. ,
then formulas (1~3) again
ral re~ resentations (!t!t), and
to utilize the integ P ,
sions ~~)? ~ .
us _to the expres (
lead
' ha and NI of our diffraction problem
The-basic greater parameters alp
' d'~tions (~3) and
e relation (3$); consequently, the con i ,
are connected by th ,
. ~ t formulas`'(~~) and;(~9) have a
o-not contradict each other so tha ?
(57) d
'? - ,
?in their~cor~ ?
Thus formulas (~~) and (59)?.,
common zone of applicability. ,
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sense'of formulas (~2) to (~9} consists in the facts
The physical
ade the ro a ation (with attenuation} of complex
that in the zone of,the sh p p g
take lace alon the meridians in a direction running from the geo-
waves p g
of the shade See Fig. F p. 80)
metric boundary ( ~
rsinfl=a,
to the axis
'di ~ with P and
These are the first components of forrmil.as (~~) coinci n,
a 61 be-
o di to formula 3 , The complex, waves on the axis ( )
Q act x ng (~ )
otentials and fields near
cons focused and in order to calculate the p
the 'axis theta = pi and on the axis itself it is necessary to apply
formulas ~9 . After focusing the taaves diverge again propagating
( ) .
cation aloe the meridians toward the boundary of shade~(60).
(with atten ) g ,
' et'on tortes nd to the second components
The waves propagating in 'this dire i po
in formulas (5~).
Thus the break down of otentials according to formula (33), come-
, p ~
s onds to the separation (fornation) of afield of direct waves which arrived
P
at trio iven point of the shaded zone from its geometric boundary by the
g
shortest way over the meridian. This field is determined by components
~ ~ u '-
P and Q . The components P and Q offer afield of waves which passed
through the polar axis (bl), i.e., passed at a given point over the very
same meridian but from the opposite dirrection; since these waves have
covered a much longer routes they normally appear weak in comparison-with
the direct waves. Only in the vicinity~of axis (61), i.e,, at theta = pi;
..
? ^~' ~- ~ ~ o a d the same order,
do we Have xi ... xil ,...- M , and both waves are of ne n .
Near the geometric boundary of shade, at theta.M'r 2 ;~,~the`~~field of,
a field of -" olar":waves .~:,
direct waves is considerably stronger th n the p
? n n ,,, ,
di of P and. t e co onerits in'the~theory
This justifies the disregar ng Q yp. ~?
a e o the earth~~where;,the .~ ~:,
of diffraction of radio waves around the surf c, ,f ~ .. ...
Fig. F. ~ STAT
94
95
STAT:
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meter a].lia ac wires unusually high values and the investigation is,
paz'a P ~? ..
' ~ of he semi-shade In our own
therefore, limited usually by the zope t
co onents'of the P and Q type cannot be disregarded;. these '
case the mp
o ex structure
co onents ~ive recisely (see, following paragraph) that c mpl
mp g p
.of the radiation characteristics which was already mentioned before.
' t' ate the fields in the illuminated zone. First
ti1e will now. a.nves ig
u n
culate. P and Q b deductions in the very same points
of all, we will cal Y
r~~ _ rc l1' _ 2 N
~~ ~~II II
rot
and weesubstitute the Lengendre function with its asymptotic expression
At the condition
as > 1 (64)
If we,have a fulfilled condition
then P" and Q" can be calculated by formulea
Forjl~ulas o5 and 6$ ineludin the entire illuminated?zone show
( ) ~ )~ g ~
t ~ b' 1 "n ?"
that the adds ions/ field o tamable from potentials P ,and Q consists
of two complex waves which completed (in opposite directions) "an~ele~ent
? ~. .
around the world" ourne ?and harts fallen wore cau~~ht in the ven int
~ Y ( g ) ~ Po
f the irradi to a hav ed verLth e~
o a d,spac , mg pass o e m ridian through the axis
of the shade~(El). Oo the "ads of the light"
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radians bt+como focused (formula
all such waves traveling over tho ne
' rod moxe than one half of the cis-
66 ). Since these waves have cove
( ) ~ ,
a small. correc-
rienced a4 strong weakening and give
cumference they exps
~ i
the fields obtainable from P and Q .
tion to
entiale P~ and Q~ produce the basin,
In the illuminated zono the ~ pot ,
in of a lane wave on an ideally
" ct" field on mating during the fall g p
dire g
. 60
the shade boundary { )
lectin a ere. At a certain distance from ,,
ref g ph
erminable P~ and Q', should transform
e eleotroma etic field, det by
th ~ ,
the law of ometric optics. Near the
into a field determinable by ~
t al P~ and Q~ aro
d in the aemi?shade zone, the poten i
shade boon ary,
It can be shown that, during the departure
termined formulas {43).
de b'Y
the illuminated zone, the diffraotion formu~-
from the shade boundary into
- com~ert into negative formulas (formulas
lee derived for the semi shade
veati ted Y. A. Fok (6)
tics This conversion was in ga by
of eometrio op ).
g
neral form /rat ue will not analyze it bore.
inage
.as for. Radiation Characteristics
Values of the Parer
,at Greater ,
us'in the ceding'paragraph allow the radian
The' formulae listed by P~ ,
,~
be calculated at
haracteriatica of~a apherical,aurface antenna to
tion c
ten-
uea Darin the differentiation of the P and Q po
greater alpha val ,g
al it'is necesa. to retain otily the main members. .
ti a , ~'
or cam x characteris-
write first the a totic expressions f P~
.We will
'(1) (2') i e
eta W theta) and W (theta) in the shade zones .~
tics W (th )~
at, ~Pi theta pi.
2
(1) (2) ~
Dor jd and 41 tits hove nioro cor~p.~e~: formulas
^ 1~~ O ( ) a sin a
t~yi~3n3 tP.e auxilif ftwcti ~~ U{1) (2) U
ny o a , U and era detormined by ex
p~assious
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an3 the fui:~*?~cr f(xi) is equal. ~ ?
? )`r ~(-)
~~ ,
At r~aa~:or osititro d values the function f (xi) tonds toward
~ k
~c;ro a~uch faster then the xi) functicn. This ~uatifies the inclusion,
g(
in the ri ht side of tho first of formulas (72) of the component -
W (theta): even though at xi r'~/ 1 this component
alpha sin theta
by its order of magnitude in times smaller than tho first ones buts
they graduallir become equal farther in tho shade and in the vicinity
of the "dark pole" theta : 0, the second component, as is shown by cal-
culations, appears to be the main one. In contrast to thin, the component
1 U (theta) is of no matorial importance during the
alpha l+i sin theta
(2) F
calculation of W functions and is writton out for symmetry. or-
mulas (72) lead to a relation (comparo formulas (13):
wll l (~1 . _? ~~/Ill (r).
Here, in accordance with formulas (33).
F r=W" -W and W it is possible to writes with the aid :of
o , .
e followin a roximate a reesiona:
formiil.as (65), th . g pp xP
i+ covY i? Jc - L'
c t
Y ~iu.H '
flr. _ Y
iR cue Y ? !?
,M Yst~~ r
"?,IR' Y
t. ~~~~, * ,,, -
1v, c
Ysinii~
Ye e t correction to the basic terms' but oal~;at,the~:.vioiaity
which of r sligh , _ _
bade bound Accordin to formulas (65) the eup~eesiomti~iii`(T9)
of the a ary. g
ill also include cam vents o rtional to f (xi=) and g (~d )~~'but" `
w Po Pi' I~ 2 2
For the basic components in conformity with ?the'Fok;formulaa
Y ,
' 6 to 28 ? it is ssib]a; to write` `
above Paragraph 3, foimulaa (2) ~ ) po_ ,...
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.~. ~ f rmulas (79) ~ (~)
Near the a uator 'i. e; ~ at theta ,~,~. ,,,p,~, o
4 ~ 2
' e radiation~characteriatics in the shade
ive a smooth congruence with th
B
I t e illuminated zones at a dis-
exam le with formula (70). n h
zone, for p ~
~ ~c uirea star negative values
tance from the equators the variable xi a q gre
and G we use the approximate formulas
and for functions F
rmulaa $0 automatically produce a contrersion
Conse uent fo ( )
Q lY~
ormulas 25a We want to call attention to
osietric o ica f ( )
toward ge pt
e the variable xi' (81) end not the
the fact that formulas (80) includ
suss for this is that the "direct" field
..;pariable,xi ,(45). The physical c
c ant wave
e is created directly by the fn ~,d plane
in the, illuminated zan
te.~ ~ ,
onal to cos theta (or z), whereas in the
the phase of:,ti+hich,ia proporti
:{.
sated
case of round-the-world waves'in the irred
shade ,zone,. (~~uet.: as in the
end
tion takes lace over the arc of the meridian
hemisphere) the propaga P
variables 45) (5b) and (~) are pro
in dimena~.onleas ( ~
the correspond g
,;t; .
. ~ _
portions/ to'the angles. ~ ,.
' e t k into con- ?
e to the feat that id the previous paragraph w o0
Du
hich ased~throu~i the "dark" pole theta = pi
sideration the waves w pa
,.
as of this agraph they have corresponding componen s
in~the formal Pte'
of ~ ~~
aired the aeibilit
rtional to f xi) or g (xil) ~ we obt po y
propo (. 1
ation charaateris-
?e ualitative a oration of the different rods _
offering q XPl:
~~ ~~ ation in the shade
c vexed in a aph 3. Namely the oscill a ..
tics dia o p~' ~';
.
e ed
me am ified during approach to the dark poles are xplain
.zone which beco pl
ro stir (with attenuation) in oppo-
by the interference. of two waves p pag, g
e e The neral
directions.and having equal amplitudes on th dark pot . ge
side
102
STAT
increase in~am litude near the direction theta - pi is caused by focusing
P -
the watres propagating over the meridians. The increase in the number of
oscillations with the increase in the parameter alpha takes place simply
because the waves propagate in first appro;dmatioa with the speed of light
and in-~this way the meridians pile up a greater number of standing waves.
To~what extent do the formulas which aro given a]low the radiation
characteristics of spherical antennas to be calculated at greater values
of the parameter alpha? In order to answer this question we compare
(in Figure 2$ to 30) the radiation characteristics for alpha = 10 ac-
cording to accurate formulas of paragraph 1 and 2 (these curves are
asymptotic formulas for the value alpha = ZO give a perfect represento-
tion of the nature of the functions interesting use Even from a quanti-
marked by the letter T) and by the asymptotic formulas quoted in this
e marked the letter A In Pi 28 to
paragraph (these curves ar by ). gore 30
1 (2)
( )
the functions W (theta)s W (theta) and W (theta) show that the
tative vie int the a totic'formulaa offer a satisfacto results
~ ~P rY
the divergence between accurate and asymptotic curves do not exceed
15-2 - .
~._
. (1)
The dotted line in Figure 29 also indicates the function U (theta)
(1 .
1 )
()
i. e.~ U (theta) and arc U (theta). A]1 calculations by asymptotic
e u1; with the aid off xi and xi func i
formulas were carri d o ( ) g ( ) ,ton
tables and Beasel function tables of the com ex variable 11
,i ~'..
Using Figures 28 to 30 as a basis we can. freely adapt the asymptotic
formulas for the calculation ofradiation characteristics of spherical
103
STAT
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d the a otia for~ulas~
e. error roduce by sYmPt
antennas at alpha 10,, Th P ,
e s e ua1 to ands consequentlY- upon an
air order of ~agnitud i q .~.
in th
Mz
decreases cite rapidly.
increase in alpha it q
teriatics W (theta) far
Fi a b to 9 show the radiation charac
In Fi ea 15 to 16 the radiation charac-
alpha = 15; 25, 50 and 100. g~
1) to 27 the radiation characteris-
( theta and in Fi es 24
teriatica W ( ), ~
(2) ~ ve Same 81 valuea?
theta are shown for the rY P~
tics W ( ~
the al a ameter wee already clearly
At values as large as these of ph 1~
characteristics at
that a char takes plane in the radiation
see ge
I
? version from light to shade becomes smooth
alpha ?-~ infinity. the con
t na and focusin take places is forcibly
and the zone, where oscilla io g
k le and the oscillations become smaller and
displaced toward the der po
more frequent.
' in these die ems are the "mean lines" of the
The broken curves ~'
acteristics. The mean lines represent the ab-
oscillat~B radiation char
.. l t 2 + '
() ()
a e of function W~ W and W , respectively
solute value and ph s ,
d hemis hero these functions are determir-ed'by
and iri`the illuminate p
,. ~ ed from ~3 -t
hero trio are deriv (4) YPe
formulas ($), ~In the dark hemisp y ,
fun tions a ar to ba the "basic" components in the
formulas. These c pI~ ,
' ~ iatian characteristics, and the "adds- '
as ptotic expressions for the red
ym
t f the graves which bypassed
ti:onal" components are derived as resul o
' ~ ant the
e here throw the dark pale. llithout this additional compon
th sp ~
tions are obvious/ not obtained but only a smooth moan line
oscilla , ~r
Thi mean line
indicatin the existence of focusing at theta.----~ pi. s
g
101
STAT
is presented by a dotted lira,
the radiation characteristics, plotted at conditions
The curves for ..
h u1d interlock at
(53) and (57), s o ,
pi -theta ^,,r 1
I~
e uith au totic forrm~l.as, the short
On the graphs, plotted in accordt:`nc "ymp
b ben curvo ive a ccnitinuation of the curves beyond
sections of the re g
a Ticabilit? asd in this w'aY~ showy the degree of
the limits of their pp ,~ a
the intorlocking curve,
era a 6 Resorsa ,ces in a Spherical Antenna. Radiation
P gr ph
_ Im~nendanco --
discussed above show the relation between the field of
The graphs
r' al surface antenna and the angle theta at fixed
radiation of a sphe is
al? ha It is also 'interesting to explain the
values of the parameter P ,
t n bet~-reon the ammeter alpha, which is proportional to the operar
rely io p
tin fre ueue and the radiation in a givon direction.
g Q Ys
ante of the absolute value of the function W upon the
Thg depend
TAY e l Acebrding to formulae. (1.ti.) this
alpha parameter is shoran in gur 3 . .
vas the radiation field of a spherical antenna excitable by
function gi .
' Thi de ndence is
the radial electric dipole situated on its surface, s pe
d uri11 the fixin of values of the angle theta, which are equal
depicts d g g .
o- 0 0 0 0 ~,ha 10"
0 whereb for the values 0 p
to 30 , 60 , 90 , 1~ , 150 , Y
alculations were carried oat by the accurate formulas of paragraph 1
the c
and 2 and b the asymptotic fornuJ.as of paragraph 5, for the values
~' ,
ethod'
difference in the calculation m . s
l0 alpha 15, Because of the .
?~
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at al ha , l0 indicating the degree of
the curves show a ceztain dump p
a roximatiou offered by the asymptotic formulas.
PP
e iven value theta we
If we should examine the curve for any on g
? , an ~ iu
find that it has the nature of a resonance cui~re - daring
will
are re laced minima rnd (aa it
crease in al ha the resonance maxima p b9'
P
cted?from the sits/ considerations) the oacilla-
ehould have been exile P~F
tion aduell become smoother and the resonance characteristics taper
s gr y
down to nothin , However by making a comparison between the curves
g ,
for various values of the angle theta we see that these resonance
characteristics virtually are almost entirely illusory because the maxima
~ minima for different ang].ea theta are oriented at different values
al The exce tion is constituted only by the first maximum oriented
I~ P
on all curves in approximately one and the same point, at alpha values
of about alpha : 1. If this maximum should not be taken into considers.
tion then it must be acknowledged that the curves in figure 31 do not
indicate the resonance characteristics but the presence_of interference.
? This interference can be easily comprehended physically provided
idered as a receivin antenna on which a ane wave is
the sphere is cons g ~
The?fluetuations in the radiation am tude?characteristica
falling. P~
al a - coast. theta.changea) and in the curves in figure 31 (theta
( .Ph ?
} the same cause i e inter`
cons~,~ alpha changes) are due to one and ~ ?~
++ ++ se the
ference of the direct diffraction wave and the wave which bypas d
.sphere through the dark,.pole, The nodes and sntinodes of the "aemi-
atanding" wave are displaced toward the pole theta - pi' and the curves
in 31 are formed when alpha increases,
lob
STAT
For a final a lanation of the roblem regarding resonances''it is
xP P
necessary-to calculate the total power P radiated by the apherical'an-
tenna or still better the ratio
where P is tho total power radiated by?tho primary radiator in the ah=
o
Bence of the sphere, If the primary radiator is ari electric dipole then ?
is can be written
,where R is the radiation im dflnce of the di le in free s ace and R ie
o Ps Po P
its radiation impedance in the presence of a sphere. If the primary
radiator is a slot (as in figures B and C) then
where G is the radiation conductivity (bilateral) of the slots cut in
0
an infinite planes and G i,s its radiation conductivity on the sphere.
? ?
Tf .the sphere is e~tcited by the radial electric dipole, situated
on its?surface~ then
u 3 -
M1
-
+In~ D do
and by substituting the series (llb) instead of V (theta, elpha)~we
I' '
_~~_~ ..
I,.(~)I
n-1
107
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R
I ~'' (aJ ~~ slu U dll
~ae~
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- ~ G by formula
alas ~ )
. in calculating the v
su7.ts obtained ,
;:The xe one
e curve has only
We 580 that th
how~n in figure 32,
~S8), era s ~ e re-
e first one. ~
aximum~ namely th
1 e . rassed resonance m ~
clear y xA exions of
ed slight infl
li tl indicat b'Y
C8S aT8 only S gh y
lnaining I'eSOnan 0 in
32 we should ke p
the curve in figure
es ifhen evaluating
the cult' .? tiona of the
electric oscilla
ies of the natural ,
~d_that the frequenc
ues al a, equa]l,ing
to the val Ph
$g) correspond
see 10 , p? 1+
s bore
p
- 0.$6 - i 0.50
alpha/ -
1 $~, -10.70
alpha2 _ ?
_ 2.77-i0.$3
alpha3 -
talaa s ace approxi-
fi 3z , ~'
first resonance on g~
Tie fact is that the
at al ..~ l?~
are noticed 1~
$6 and the two following
ate/ at alpha : 0.
m y
2.77.
~~` t
di a oriented on i s
a magnetic pol
e a ere is excited by
When th Ph
he ma etic waves slab
di les , as well as t Bn
acs then the electric, po ~,
surf ~
re resented in the fo
the value can ~ p
excited. Therefore,
become
where
N
IKy1
1 ~~ are presented in
as weL1 as their sum
The function ,and,
W8e See that the oSCilla?
to the variable alpha. ,
fi re 33 in relation
~~ ~ t
slot de nd upan the ~STAT~n s
n conductivity of the Pe
tions in the radiatio
108
~2
?~' ,~ a 88 which has zeta! al hay
the aeries of which are as s rios ( ) ( p }
? .. n ~ ,
in the denoroinater, The resonance ~nxima at the slet?are oriented
approximately at a point where their presence appears at the electric
dipole (figure 32). In this way the resonances at;frequeneies
corresponding to the natural magnetic oscillations of the sphere do
not take place at all. This is explained by the fact that the natural
r
ma netic oscillations have ~atteriuation-that is too high (compare 10 ).
g
When al ha infinit the value I 'in both instances shows a
p -?-?3 Y
tendency toward 2. Physically, this can be easily understood since,
as is known, the radiation impedance of a vertical dipole on a surface
(which can bo considered as a boundary case of a sphere at alpha Q
infinity) is doubled in comparison with the impedance in free space
and the radiation conductivity of the slot should tend toward the
"unilateral" conductivity of the slot cut in the surface, i,e,, toward
1
2 Go.
Paragraph 7. Radiation Characteristics of a Spherical Antenna
Exc~.taJr~.?e~.~ Complex Primary Sources.
During a theoretical investigation oP the radiation ebara~ete~r-
istics of a s herical ,surface antenna we studied a sphere excitable
p
by simple primary radiators -elementary electric .or magnetic dipole.
Such an arrangement of the problem was due to the fact that
we were interested primarily in hoti'r the sphere affects?the radiation
field and this can be explained quite, well but only, if the':primary.?
field has a simple form;
109
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diators an entire series of
In the caee~of none1ementary primary ra ~
' ~ r not ea a result of diffraction
radiation characteristics may appea
It of the complex nature of the
on the sphere but icather as a reau
rimary irradiation.
p ,
re com lax primary radiator can
In spite of all that the anti p
nes and in this; ,ray
en down into a combination of ord~.nary o
be brok .
oharacteristia of a
ddition or integration) the radiation ,
(by a
determined, By an addition
herical surface antenna may also be
sp
? ow not only the amplitude but also
of this type it is necessary to lm
ations not only the
haracteriatic, From these consider
the phase c
? (1) (2)
i functions W, W and W are
absolute value but also the phase o
? d awin s for example, allow to find
even in Figure 1 to 30. These r g ~
6
a?s here excitable by a system of
the radiation characteristics of p
Figure G
often roven, however, to be direotly
Our results have p
re excitation, So not an elementary
'e plicable 'to real cases of sphe
p
? al ha values that
? elf-wave slot is, cut in the sphere then at p
butah ,
' ic' should not be
are not too'small the radiation characterist
noticeabl -affected by this change, It is exactly the aeme if the
Y
sphere is excited not by the elementary dipole but by a quarter-wave'
vibrator arranged on the sphere in a radial direction in such a
manner that, together with its reflection, it ;forms a half-nave
vibrator and the radiation characteristic of such a system should
not differ..much from the function W (theta),
Even though the latter assertions are almost apparent it would
be desirable to verify them quantitatively on any given example.
We shall disouss a case of a quarter-wave vibrator (see figure G)
elon which a current I r is distributed accordin to the law
g () g
Where Io is the current amplitude, and b is the radius-vector of the
final point of the vibrator (dipole), whereby
Acoording to formulas (11~} each element of the current I(r)dr of the
~ ~rkrr. ~3
fo--Ff - ( ) f~,
4 f ~
where, according to the $9rat of ?hp forlnt~las (9)', we have
V (d~ r)- ?
---- (2n 1)(-~)" - P ~ (cos ~). (94)
- : ~"' ,~
~~t ~a~ '
~~-t
Here criteria (12) 'and (31) are introduced and R? designates the
distance of the observation point from the center of the,~sphere,.~ ;
The ovc~rrall radiation field of~a quarter-wave ,vibrator is liven
b formula
Y - ~ ?
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e-~
~5j .
In the tees of nonelementgry primary radiators, an entire series of
radiation charactaristice may appear not as a result of diffraction
on the sphere but rather as a result of the tom lax natur of t
p e he
primary irradiation.
In site of all .that the e
p ntire ca~nplex primary radiator can
be broken down into a combination of ordinary ones and in this d
ay
(by addition or inte anon the red'
~' ) cation characteriatie of a
spherical surface antenna may also be determined, By an addition
of this type it is necessary to know not only the amplitude but also
the phase characteristic. Fran these considerations not onl the
Y
absolute value but also the base (1) (2)
p of functions W, W and W are
given in Figure 1 to 30. These drawings, for example, allow to find
the radiation characteristics of a sphere excitable by a a stem of
Y
slots.
f figure G
. Our results have often proven, however, to be direetl
Y
applicable to real cases of sphere excitation, So~not an elements
but a half-wave slot is,?cut.in the sphere then at alpha values
. that
are not too small the radiation characteristic should not be
be desirable to verify them quantitatively on any given example.
We shall disouas a case of a quarter-wave vibrator (see figure G)
along which a current I (r) is distributed according to the law
1(r)=lo cosk(r-n) (n;~u a~r~b), (91)
Where Io is the current amplitude, and b is the radius-vector of the
final point of the vibrator (dipole), whereby
(9l)
Aeaording to formulas (1!}) each element of the current I(r)dr oP the.
radial vibrator gives a radiation field
~~.. ,rkir. ~ ,
-_' r ` l' 0
4 f ~ n .
,(~3)
.where, according to the first of the for~nulaa (9),"we have
P~ Kn{a) "? ~ .
!i - t-
Here criteria (12) and (31) are introduced and R designates tae
0
- (2n -}-1)(-~) P~ ~ (cos 8) ~94
( )
The'ovESall radiation field of ~a, quarter-wave,, vibrator:~is: given
by formula
noticeably affected by this change, It is exactly the same if the
sphere is excited not b th _
y e elementary dipole but by a?gaarter wave
vibrator ar""ran ed on the s~here i
g p n a radial direction in such a
manner that; together with its reflection, it :forms a,half-wave .
.~
vibrator and the radiatibn?characteristic of such a a stem sho d
Y ul
not differ much from the function W
(theta).
Evan^though the latter assertions are almost a arent it would
Pp
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'Substituting the series.(94) in this expression and applying formulas
of the type
In conformity with paragraph 2 it is natural to introduce a
,function
and the radiation field of a quarter-wave dipole can then be presented
whore R~desi~ates the diatance?from the vibrator (dipole) base.
int r - a theta - o to the Dint of ,observation
(1~ - ~ - ) P
by formulas (97) to 199) at alpha - 5. For the purpose of oomparieon-
we lotted a dotted line which also re resents the funetion~w, theta
p p ( )
for the-elementary dipole (broken curves repeat Figure 3)~ Aa is
.. ?
evident, the radiation characteristics W (theta)~and~ W (theta) differ
from each other Daly slightly. The arrangement of maxima and minims '
in these characteristics is perfectly analogous. '~
If?we should turn to the directivity diagrams, with which one ,
usual/ confines himself Burin experimental'end~theoreti~cal studies
Y 8
of antennas, i.e., to reduce both amplitude characteristics to .a
? "r .~.
eoamaon maximum then the difference between both cases?will be,.elmost
completely obliterated (eliminated). ~ ? ; ?
During an increase in the parameter alpha, the differanee~
between both cases should also decrease because the radiation character-
istie of~a uarter' wave di ole on a s here in_the illuminated semi-e ce
q P p 1~
~0 theta i should a roach that characteristic which it has in
< pp ~ -~
~ :.; ,
' ~
`
,
from the
e a ter~~i 1 sli htl different
emptiness (vacuum), Th 1 t s oa y g y_ .,
f el nta di' le, ~In the zone of shade???and.' .
characteriatio o, an ems ry po ,
'ar ris is sh ?ld~~oonvert into function W.,theta
.semi-shade the ch sate t ou
because the "dimensionless height" (39)~of a quarter-wave';_d:
, ~ y i ,~
.,~
d W rd zero Burin ?an.increase ~in_~al" ha ~ ~ , ..;V? r:? :: ~`? >~
ten s to. a _ 8. ,. A ,. ,. . .
values of the alpha parameter not~~only.,this~~vibrator~:but>,an~entire~``oeries
of composite radiators are similar to elementary dipoles .when>~t}ie?
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? ,
radiation characteristics are in the semi-shade and shade zones
where radiation is determined by diffraction on the sphere), In free
(
~epace, as well as in the illuminated zone above the sphera, these
characteristics different from
composite radiators may have radiation .
dipole radiators.
The explanation on the origin of, the lobes and other features
of the radiation characteristics would be more 'realistic if the
d
which a lane wave is
' sphere is considered as a receiving antenna on p
~~
The total results obtained allow a canplete quantitative
~re resentation about the radiations of a apherioal surface antenna to
P
be prepared.
ideal/ refleetin s herical surface may
The presence of an y g p
ti .. ':Se,
characteriatio of a di ole or slot
bring a radical ahange into the P
which .excite the surface antenna, If the radius of the sphera is
compared with the wavelength then its radiation characteristic has
,t ~- ~ -
lobea the number oP which increases with the reduction in wavelength.
elfin The lobes can then be interpreted as the interference of
f g.
' i d b" the incident wave with a
a direct diffraction field exc to y
., ~ ,
"round-the-world" diffraction wave which by-passed through the
.:"dark' ole" being a point on the sphere oriented at a distance from its
:p -
d art At su ficientl hi h values of the ratio al ha the
.illuminate p ~ y g ~,r
' ' - lambda
on?characteriatic~has minute oscilla~tions~onl in a small section
~,~sediat,i ;y
` c nn in the remaini
.,(,theta pi ). and aciuires a simple fo ng part.
A spherical antenna has almost no resonance characteristics.
Exceptions aro constituted onl b values al ha 1 where resonance
Y Y p
occurs near the natural frequency of the basic electric oscillation,
The radiation characteristics of ~a spherical antenna excited
by composite primary sources pertain in their mayor part to the
characteristics of a sphere excited by elementary dipoles, ,
The phenomena mentioned above should take place also in other
more complex (c~pounda) surface antennas. In view of the.fact that
these phenomena can, to a large extent, change the directivit die am
Y 8'r
(radiation pattern), they must be taken into consideration durin
g
practical utilization of antennas operating in conditions of electro;_
dynamic interaction with massive conductive bodies situated nearby. ??
1. Frank, F, and Mizes, R,: Differential and Integral Equations of
Mathematical Physics, Moscow-Leningrad, CNTI,. 19;7
2, Born, M.: Optics, Kh - >I, ONTT~, paragraph 70', 1937
3, Fok, Y. A,: Journal of Experimental and Theoretical Physics`
'
1949; Yol. 19, Edition 10, p. 916
4. Tables of Associated Legendre Functions, N.Y., Columbia University
Press, 1945
Tables of Spherical Hassel Functions; N.X.,,~Columbia University
Press, Yolume I-II, 1947
6 Fk~V A: Pr ss fPh is S `
. o , ogre o ys al ciences ~ 1 8~ ~ Vol - 6 08
. ~ .. - , . ~ 94 ,. . 3 ~ .p. 3
7, Fok,.V. A.: Journal of Experimental and Theoretical Physics,
1945 Yol. 15,. Edition 12, p. 693 ~ - - ? - '
8 Vvedenski B A : Princi les of`the T ~
y, ,p henry of Radio Vave
Propagation, Moscow-Leningrad,' ON'TI, 1931}~~
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 0?
.#
7 d0 99 !10 ~!
0
td
STAT
Picture ~.
? 116
ip
90?
60 9Q !?D? vo? rso ~
-,
~
w
11
1
l
90'
.
W
d)
(
cu
d=1
d
Z
d~3
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9?
? ~ ~ and the Surface of
ction of~Radio naves Aro Sciences,
Fok lf. A.; Diffra n of the Academy of
' ocr-Leningrad, Publieatio
the Earth, Y4os c
USSR 194b
,
?~~oecow?Leningrad,
? Thaor of Electroma~ettam,
Stratton, J. A.. y
lo. dat 194a
OGIZ, Goatekhiz ,
? and J (z) for Complex
.~ Functions 7o(z) 1
Tables of the ll.asel ega 1943
11' ~ C lumbia University ~ ?
? Arglunents, N.Y., o
Picture 4 ~ ?
Picture 9
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STAT
1~*2
R.4DIATIOI~i CHARACTERISTICS OF AN ELOi!iGATID ROTARY ELI,IPSOTA
by ,
1?S; G. DeLcin
Pa.r:'l; ~1ave Functions in Spheroidal Coordinates.
this re ort the roblem of diffraction on an elongated
Ire p ~ P
- onductive s heroid excited by an electric dipole situated
ideally c p
irected alon this axis
in an iven point on the spheroid axis and d g ~
Yg
is discussed.
roblem is solved in a spheroidal system of coordinates
The p
xi eta and hi connected tirith the Descartes coordinates x~ y, z~
~ p
by the ratios
~ (
- c 1( i) ~
y
i
z-= fi;~
tamable rotating the elliptical system of coordinates
and ob by .
the eater axis of 'the ellipse family. The areas in this
about gr
id elori ated
~ _ const are elongated sphero s ( g
system of coordinates _
' ate xi than es from 1 to infinity
rotary ellipsoids). The coordin B ,
'' dinate eta than es from -1 to ~1.~ The sign 2f desig-
and the..coor g _,
tes~the distance between the foci of the spheroid family -
na
_ 'tel thin rod
const= In particular, when :ti - 1, we obtain an infini y .
the heroid xi - const the valuo eta - 0
with:a length of 2f. On sp _ _ _
s nds to the e uator of the spheroid '(its intersection`~t~rith the
carre. po q a ... .,. ,
_~ a oriented on the
lane z`= 0) and the values eta _ 1 are its pol s
P ~
143.
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axis z when z - ~ fxi. The metric coefficients of this system have
~, ~ the form of
If one would look in the spheroidal coordinates for the .function
at ri the wave a cation
Pi (Russian letter P) s isfyi g q
? ~ 1.3
SCI k n-0 ( ~
R = R ~ S cos map
() (~)
stn rrt~
then for R(xi) and 5(xi) we would obtain equations
where the parameter
is proportional to the ratio in the distance between the foci of the
ellipsoid family to the wave length,
olutions of these r blem w re in a t' t e
S p o s e v s iga ed in th report (1).
lde shall write out here these results of this report which wi11'be
necessary to us in further investigations.
The angular functions?5(eta) appear to be the functions
the associated Legendre functions, determined in such a manner that if
Pm COS ~} = Slll"'
n
I
d"`P" (cos d)
wYtere P (cos.') is the Le endre 1 omial The coefficients
n g Fo yn
ml
d ~ (C) are determined from the trinomial recurrent ratio
n
c~rhich bind m,l 'th the ' d' ?
s do wi in ices n of one and the same evenness.
m,l
Consequently the coefficients do differ?from zero only when the
indices n and 1 have one and the~same evenness and in this connection
a rime ~t.the si re resentin ''the sum in formula 1
P ~ ~ P g ( .7), and indicate
eve here below that the summation is carried out
.~ in accordance
with indices n a 0 of the very, same evenness as 1, The values
Cm are fixed by formulas
n .
m (n -~- 2m) (n 4- 2m - I )
A = ~, -
n (2n -}- lm -F-1) (2n -~- 2m --~ I)
m,i _ 2ni -j- 2'n (2m -{- I) -~- 2m - I
B -}- .
" (2n -f- 2m -}- 3} (2n'-f- 2n: -1)
(n ~- 2) (n -f- ~ 1
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r'or future reference it is convenient to introduce a designation
rn ,
and b 1s connected with the constant A included in equation (1,5)
1
by the ratio
b'".--- Am -m m 1 . (1,13)
1
l
m,l ~ d ? ueh a wa that
The coefficients d are standardize sn s y
n
)
~ ) ~) ~
m
l ! ~m
,
d 1) d m
S~ c = -- P
~~ m,l dq 1 ~ m
~-0 ?~-0
1.1~)
(1)
At a fixed value, m, the angular functions.5 (C, eta) foam
m,l
:i.n the interval (-1,1) an orthogonal system with a norm
l - ~ieuelriioe),
The radial fwictions R(xi) of the wave equation are fixed by terms
in the form of
also be presented
~~.22)
(1.23)
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The double rime at the sign of the sum in formul.as'(1.22)~ (1.23)
p
'indicates that the summation is carried out from n ~ -infinity to
n = infinit in accordance with indices n of the very same evenness,
Y
ws 1~ .and the ?eoefficients n~ and have the form of
1 ; ~n,1
(!?~teTxoe)
The associated Legendre functions at ~a. ~ l are detertni~d in
?accordance with Hobson (2).
We wish to?mention that dur9~tg the calculation~~the xepresentation
l ~ ~ ~ 2
() ()
H is convenient to use in the form of (1.19)' and R - in the,form
m,l m,l
of. (1.23). ?It is immediately evident from these terms that ~~'
~. ~n 1
? (2 ~
(c, xi) does~not have ,any characteristacs whexeas (.r xi) has
Htn 1
The decomposition of the plane wave
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1 l -1 - (n + 2rrr)l (I?~teTxoe)
?22rn I?t rrt ' ~ m 1 dnt,!
?I- 2 2 -~- ? nl .
.~
1-t
~cin?1.1 drr,,l 2st l~ I '--^- ,
~ ( ~'?) 2
(~?tteeerxoe
3 ! (n + 2m)1 )
~- rn ! d'n'1
l l + ,t n
nr ,n,l
~c do ~2~! ~ !
,~~he radiation condition on infinity is satisfied by the function
trhich-at cxi > infinity has an asymptotic representation
-i omega t
t?1e select the time de ndence in tho form of e )? .Fina]1y~
,: (1) (3)
the Vronskiy determinant for rad~.al ftiu~tctions R (c~ xi) and
Rm 1
m,l ~
(c, 'xi) equals
Par. 2. Decomposition of a Plane and Spherical Wave Accordintr~to
Spheroidal Functions.'
Next we shall search for diffraction fields and theix potentials
ilt the form of decompositions (brealulot~ns) according to spheroidal wave
functions. For this reason we shall first analyze the primacy fields
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should be sought in the form of
because the function (2,l) does not de nd u n and ha
I~ I~ s no
characteristic at xi - 1, Utilizing the ortho onalit of the an
- g Y gul.ar
(see for example (3) a 8 5 we will
~ P ~ 9 ), obtain
During the decomposition (b~ealcdot~n of a s heric
p al wave
orif;inatirig from point xi - eta ? l z -
x~
~ _ (
necess to
ary distinguish,tt~o eases,
~h~uld have the form of
because of the fact that at xi --~' inf3nitY the condition of
'ati n into infinit should be fulfi]led. At xi xi the
radY o y 1
ition breakdown should be sought in the foam of
decompos ( )
because it sho~ild'be right at xi - 1~ where the spherical wave
2 5 has no characteristics. It is necessary to assume in both
(.)
2 does not de nd u
cases that m 0 because the term (.5) 1~ I~
i
In order to determine the coefficients D ~ we will assume that
l
1
the values xi and cxi in.the ratio (2.6) tend toward infinity. Then
h e R n/' ~ are s erical, and r~ ~ z are cylindrical '
w er , 1~ ~ ~~
~ ~
the xatio 2~ 7
b ervation int. B converting ( . ~
coordinates of the o a po Y
cos nT ~ we will obtain
o,~ - cU
(t of
~ No,t
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Thus for xi ~ xi ,the decom sition of the s her'
l Po p ices/ wave has
the form of
? oo I1 j
eikR 50,1(1', 1) 1 3 . i '
2ik RO c ~ R`) c ~ S') c `l.l
(~t) (,) .(r~) 1
o,t of of (
R No,t ~ ,
t-o .
As we turn to the'determ5nation of ~~
the coefficients D ( )~ ,
1 ~'`].
we wish to mention that at xi -~ and eta
1 the s herical,wave
- p
(2.5) does not have a characteristic and the t
ern (2.7) should coin-
cide with (2;]2), This condition a entl i
pPaY' y, s fulfilled provided
we write
St1t (c 1
? 0, t ~ ) (3)
D (fit)= 2rk R c, ~ ,
t o ( t)
. No,t .t
i. e,, if for xi
(2.13)
ikR~' ~ Il)
e So,t (~~ ~) (~) ~ 1
-, =2ik R c c Rc) c ~ St) ,
ot(~t) (,) c, ). 2,14
R N o f o ('~ ( )
o,t .t
t=o
For the
?, . purpose of verification we shall calculate also the
? ~.
fluctuation of the derivative in xi from the functi ~'
on e
R~
during the passing through the surface of the li
. sp eroid xi -.
A l in the r ?
pp y g sties (1,2$) we wi11 obtain
d eikR-
E R,
d ~ikR,
2
d
~~ _ ~ac R, _, 2 '~ (~)
2.15
E
( )
152
STAT
00 (1)
Cl S~ t (c, 1) (1)
S c, = 2S ~ --1),
( ~)
a,t
No,t
because at any given value 1
t ~ '
lt) f1)
(~) (~ ~) ~
n nt
.t ?
1
' 'e t in the com lets ortho oval
and such decomposition eoeffic~,n s ,p g
(1) de ta-function at
system S (c, eta) are contained in the double 1
0,1 _
the point where eta?~ 1. .
ikR~
.Thus d e ,and it indicates that also the normal derives
tiye, differing from the xi derivative by the multiple
( i t2 - r
when passing through the slarface of the spheroid xi - ~3 is con-
- l
tinuous everywhere e:tcept for point eta ~ 1~ where the normal'deriva-
tive eh~ieriences a drop Navin the nature of a-delta-function, a fact
which was_to be expected from physical considerations. ?
Finall we wi11 obtain the decomposition (brealtdown) for the
Ys
~: e
magnetic field of tho vertical electric ipol
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,
-
~
1
' ~,iltk~ - . ~
~ IZ .rltll
siufl, (2:19)'
153
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orie
nted in the point where xi ~ xi ~ eta , 1 and directed along the ?
1
axis of the s heroidal to ?
? p sys m of coordinates (along the axis z). Since
'
0 0
Hr_H
J
0 0 indicatin th t '
g a,
,~ ~) ~
~ll.- --11 sand,
,t ~
~2.20~
and the Descartes component H?~ satisfies the wave equation ?Ehen H? h~
x P
should be de
composed (broken down) according to s herofdal functions
p
frith an azimuthal index m - 1, whereby for xi ~ xi it hould
- , s be
~ikRr
- .;- ~ -~- -sin;/.., p ~ ~s~ ,la ,
l l.a 1 ! ~ I)
. cy
r-~~
Assuming that in the term (?2.3}
~, = cc~s 0, 2,22
( )
then, by differentiatin it b d c
g y ~` ~ onsequent tom arisen with the
? p .
term 2.21 in whit '
( ), hit is assumed that xi ,???~r infinit
Ys cxi -r-'!
infinity, t~re will obtain the correlation
1 Q ~ .11=~
Hence b . {~ ~~~
y. applying formulas (1.18), {1.20) and {l.lb) we cazi determine
,the coefficients D
? 1
The decom sit
po ion of the~magnetic.field H?hifor the case. where
xi?. xi is then obtained 3.n the form of
1
l it _, 2
~ zp ._~c!
r 1V l,t
iV~-! !r
1 ran ,
~) Rai! , na
1 1~~~,~S
c?
11~, ,
r ~)
STAT
154
M
i~ a (, -r 1
1 a ~) 2n
r~ ( ) 1
)I
from the very same considerwtion as mentioned above at xi
one can write
Par,__3. Radiation Characteristics of a Spheroidal Surface Antenna.
l?!e will consider a problem regarding syrmrletrical excitation of an
ideally conductive spheroid ~. ~ :d with an elementary electric dipole
o ,
oriented on the a,fl.s of the spheroid.at the point xi - ~ eta - 1
- -
~~a having moment p directed along ?the axis (see T~'ig. 1).
I'i ~ure 1
~ ,
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' The pri~ary field of this dipole is given by formulas (2.19)
(2.210 and (2;25). The secondary field should satisfy the condi-
Lion of radiation into infinity and in addition at xi - this
field does not have a characteristic. Therefore the decomposition
of this field'evexywhere has?the form of
Since the magnetic field.has only the H h~ components the
P
boundary condition will be
By applying it in the sum of fields (2.25) and (3.l), we obtain
tai ~ i?
E
The complete field at xi ) is given by the, sum'of fields
{2.210` and (3.1),, consequently , ?
STAT~
F (cl, ~o)
1
rt) d ~ - i ,
,~ d, i,t
d z (3l
-YE-1R c?
d ~,~ (. )
~ .a,o
In articular when the dipole is oriented on the very spheroid
P ~
i; e., if ~ - ~? then as.a result of (1.2$), the formulas become
- 0 ~
sim lified and we have
P
,, - d ,
~ E~-I N -- ~ 2_ cs~
~Q:
Thus at - xi
- 0
o
4 ~ ~ R~~~ c? Sl~l
~ /~ p
~ jc (E; - I) d -- ~
~ ~ 1,.u N - }~:z --1, Rt3 c, E
;ago
(3)
k'inally in the wave zone, where the function can be substituted:
b its as totic representative (1.27) we can write
y ~P ,
ere at'~xl - xi the function V. ~') is~given by the term
wh (
1- 0
i57
(3
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The function V n9' we also ca]1 the
. ( ) (complex) radiation characteris-
'tic of an elongated spheroidal surface antenna.
For the case where c ~C 1, i. e., for the case of very long
.waves, formula (3.9) acquires the form of
sin ~ ,
( n ,10
--- g ~ o~
' 2 ail ~n -{- 1
(Eo-1) - In -I _
2 ~~ -1
i. e., it represents a radiation characteristic of.a di 1e in a free
Po
space (sinusoid) multi lied b the coefficient
~ P Y g (xi ), determinable
0
by the form of the spheroid.
Par,_~. Results of calculations.
The radiation chars
cteristic of an elongated spheroidal antenna
.de nds u n two ammeter
Pa Po P s. c, proportional to the ratio of the
focu
s distance of the spheroid to the wave length of the xadiator
(emitter)
? _ ~,r_ 2nf
and xj. bound with the r
0 atio of the spheroid semi-axes a and b
(al ha ~ ~b formal
P ) by a
-- 1 - ~ z
u
These parameters are included in all formulas and should therefore
be accepted as basic. However, during the evaluation of calculation
results together with the basic parameters, we will also always have
in mind the ametera ka and kb connected with c and xi formulas
~ ~ 0
Ira = cc ,
0
!rh = c ~
E ~ ---' 1.
These parameters are interesting in the 'fact that, if f --i
0,
and xi ~-~-sj infinity, the product fxi remains constant, i. e.~ i~f
0 0
the s heroid asses over into the sphere of radius rho - fxi ,both
P P _ 0 ?
parameters (~..3) convert into krho, which is an ordinary parameter
of the sphere.
The calculations were carried out in accordance with formula
(3.9) for all pairs (seta) of parameters
~ )
a -- 25,
10; 5,07; 2; ,1 5.
r, ~ 4.6
( )
Furthermore, accozding to formula (3.10.), we obtained the
- 0 f r the ve ame values of the ar
characteristics for c _ and o ry s par
_Fo - 1,000801; 1,005037; 1,02; 1,154700; 1,341641 (4.5) .
meter xi (or a ). .
0 ~ b
The calculation results are given in~Fig. 2~ wherein the polar
STAT
,158
tem of coordinates re 'resents the directivit di ams i; a 'the
sYs P Y ~' , ? ~
tion characteristic of a s`heroidal`surface~'
r. .
e maximum. A11 diagrams oriented on one~line
159
i~
.~
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correspond to one and the same value of parameter c~ marked on the
left. All directivity diagrams oriented in the column under the ellip-
,soid'correspond to the foxrn'of the rotation ellipsoid xe esented in
the upper line where the ratio of 'its semiaxes~ a as well as the value
` ~ b
of the parameter xi are also indicated.
0
Near each curve the corresponding values of the ammeters ka
p
and kb are also indicated.
The radiation,characteristio is obtained from the directivit
y
?diagrams by, multiplying by the value max V (~) An idea about
the characteristic can be obtained by simultaneous/ stud Fi 2
y 3'~g g.
and the graph showing the dependence of max V ~ ? u n the aram-
~ ) Po p
eter c during the fixing of ratios of the spheroid semia}oes a
b
(Fig. 3, a) or upon the ratio~of the semiaxea a Burin th
,.,., g e fixing of
b
values of the parameter c (Fig. 3, b). It is immediate/ evident from `
. Y
Figs 3, b that at any given value of the parameter c the am etude
~ ~
maximum of the field r '
adiated by the antenna increases sharply during
an increa e
s in the ratio of the semiaxes a,,.
. b,
A study of Fig.~2 shows above all that~the member of -
lobea in
creases with the increase in the parameter lsn - cxl ~ An'analo ous
- 0 g
phenomenon was observed also in the c '
ase of a spherical antenna during
the increase in'parameter alpha : krho~ where rho is the radius oft
he
sphere (see F'i 1- '
4 g. 5). Together with the curves for a -
_.. -
'~ b
1.5 are also `given (by dotted line) the directive dia ams for th
t3'' . ~ e
sphere, corresponding. to the ,parameter alpha , krho : c ditto Fi
( g. 1 !~
=13
=10
=3,01
Z =1
i ?=IS
~,
~?1~7080r
,i;=!,003031
j=1,01
j=!,!34100
S?1,341641
ka=k0~0
ko?k0=0
ka=k0.0~
ka?k0=0
ka?k0=0
C?D
'
ka?Q98
kD~Q04
ka~=0,99
kD?0
10
k?=I,00
k0=0
?0
ka=1,13
k0=QS9
k?=131
k0=6d6
,
,
C=1
i'- ~ ?
~
ka=,~46
B'173
,' ~ ~.
kq=j
D= IZ
q
k =
4
ka~30
kB-.h
~=
~
kB=
C?3
~
Kq =J
kd=
~?
ka=30
-
kd - ~30
ka=5/0
-
kB- I
DO
ka=5,71,~8?1,d9
~"~
,
1
,I_ ?6,11
kB=4f1
ka=9,39
ka d08
,
.
..
kd = ~e,04
,
C=7
ka=101
kd=d,7d
ka?103
kB=(~~
ka-7,14
~, - .
?
kB=1,41
~ y
~ ~ ,
~
~
J
' Figure 2
161.
1
~1
68
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Thus we compare an ellipsoid having a focus space 2f and a sphere
of the radius f, As is evident from a'comparison of the curves
oriented on one lines Fig. 2 (c = const)~ with the reduction in the
elongation of the ellipsoid, its directivity diagram approaches the
diagram of the sphere, which could have been expected from physical.
considerat '
ions. This approximation however depends upon the parameter
c (the greater the parameters c~ the slower the a roximation . And
Pp )
soy for c : 7 and a : 1,5~~the~eharacteristica of the sphere and
b I
ellipsoid have almost nothing in commons while at c 5 and an
identical ratio of semiaxea~ the characteristics of the sphere and
elli soid are ualitative
p q 1y quite close to~each other.
A~characteristic feature of an elongated~sphoxoidal antemla
a ars to bo a strong radiation backward^`. In this case} each con-
P~
' f h acteristic a ars to be
secut~.ve lobe in the entire series o car s ppe
stronger than the preceding one and the lobes are separated from each
other by~deep minima, which reminds one of the characteristics of a
traveling wave antenna. Particularly clearly expressed is this
phenomenon for c _ 5 and also at c _ 1~ when the radiation diagram is
little different from the sinusoid (e = 0); the directivity phenomenon
in th4 rear semi-space in the case of a , 1.5 is already clearly
noticeable.
For the purpose of comparison we formulated the characteristics
of a traveling wave antennas consisting of a section of thin wire
with a length equal to the greater axis of the spheroid. It was
assumed thereat that the current wave propagates aion this wire with
g
the s ed of li ht in the direction of the ne ative axis z i e~ \
I~ g g ( . ~
from the le in which the~excitin di le is situated t the
po g po 0 opposite
end of the spheroid).
'From the viewpoint of geometric optics the entire space is divided
into two semi-s aces; the illuminated 0.
P ( . ) and shaded
. ~
.._. ~ see Fi
l ) ( g. !~) . The bacln~ards direction is called
~ - .. .. , .
the direction leadin toward the shaded. semis a
. g ^ p ce.
i63
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Figure !~
the end z -
This simple model offers a perfectly
identical arrangement of lobes as
the spheroidal antenna with the ex-
ception that its lobes increase more
rapidly with the increase in the
angle
For these cases we also computed the
radiation characteristics of a
standing current wave, under the
assumption that in the section of wire
mentioned above - a ~ z ~ a the
1
traveling wave of the current is re-
fleeted with the coefficient - 1 from
-a. The characteristic of such an antenna is'symmetrical
relative to the equatorial plane - ~
2
A comparison of the characteristics for the values e - 3 and c - 5
is shown in Figures 5 and 6 which also bring the directivity diagrams
of the spheroid for the ratios a _ 25; 10; 5.07 and the diagrams
`? b
of the traveling (Russian abbrev. Beg) as well as standing (Russian
abbrev. St) waves of the current. A study of the drawings shows~,that
the curves for the spheroid occupy an intermediate position between the
curves for the traveling and standing wave, whereby an increase in the
elongation of the spheroid is follotired by a change in the directivity
164
. STAY
diagrams tending toward the traveling wave. l~le wish to mention that
the parameter, ka, for the given value, c, remains practically unchanged
ka : 3,00 - 3.06 for c : 3 and kc : 5.00 - 5.10 fox c , 5), whereas
(
the ammeter kb changes considerably (kb . 0.12 - 0.60 for c . 3 and
P .
kb : 0.20 - 1.00 for c - 5). Thus during a reduction in the~parameter~
kb the diractivit die am draws closer to the diagram of the standing
Y 6'~'
traveling wave.
? Ivl
wave, ands during an increase in kb~ approaches the diagram of the
ca z i z
her
' 1
.'
3 1
m
QS
I Y V it ~I~~I
~ ~ f =13
~ 1 1 ? =t0
Q~ o ~ o o
0? !0 60 ?90? ~ f?0
! 3. ?S,OI
6e~ Cm
~ Figure 5,
165
`STAY
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'The indicated analogy liettaeen the directivity diagra>>s of the
sphoroid'on one hand and the standing or traveling wave on the other
hand does not ortend to the, phase characteristics (tahich tre trill not
discuss further here). Namely if we should calculate the radiation
? from asemi-standing current wave, then, by proper selection of the
standing wave coefficient, it wi1? be possible to attain~an accurate
conformity with the amplitude chaxateristics of the spheroids as
given, e, g.~ in Figu~~es 5 and 6; however, as shown by calculations
the phase characterist~.cs of such asemi-standing wave are much
different from the phase characteristics of tho spheroid.
~Y
ll
40
I 2 Bee
3
C
C?3 B
e ~,
?1
I v
' 2 ' t ?15 E ?!,000801
b-
3
I
' 3
~i
3
2
for c _ 7 and a + 1.5 (lower right angle), for which the parameters
cidence of a plane wave on any arbitrary convex body provided-the
radius of curvature, rho, of the surface .section of the body is great
In the caso where c : 7, the radiation characteristics have a
somewhat special character soi7etimes very reminiscent of the charac-
teristics of the standin wave a - 10. 5.07). It a ars that
g (--~. - , 1~ ~
b
togother with the previous picture of. radiation into the backward
semispace, we have here also the screening characteristics of the
ellipsoids i, e.~ & darkening of the rear semispace, We wish to
mention that the latter effect in pure form practically does not
appear for the cases investigated by us. An exception is the curve
b
~ e 9.39 and kb ; 6.26 appear to be the very highest of all these in-
vestigated by us.
An idea about the picture corresponding to the very high values
of these parameters is offered by the V. ?A. Fok theory (5). This
theory allows one to calculate the field originating during the in-
in the plane of wave incidence in comparison with the?wave length,and
distance from the body on which the field is investigated'.
Accordin to the rinci le of reci rocit the field in the ,.?
g P P p Ys
distant-,zone in direction is (for our case) equivalent to?the
. ? I 1 ? =10 t ?1,00303
STAT
y~ ~ / 3 1 Bet 3 = 0? ~ 01 ?
~ Be 5, ~ f
90?
~i~L?v 6
.166- .
150" IBD? ~
field in the pole of the ellipsoid produced by the plane gave falling
at an an le l~ -- ?
g
i67
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STAT
if only in general outlines it is necessary to reach values of the'
n.- 0 f r h ch a en at - l it is necess
`~'
parameter krho l ~ o w i v - .5 ~'y
to take c ..~ lb. At such a value of the parameter c in the (3.9).
ate a roximate 0
series,. it ti~rould have been necessary to caloul pp l~r 4
members terms.
U 'n com lotion of this report reports have appeared by (6)
Po P ~
and (7) devoted to the symmetrical problem for a rotation ellipsoid.
Calculation results are available only in the second reports namoly~
for the electric dipole oriented in the very pole of the ellipsoid
but for a much narrower zone of parameter changes. Wherever the
results of report (7) and these of our own article can be comparat~
they do coincide.
Literature
1. Stratton,, J. A., Morse P. M., Chug L. J.~ Hutner~ R, A.s
Elliptic Cylinder and Spheroidal Wave Functions U.~S., A.~ 1941..
2, Gobson, Ye. V.: The Theory of Spherical and Ellipsoidal Functions.
Publication of foreign~litexature~ Moacow~-1952. ,. '
3. Frank F. and Mizes~ R.s Differential and Integral?Equations of-
~ ~
Mathematical Physics, ONTI (United Scient. Techn. Publication),
1937.
~,, Delkina, N. G,, and Vaynahteyn~ L. A.s Radiation Characteristics of
' Surface Antennas (see this collection~of articles).
5. Fok,. V. A.: Distribution of Currents Excited by a Plane~~lave on the
Surface of a Conductor. Journal. of Experimental and Theoretical
Ph 5ics 1945 12th Ed.) Vol. 15.
Y ~, ~ ( :
P4eixner, J. Ann. Physik. (6), Vol. 12, Nos. 1~-b,,p. 227 July 1953.
7. Hatcher, E. C~, Leitner, A.: ~Journ. of Appl. Phys., Vol: 25t No~. 10~
~Octobor, 1954. ~' ~ `
According to V. A, Fok, the field in the zone of light can be,
under the c di n e umera ed above c 2 smitted
on dos n t , suffi iently wel trap
b ~eometric o tics and~in the field of semi-shade directions close
Y~ P , (
-~g0? ~ it decreases ra id and smooth/ where the
., P ly Y by
rate of reduction de~enda only upon the parameter, lo;ho.
. The radius ot``curvature of the rotation ellipsoid in its pole
~2 ,~.
is rho ; _.._,_, and it uan be considered as changing but little up
a ~ -
to direction of Ilse order of 220? whereu n i. e, ahead in
Po : ~ Y
the shaded ne ft be' t e B
zo , gins ovary oonaid rably. , ut at a greater
krho~ the field in-the gone of the shade is very small and is of rio
interest.
Thus, we can say that, at very great values?of the parameter
b2
krfio _ k , the field of the elongated spheroid ~ directions
~9'
a a
not exceeding approximately'~0 , is the 5ame,as the field of, the sphere
with a radius rho =
This field was calculated in a previous
report: of this collection, where Figures 6 - 9 show the radiation
. , ,
characteristigs.of the sphere for values of parameters alpha , krho
,- 15, 25; 50, and .160. ~In thi.,s report, on Figure 28 there is shown
- ,
`how the curve calculated in. accordance .with the V A F k t e d
o h ory oes
noi;?xeproduce the nature of reduction of the radiation characteristic
and gives only the mean line of the oscillating curve. It is evident
therefro t
m hat even for the very greatest of. the investigated values
? 2 ~ - -
of the parameter krho = k b ~ : 4.2 (c,= 7, a = T.5 the icture
. )i P
~' T
obtainable b t
. y he V, A. Fok theory for very great krho values, is still
? considered'as unrealizable. In order to obtain such a picture,,'even
6
18
169
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DIFFRACTIOII OF ELECTROI3AGI;ETIC ~lAVES BY A DISK
by
? ~ I~I; ~G. ~~Belltiria .
,~
Par. 1 Radiation Characteristics of an Oblate Spheroid and Dish
during Their 1~xcitation with a Vertical Electric- Dipole.
The solution of the problem concerning the diffraction on an
oblate rotation ellipsoid can be obtained in the following mannox.
~!o will analyze a system of coordinates xis etas phis con-
e D arts s stem b ratios
nected with the esc s y Y
fl ( )( i) ~
.l 1 ' n
yi
z ~? 'r.
f~ i
This s stem is obtained by rotating the elliptical system of
Y
coordinates about the small axis of a family of ellipses, and we
shall Dell it the oblate spheroidal system of coordinates. In this
fr m 0 to infinit and eta varies from -l to ~ 1~ the
case xi varies o Ys
? ,~ ,
- corist is re resented~b an oblate spheroid
coordinate surface xi _ p Y
- di k with the radius f and 2f is the
xi-_ 0 is?an infinitely thin s ~
d' tance~between the foci of the meridional cross section of~the
is
1 The value eta - 0 corres onds to the equators of
spheroid farm y. - P
ex id' and eta - 1 corres onds to their poles.
thesh o s _~ P
P ~
.Substitution
STAT
formally transforms the system (1.1) into an elongated spheroidal
s stem of coordinates (see (1) Par. 1) -
Y ~
x = ~ c'2 --1 1-- '~ COS
f ( ) ( ~) ~~
- - '2 _ 1 l - '~ 5in
z_ir- ?
. f~~,
and the wave equation in the system (l.l) -into the wave equation
in system (1.3). In this case the parameter c ^ kf~ included in the
wave equation, transforms into icy. Consequently any solution of
the wave equation in an elongated spheroidal system of coordinates
(1.3), and any formula equitable in that system convert into a solu-
tion of the wave a uation in the oblate spheroidal system (1.1) or
q
into a formula equitable in this latter during the substitution
--+ is
' 170
?~ '.~ '9
~;-- -i~
Further we will make use of all the formulas of report (1)~,by '
changing?(1.~.) in these formulas.
ferences.
This chap a will be '3tn lied in re-
g P
And so b char in (1.1~.)~ in forr~iula (3.9) of report (1}, we
~ Y g g
radiation characteristics of an oblate s heroid xi -?xi
obtain ?the P - Os
ric di 1e oriented in the le of the
excitable by a vertical elect po po
spheroid (Fig. 1).~
' dl (-i)~c (-- is S~~t - is cus~li
V~).-~
~d~ ~ .
.1
171
~ ;.'e
(1.5)
STAT .
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' i, --~ e oblates heroid changes into a disk with
,(hen xi 0th p
.0
unit at :fl. --i 0 in forrnzla (1.5),
radius f. By changinb to, 'the 1
0
iation characteristics of an infinitely thin and
we obtain the rad ,
di ole
'deal/ conductive disk excitable bs a vertical electric p
i y ,
its center Fi . 2)
oriented in ( g
00 -1 la - is 5 t'~(-!c, ~)
41 ( ) i, t( ) t,t
6 (1,6)
v~ )!
~' ~ ~3~ -
~vo do E..o
V(8)=sin8. (1.10)
Figure, /
It can be sho~m that. at any given value of the parameter c
on the disk takes lace
for the vertical electxic da.pole p
Ph site// this is explained by the fact that in the direction
Y, Y
f `~ the currents on the disk, because of their symmetry
0
2
do not radiate.
STAT
172
e oblate s heroidal antennas we obtain from
At c ~ 1 for th ,, p ,
a 10 of re rt (1)
formal (3. ) 1~
~_ s~" !i -- i sin 8. 1.8
E ( )
_ g o)
(;- -}~ 1) (- ;~ arc ct~ c? + 1) ,
0
If xi -~ 0, then
0 --
g A~1 = ~~ (1.9)
and for the disk at c < 1 takes place
nse uentl the radiation characteristic at c
Co q y,
1 for the
d for the disk (as well as for the elongated '
oblate spheroid sn
enta a sine curve (sinusoid), i. e.,
s heroid see (1) par. 3), repres
P ~
tic of the di le in free space, mu7.tiplied
the radiation characteris I~
e nds u n the eccentricity of the spheroid
by the coefficient which d pe po
cal extreme) case
the disk. Thus in a spe i
and is equal to one for , ,
oes~ not, affect the
tel lon waves the presence of the disk d r , /,
of infini y g , .,
. .. .. ~,
r ~ - riented~. in
cteristic of the vertical electric dipole o ,` ,
radiation chars __ ,
the center of the disk.
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We made, calculations for values of~
. ~, ~ ?, ~~ ,~r' .. ;
cCordin
the parameter c. ~ 1~ 3, 5 a .g
- ~/;.. .
1 6 ' .The -results,of~
to formula (. ).:
,~culations are iven
these cal g
ch value
Fig. 3, whereby for ea ,,
~. ~ ..4:
.
.
~
.
a
? ton characteristic:
~
the amplitude radix i - -. , ,; ,, ,~..
X173
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of the disk V (. ) is given (continuous lines). The dotted
lines represent tho radiation characteristic of a sphere, excitable
by a vertica]: electric di ole oriented on the suuface of the
p , sphere
for values of the parameter alpha - krho = c, where the radius of
the sphere,~rho is~taken'as equal to the radius of the dislc f (see
(2), Figures 1 - 3).
C-11 C?3
C?5
Figure 3
Tt is evident from Fi ~ ,that f r c -
g. 3 , o _ 1, the radiation , ,
characters tic '
s of the dipole on the disk 3.'s practically the same as
..
in the case of the dipole in free space (or on the disk at e - 0).
4Jhen c : 3, the radiation characteristic is already divided into two
lobes whereupon the rear lobe is approximately one-third the size of
the forward lobe, j~)hen c - 5, the radiation characteristic also con-
,?sists of two lobes. But the shading phenomenon here is ahead uite
. ~ Yq
'the shaded zone takes place.
Comparing with~the radiation characteristics of~a s here we see
p :
.tha _ ...
t for alpha _ 3, the sphere does not quite roduce a shadin an
p g, d
.17~+ ~ .
STAT
for alpha - 5, this phenomenon is expressed very poorly The smooth
conversion (without oscillations) from the illuminated zone uito the
shaded appears in the sphere only at values alpha N 15 ( ~ 2~ ~
Fi 6
g. )..
'Thus, at one and the very same (investigated by us here) values
of the parameters c and alpha, proportional to the ratio of the body
radius to the wave length, the disk gives a considerably greater
shading than the sphere. It should be mentioned however that the
problem of comparative effectiveness of radiation shading by the disk
and sphere at greater values of c and alphas requires additional in-
vestigation.
The oblate spheroid should occupy, an intermediate position be~-
tween the disk and sphere. The calculation of its radiation charac-
teristics during excitation with an axial electric dipole (formula
1.5) does not represent any great difficulties. .
Par. 2 - Horizantal.Ma~netic Dipole on a Disk.
The conductive surface can become excited not onl Frith the ai
y d
of an elementary electric dipole but also with the aid of a slat or
system of slots. It is therefore interesting to explain the type'of
. .~
characteristic of an elongated or oblate spheroid excitable an ,
by
elementary slot, cuts for example; near the 'apex eta - 1 of tho
heroid This element lot can be n id red ~`
sp ary s cos a as an, elemen~ary
magnetic dipole having a moment
!rl - lily, ,
175
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directed along the axis x and oriented on the axis z at the point
xi _ xi ,'eta - 1 at xi ~ xi (xi - xi is, as above, the equation
1 - 1 0 - 0
of tho surfaco of?the s heroid).
p ?
However, a solution of this problem for an arbitrary xi , as well
o
as a solution of the problem concerning the diffraction of a p1ano~
electroma etic wave on an ideal/ conductive?sphoroid, has not been
~ Y
obtained up to this time. ?
the roble for xi 0 i. e.
It is possible only to solve p m 0 ~ , ~
the problem concerning the diffraction on an ideally conductive dish.
It is assumed in this case {see
Fig. 1~) that?the disk is excited
by the magnetic dipole (2.1)
oriented on the paint xi . x~,
1 1~ - -
~fe have confined ourselves to
this arrangement of the magnetic
Since the components of the Hertzian vector of the primal field of
the magnetic dipole are
where
R'= %~r2
l -}-
(z
zt)2, ~ ~ 2.4)
and r is the radius vector of the observation point 9n the c lin-
Y
drical system of coordinates then the Hertzian vector for tho entire
field has the form of
Figure ~r
dipole because within the boundaries
produced by the dipole a case of an elementary slot on the disk Zs
derived ( . 0) and plane wave normally falling on the disk (xi
? 1
infinity), which interested us.
In. the role of potentials we select, first the components
'~? x
and ~ f the Hertzian ma etic"vector ~ bound with the
z ? ~ ~ ~
fields E aizd H by formulas
E = ik rcit [1
? ~ d~~?fl?I~ /, H
STAT
~ = no i
x x n ~
x
1
Tn this case the index designates the potentials 'of the '
secondary field formed. by the currents originating ( enerated on the
g )
disk,
Inasmuch as
secondary field,
on phi, i. e.,
the com nent o does n t
po o depend on phi then the
x
by virtue of the, dislc et should
symm ry, also not depend
i77
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The boundary conditions
whence it is derived that on the disk
8y integrating equation (2,13) and by utilizing any of the given
ratios 2,12 we will
{ ) obtain the bound conditions in the form of
a~
-=C
dz
where C is a certain constant, subject to determination from addi-
tional conditions. Thus in a c lindrical s stem of coordinate t
Y Y ai
is sib
pos le to obtain simple boundary conditions for the potential
Phi and Psi~~ which we will now seek in the form of decompositions
in accordance with spheroidal functions.
The fields and boundary conditions in an oblate spheroidal
system of coordinates are written in the form of
E~ = i k cos ~p { f cep + nr~ ~ ~ (~2 -I-1) ae -FF(~ -Y~2) a,~~ -
Y(E'-I-~)(~-~la) r dW dW1l
H.P==sin~(f,~~217I9,~Eaa.-..~ a^l+ (2.15)
l ~ J
a~ =~f~
(E=0).
y`=~fV ~ -~2 ~
STAT
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'wiU. be sought in the form of
The component ~
' ~ ~ ~ ~~r r z cos ,
Ct ~ ) ~P
(
_ .~
.~ in the c lindrical coordinates
The magnetic Hertzian vectox Y
wi11-then have components
._.. _ ~ ~ -- iF cos 2.8
.._ ~ sin TI -
n ~-~COS~, ~ -' ~r ,x ~
r 'F
d psi in the following
?and the 'field will be expressed through an
where for the purpose of brevity, we designated
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Since the Descartes components of the Hertzian vector satisfy
the wave equation, then, by virtue of (2.6) and (2.7)~ the functions
1 ~ ~ 't
phi (r, z) and psi (r, z) should be decomposed ~n accordance w1 h
the s heroidal functions with azimuthal indices m _ 0 and m ;, l
P
respectively. Since with respect to anfinity, they should satisfy
the condition of radiation- these. decompositions should be sought in
the form of
'd-I c - ~ Rlat -- is i= StU -- do `
0 0 ( ~ ( )
,! .t
t-u
-~ ll
,~ , t ~ (
.
t-q
Q
As to the potential of the secondary field Phi (xis eta),
then according to formulas (2.12) and (2.110 of report (1) we have
PhiO xi eta - 2ikmx.times
( ~ )-
~? I ~ ,
S~~ ~ (- tL, ~ p I ~r 111 I Il
. R ) ( ) t
X r( l o t o,
No, t (_ tc) o, ,
(c > Ei) (2.19)
(
11j ~ '
/. a ( n, t
No t (-- ic) ~ t
!-U
S11j -- is .
( , ~7)
n, t
c (2.2U)
.(< t)
The coefficients A and B' are obtained from the boundary con-
. 1 1
dition 2 16 b uti].izin the ortho onelit of the an ar functions
(. ) Y g g Y t~
in the form of~
i81~
npi~ r= f, z=0 (2.24)
STAT.
STAT
180
_ - 2iknc
lit ~:
I
191 ~F .c d 111'
.
do
d~~ t (- is
t )
~ `~ vt d 1~}
Noy t (- ic) Ro t (- !c, i0)
d.
dl~t -!c
4 ~~ ( )
Bt(Ft) -_ -Cf
t, t t, t
(2:21)
Here and everywhere below it was design~tgd ,_
. ~:. ? -,
d _ . _ - -- tc t 2.2
- tc r0 -- . R ~) ( )
R
d; d; ~.~
? (1)
We like to' ca11 attention to the fact '~ia~ d R (- io,,
j? dxi 0, l
30 convdrts into zero at an even 1 (see (3), pp. 74-71). Since
OIL
in this cased is different from zero onl at an odd Z and
1 y ,
1,Z
d -only at an even
0
~ then
A .=0 npe l=U, 2,...
t
B =0 ndtt 1=1, 3, ..'. '
t ~,
Consequently, the line (2.17) is summarized
(2.23)
actually onl`.y by
'- nl the even indices Z
the odd, and line (2.1$) . o y by
In order to determine the constant C, it is necessary that the
radial component of the field-free (complete} current on the,edges~ of
the disk should convert'into zero, i. e,- (expressing the current
densit in CGSM units) in order .that ~ '
Y
1 - +
.- --- H -- H ) ... U
1
~ 4a q. ~
-or that ditto
H -- H
~;o p
r?-0
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By. using formulas (2.15), we will derive that tho condition
(2.55) takos place provided the.term
~ .
dq ,d~
1 a~ (0~ ~I) d~~ (U~ - ~)
___ .~..
~" dE
1 r~ dE
~ i
~2.2s~
at eta?=~ 0 is an infinitely small value of much higher order than
delta hi 0 -eta and so on written hexe
eta. (Under the sign ~
e sea
and below, we mean the derivative according to eta, in which instead
f eta it was substituted with -eta
o )
e t HO of the rim ma etic
>11e want?to mention that the eompon n p ary gn
psi
field is a continuous function at z z , paxticularly
1
N? = N?
~ z- ?~?0 ~ x--0 9
(2.27)
and consequently, the condition (2.25) for the primary field is ful-
filled automatically. Consequently in (2.2b) under Phi,~we can under-
1 '
stand the function Phi corresponding to the secondary field and the
' determined frdm the re uirement that the?value
constant C is q
at eta --~ 0 should be infinite/ '
y smal], of much higher order than eta
. (1)
Since . d ._. S (-ic eta at an (1)
) odd Z and S
d e t-iC, eta) at
to 0~ I 1, t
an even j 'are the even functions f
o eta (formtLla 1.7 in re
~~ (1) ),
then they decompose into 1?iaclaur ~
in s,series having onl even de e
Y gre s
eta. Conse uentl
q y, the term (2.28) will satis our re
fY quirement~
provided its first member (containin e
? g to in zero degree) of decom-
position into the Maelaurin's series c
onverts into zero, i. e.~ if
~ (~) d ftl
,R (--rc, r0 - S -.1
o,~ ) c U
(
t o,r )
r n "~-
(2.2s~
Using the ratio (1.28) of re rt
po (1) and keepin in mind t
g hat
(1)
R (-ic, i0) at an odd conv
p ~ erts into zer
~ Z ? (3, pp~ 70-71} we
will obtain the constant C in the form
of
C=3~~ Ci ~~i)= ~y Ci ~Ei)~ (2.30)
where
A Rt3~ - is
1 O1( ~
d tt~
? d?~ o,t
d (~) '
-~- S -lc
( ~ '~) -~-
d,1 o,t
~ d ~ ,
-F' ? R - tc i0 S !c
~ -I--
r ( )[ ( ,~)
ir l -- r,~ d. ~ 1,t
-~-.Si~i~-icy -~1)1 (2.28)
STAT
_ d (a~ tip
~' 13 -
d- ~,r ) r ( , 41 0,
\ ~~/. /
l~~~~,~,.,.
E
.. i~ . .
-!c 1 3)
o,~ ~) d
-' Stet
No,t
.C
(-ic
d
)
,
- tar ? dry o,i
d Ro,t (-!c, ~0)
E
T ,
(2.31)
STAT
ib3
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l~1,3,.,,
c~ (~i)
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Tn this w the coefficients and B of the decompositions
a3'~ 1
of the second field aro now
(2.17) and (2.18) of the potentials ax`Y
completely determined. '
Tn the~wave zone i. e.~ at xi ~--~- infinity and cxi --} an-
finity, the fields (2.15) have the form of
'tutin in these formulas the decompositions (2.17)
By substi g
1 ~ .
r i and Psi ~to which the asymptotic formula (1.27)
and (2.18) fo Ph ~
e gill obtain terms in the wave
of report (1) should also be applied w t
e second field excitable U the magnetic dipole (2.1)~
zone for th ary Y
the disk in int xi - eta , 1.
oriented on po - . ~
into infinity so that eta ?- 1 ----~ ~~'
1 - , ity. Another condition
necessary will be that c ?--~ inf ?H
_ inity. The ertzian vector of the
and consequently in this case the primary field re reante a lane
P p
wave normally falling on the disk
- ~~ - ~ ? 2 ~ -lkz
/! -- Ir Il = Age
Y x x
with an amplitude
~ l kz~
M = l~211t
a,
Tn order to obtain the diffraction field of a.plane wave' it'is
necessary to write .~,~ infinity c _.__~ infinity into the
terms for_ ( ) and C {, ) of the previous pare a h. Then :talon
1 ~' .p, ~ B
into consideration (1.27) in (1), we can write
jJe 1:11 assume that the horizontal magnetic dipole oriented on
' in rientation de arts
the disk {Fig. /~), stands on axis z and retain g o p
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and the value T is determined by formula
? (2.30)
(2.31a), whereupon according
C = -- C1= - ,-z 1110,
f c z~ f c~
2M
`"' ~,ks
in this case
2111
cl~~
lt) d_ U n,t
(-?i~ cS (-!c, I) - .Ill (-ic, !0) ?~- C1d
b,t d. u,t
- - ~ -- --
u,! d; u,!
~M
Lit (oo) _ - - s' C2bt
ck
C c!''t
4hf _ _n__
-ck~ ~I:t~ ~ '
Nt,tlti,t (-~c, tU)
By substituting the terms (2.7) in formulas (2.32) we will
obtain the?,secondary field of a plane wave in the crave zones in the
form of,
- - A4c~! ~tkR t
- H - E? - Vt } >4 cos
n ~) 'Q~
r l kR
R1c2i
E~} = H-
~ ~ z
placed,in the beginning of the coordinates and directed'alon ~the~"-
g
1~'~ t ~ ig - 4! p
T-' -' -- C O5 V
c~.
? t?t. 3,
-}- si n ~`} ? 2C
2
ts~i Li .
`~l?1 ~ -- 4i
) -,
~?a
e`kR ~}
Vt ~ sin ~
() ,,
~2
where the coefficients ?and b are fixed by ratios 2.7 . The
al ( )
1
(1) (2) .
functions V ( ) and V ( )are the com lex characteri tic '
P, s s
of the secondary field of the plane wave falling normally on the disk;
It should be mentioned that our adopted standardization of these func-
tions~ as will be shown below, (see 3.23) warrants at t
tendency of V`j'(0)?~ V`"'?(p) toward 1 when c infinit .
. ~ ~-~- Y
When c 1 formulas 3.8 ac uire the fo
~ ( ) q rmof
~}~3 ~ikll
-- N = L = - N[ ,~-' cos
n ~ 31< k '~~
4 j3 ~rkk
L:,, = FI = - -- A~t;~'' - cos ~~ sin
(3.10) ':
i. e., for infinitely long waves, the secondary field of the lane
P
incident wave in the wave zone is such as if it would be radiated b
.y
an electric dipole with a moment
~.
y-axis.
The currents excitable on the disk b t ' '~'~~'~,
y he secondary field of the
-i tGSI~} -ic
~ ) , cos 8
I ~ ~ r
,l )
~-~?) cos
t o ~ ,
,t )
ant ilk.,
STAT
X86
cos3)-~- l
I
incident plane wave, are expressed`, (in the: CGSM system)' in the'followin
? ~ g
? manner,
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 ' or in cylindrical coordinates
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~rhere f is the radius of the dish.
At greater magnitudes of the parameter c, it is possible to obtain
approximated formulas for the secondary field in accordance with Huygen's
principle. For this it is necessary to take into consideration that
the ma etic field on the dish has the very same magnitude as it i~rould
have had if, instead of a disk, we i~rould investigate an infinite sur-
face, i. e.,
This formula is known to be incorrect on the edges of the disk, but
at .distances of the order of a wave length from the edge, it can already,
be 'considered as equitable. If c _ kf is great, i. e., if
takes place then the tiridth of the ~band~ where formula (3.110 is 'its
correct, is small in comparison with'the radius of the diak and .in .
these"directions in which the secondary field of the diak is not close
~to zero "the edge effect" is small in comparison with the basic one
which depends upon the area of the disk.
In the
hypothesis of (3.14) the current on the diak in the CGSM
s stem is
Y
and a certain potential in the point of obaervation:P :will-have a
0
single component
where R' is the distance from the int of observation to the .
Po
t f t ~ '
poin o in egration, and the integral is taken according to the, upper
where R,~
r sin ~ cos
~~
~P)~ ~ 3.18 ~ -
~~ Y
si are, the s herical coordinates of the obse at
P P rv ion
~P and r and ' are the c dri t '
point , psi ylin cal coordina es of ,the integration
o .. ~.
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int on the disk and by using the knot~n formulas
we will obtain
and the approximated formulas for the field will havo the foam of
e~kR ic~~ (c sin D
- N = E = M co
e '~ kR sin fl s ~
etkR Icli (c sin d
~ ,
E ~_ N = M , co 8
e `~ ~ s1 n d S SIn ~.
Fina11 comparing formulas (3,21 and (3.$) we should write
y~ ,
It should be mentioned that our selected standardization?of
(1) (2) ~ i.nc e
functions V and V gives in the case of the Huygen s pr ipl ~
which also justifies its introduction.
'The~~radiation characteristics of the dish which roduce t
p he electro-
ma- etic field
Gn ~n H- and 1r- planes of the incident ~~rave respoctively..
wore calculate _
d by us for c _ 1..3 and 5 according to formulas (3.9.),
and for c -~3 and b = 5 also in accordance with the a rbximated f '~
.. pp ormulaa '
(3.22). ?The xesults of these calculations are iven
. g in the Descartes
system of coordinates in Fig. 5 - 7, whereby the dotted 'lines re resent
p
_ ,
curves calculated according to the approximated formulas (3.22). The
Descartes system of coordinator is more favorable in this ease because
it gives abetter representation of the behavior of the radiation
cl~aracteristica in the vicinity of the minimums.
Starting with analysis of characteristics, we wish to mention that
all characteristics are symmetrical relative to direction _ o
_~~
as also should have been the case for th
e flat and infinitely thin
dislt? in t i (2) .
h s case, the characterista.c V ( ) converts into zero
~_ o
at = g0 ~ The explanation of this circumstance is evident from
Fig. $. The incident wave (3.2) generates,~by virtue of the a et
. ~ rY~
such currents on th d'
e irk that their component, along axis x are able
and the unabsorbed components according to y along axis y are non-'
radiating. In the H- lane in direction ._ ?~ the~rad ati~
P _ 90 , i, on, is
(1
V (g0) 0 .
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Itizrther, everywhere are the ratios
~) V
~ .) ~
(), . 3.1 ~
( )
which is immediately evident from (3.9), since at - 0 and
_ (1)
~'
_ ~ the second item in Y disc ars.~ In directions
~ ( ) ppe
_ (1) (2)
_ 0 and - n ,the functions V ( ) and V ( )have
maxima and'an~increase in the parameter c brings these maxima close
to unity
The approximate functions (dotted line) are always equal to one
at _ 0? and - 1$0? and have a course ual:itativel ve close
_ q Y rY
to the slope of accurate cuxves. The maximum difference between the
~,
approximate and accurate takes place, as it should have been, in the
minimums whereu n the
po_ approximate functions in the minimums convert
to zero
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(r)l
~
f
?
J
H- ~~-D
?
0
30? 60? 90? r70? 150?
. r, ,;~u e 5
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C~3
I v~ I
r
~~;
6
30?
? 9
0? " 1
?0? t
0?
td
0?t1
1D
ao
,~ irtiu~e 6
194
STAT~
6? 90? "~?~, ~ ~ ~~ ?
0 10 ISD 1d0 ~
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196
bt ' b H ' r c e for the norms]. fallin of tlie~ '
are o aanable y uygen s p in ipl g
have on the click.
1,13 "1,'18
1,U~1 1,QH
A evident from the table at~c - 5 these ratios are"dread
a is , ,. Y
' r ~ rea e' iri ameter n `the
quite close to unity. IIpon a Earths inc s , par ~;,
a roxitnate formulas should eld atill'mbre accurate results: ,
PP yi
In this'w "the Hu n's Prinei le" a licable in the classi-
~ Y~ P ~ PP ,
r blew concernin
cal theory of diffraction for the solution of the p o g,
the diffusion on a diak~ offers quite satisfactory results for values
of the parameter c ~ 5.
Par,/. Radiation Characteristics of a Slot on a Disk.
Let ua assume that the dipole with the magnetic moment (2.1)
is now oriented on the very disk in its center, i, e., at
0 eta - l Fi . which corres nds to?
~ .. ( g 9), Po
era/ ele e t lot slotted?on
a unilat . m n, cry s ~ ,
the'disk along axis x. In order to.
'~ econd field in this case
obtain a s cry
t
it is necessary to write .- 0 in the
a .for A xi ) 'and B ~ of,
formal s, ( ( ?) _ .
- 1 l; 1
Par 2 FIe wi.Il 'then obtain
. .
,??
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''.le ~ai;ih to mention that an 7ncxease in para~etor c will be follo~~red
b;/ a sinking in the minimums of the accurate cures. The divergence
e ' ' c co ted ili accordazice with
~in tie calculated values of charact rista. s mpu
Figure $
accurate and appro~natc. foi7nul.as in the'
vicinity of maxima at e ~ 3 does not
?' at c - . does not
exceed 15~, and _ 5,
exceed 5N. .
It. is also interesting to compare the
results,obtained during the calculation
of the diffusion coefficient alpha
(c:alculated according to accurate and
approximate formulas
IEI ~;
( )
~ L. I
where E? is the amplitude of the plane incident wave, and E
is the am litude of the reflected field in direction o osite to the
P PP
incident ~aave (in our case ~~ 0) and effective zone of diffusion
sigma .
tide will bring fortfi~the ratios of values alpha and sigmas calculated
accord'in ?to accurate formulas to the values alpha and~si where
g P, 1 ~~,
S _ ~~~
a - :.~k-
ks '
(3.L')
STAT
at =
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o,t ) ~ O I
31
NO,t e
d,
4h C (0) dt,t
8,
(- ic,i0
l,t )
13 0 .- rrr .__. ~ ?
1()
1 I,l
However, the term fer the constant,C (0) cannot be obtained in this
1 ,
manner because if we should aixite xi , 0 in formula (2.31) for C
1 1
)~ then the set obtained 3.n the numerator wi11. be unsuitable for
calculation The method fox effective determ9nation of the constant
C (0) will be described briefly,beTow.
1
' The second field in the.wave zone (see 2.32, 2.17, 2:18)
~'3'
has the form 'of ,
`' where
'~rAe
I I ~ ~llr1~' (I)
-~- H, _ E _ --- lr?!tt V ~ cos
+ () ~>
~ R I
~' -h k tlt ~V >~ sin
p, O ~P~
~ R I
l-1 x 2 '
I ), s
c a S cc, cos S --
t o ~
,t )
t-1,3..
' q'
-sin 8 2C 0 _
t () tc cos ~}
I,t )~
:~
t-0a:.. .
V -. ? S -- i
I t ( c, cos $ ,
c o,t )
1-1,3...
19S
4.3
STAT
-- _lt
R
~ )--'
-
P 3
t
I,r
,
entod b formulas
ents of the rim field are pres y
The compon p , ~9 ,
tk/t
-- ? - ? - - k~m ~ cos 8 cos
-N-E- ~,
~ r R ~ (4.4)
ue
and finally the entire field is ,
ex radiation characteristics of the-disk with a unilar
are the comp/ ,
texal elementary slot cut in its center.
T. the conclusion of the formulae for-~tlie
- ,Je iri11 now discuss
'e the rinci le' of,
constant C 0). lde wi7,T,apply for this piirpos p p ,
( ,. .
1 .
' form of - . ', ,,; , ,,,, ,, ,,,,
reciprocity which can bo written in the
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2krn
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;P2 (xi - x'~. ~ 1~ e'~a = 1) the di~olea with equal moments m~
directed along axis x (Fig. l0). The x?atio (~*,7) will then be written
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By substituting formulas (2.15) (2.17) (2.18) and (3.?) in
ratio (~..9) we obtain a term already suitable for calculations
CI (0).
(4.1 O)
o,t 1)
dl S~,t(-ic,l)
1..1,3.... NOI, ~ 'RO 1(-'
~- Cs
..._ ~ t~-I S~!a (- it,l) d~ Sol (-~ ic,l)
L - ~j (- c) d -{-
do't 8(31 (-- ic. iU f d Stlt is i - c=Stiff -- tc 1)
d
3) -~ )
Nc.t ~l; ~c.t ( tc,iU
- 2 ~ (:~I
(4,11)
.At c