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UNCLASSIFIED
AD 4 447 9 2
DEFENSE DOCUMENTATION CENTERFOR
SCIENTIFIC AND TECHNICAL INFORMATION
CA14ERON SiATION. ALEXANDRIA. VIRGINIA
UNCLASSIFIED
NOTICE: When government or other drawings, speci-fications or other data are used for any purposeother than in connection with a definitely relatedgovernment procurement operation, the U. S.Government thereby incurs no respousibility, nor anyobligation vhatsoever; and the fact that the Govern-ment may have formulated, furnished, or in any waysupplied the said dravings, specifications, or otherdata is not to be regarded by implication or other-wise as in any manner licensing the holder or anyother person or corporation, or conveying any rightsor permission to manufacture, use or sell anypatented invention that may in any way be relatedthereto.
ARPA ORJ(-) . ,f
.{M, 'T '4-ARPA
THE SCATTERING OF* "ELECTROMAGNETIC WAVES FROM
PLASMA CYLINDERS: PART II
Phyllis Greifinger
PREPARED FOR:
ADVANCED RESEARCH PROJECTS AGENCY
SANTA MONICA , CALIFORNIA-- --------
ARPA ORDER NO. 189-61
MEMORANDUM
RM-3574-ARPAJULY 1964
THE SCATTERING OFELECTROMAGNETIC WAVES FROM
PLASMA CYLINDERS: PART IIPhyllis Greifinger
fhis research is supported by the Advanced Research Projects Agency under ContractNo. SD-79. Any views or conclusions contained in this Memorandum should not beinterpreted as representing the official opinion or policy of ARPA.
DDC AVAILABILITY NOTICEQualified requesters may obtain copies of this report from the Defense DocumentationCenter (DDC).
1700 MAIN ST * SANTA MONICA * CALIFOmNIA * 9046
-iii-
PREFACE
This Memorandum is one result of a continuing investigation of
the radar returns from missile wakes. In a companion Memorandum,
RM-3573-ARPA, The Scattering of Electromagnetic Waves from Plasma
Cylinders: Part I, a general theory was developed for the short-
wavelength limit of the electromagnetic radiation by radially vary-
ing electron-density distributions. This Memorandum considers the
techniques for calculating the long-wavelength limit of the scatter-
ing.
The work is part of RAND's study of midcourse phenomenology,
conducted under the Advanced Research Projects Agency's Defender
program.
_v-
SUMMARY
In this Memorandum we consider the long-wavelength limit of the
scattering of electromagnetic radiation in the transverse magnetic
mode by infinite plasma cylinders with radially varying electron-
density distributions. Techniques for calculating the scattering
cross section are described for both increasing and decreasing
electron-density distributions. Resonance phenomena peculiar to
increasing electron-density distzibutions are discussed in some
detail with specific numerical examples.
Finally, we present a resume of methods one can use to calcu-
late the scattering cross section for radiation of arbitrary wave-
lengtb for monotonically decreasing eleatron-density distributions.
Examples are given for the Gaussian electron distribution.
-vii-
ACKNOWLEDGMENT
The author is deeply grateful to Jeannine McGann for programming
all the machine calculations and for preparing all the figures.
- ix-
CONTENTS
IREFACE ...................................................... iii
SUMMARY ...................................................... v
ACKNOWLEDGMENT ............................................... vii
SYMBOLS . ..................................................... xi
Section
I. INTRODUCTION ......................................... 1
11. REVIEW OF PARTIAL-WAVE EXPANSION ..................... 3
III. SMALL-K PHASE SHIFTS .................................. 8Finite-Radius Cylinder Solutions ................... 8Evaluation of 1 for Arbitrary Electron-Density
Distributions . . .. . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 10
IV. RESONANCE SCATTERING ................................. 21
V. CROSS SECTION FORI < 1 AND K < 1 .................. 31
VI. RESUHf ................................................ 37
-xi-
SYMBOLS
B - amplitude factor in wave function *oK(Y)
C - Euler's constant
Cn - coefficient of Lny in asymptotic form of Y n(y)
c - velocity of light
Dn - constant term in asymptotic form of Y n(y)
E - electric field
e - electronic charge
iFl(a; b; Z) = confluent hypergeometric function
2If(c)I = differential-scattering cross section
If B(9)2 =differential-scattering cross section in Bornapproximation
2A.G(p)l = geometrical-optics-scattering cross section
g(y) = electron-density distribution; defined by
n -ng(y) _ e ew
n - neo e_
J (Ky) - Bessel function regular at origin
K - kP0
p.k m .
c
m = integer
m - electron masse
N (Ky) - Neumann function
n - electron-number density
S(T) - angular distribution of scattered radiation
W (y) - wave function for m - 0 and 0K%
-xii-
Yn(y) - coefficient of on in power-series expansion of 0o(y)
y = P/P0W 2 . w2
CY= PO Pa2 W2
Po
= 0
[ 1
2. constant term in Wo (y)
9 - 1 when m = 0; = 2 when m 0mth
- phase shift of m partial wave
P - perpendicular distanc from cylinder axis
Po W characteristic radius of scattering region
cp - scattering angleth
*m(y) M m partial wave function
w - frequency of electromagnetic wave
W - plasma frequencyp
W P plasma frequency on cylinder axis (P - 0)P0
W = plasma frequency as p -
P.
-1-
I. INTRODUCTION
In a previous Memorandum, hereafter referred to as Part I) we
developed a general theory for the scattering of electromagnetic radia-
tion by radially varying electron-density distributions under the
conditions that the incident wave was a plane wave traveling perpen-
dicular to the cylinder axis and polarized parallel to the cylinder
axis, the electron-density distribution was monotonic, and collisions
were negligible. Techniques for calculating the cross section in
the short-wavelength limit were described and applied to a few
specific cases.
In this Memorandum we will consider the long-wavelength limit
of the scattering. The conditions of the scattering and, therefore,
the formulation of the problem are the same as in Part I. However,
the techniques appropriate for calculating the scattering cross
section differ in the long- and short-wavelength limits.
In our consideration of the long-wavelength limit, we will
place special emphasis on the scattering by monotonically increasing
electron-density distributions (analogous to the scattering by
attractive potentials in particle-scattering theory), since distri-
butions of this type have not been previously considered in the
literature except in the small perturbation limit. Such distribu-
tions can give rise to anomalies in the scattering not encountered
with decreasing electron-density distributions. These anomalies
are associated with partial-wave resonances, familiar in the
Phyllis Greifinger, The Scattering of Electromainetic Waves
From Plasma Cylinders: Part I, The RAND Corporation, RM-3573-ARPA,August 1963.
analogous situation of the scattering of low-energy particles from
spherically symmetric attractive potentials.
Despite the emphasis placed on the increasing electron-density
distributions, the methods described for calculating the cross
section in the long-wavelength limit will be equally applicable to
increasing and decreasing electron-density distributions.
-3-
II. REVIEW OF PARTIAL-WAVE EXPANSION
The formulation of the problem, the expansion of the wave into
partial waves, and the expression of the scattering cross section in
terms of the phase shifts of the partial waves were presented in some
detail in Part I. We will present a brief review here, with a few
changes in notation which are introduced for convenience in consider-
ing the long-wavelength limit of the scattering.
Dropping the time-dependent factor e iWt, the wave equation in
cylindrical coordinates p, Z, and qp for the electric field E - Ez is
+.1 l2E I P1EE
4rrn ew - = - plasma frequencyp e
Letting
y - P/Po
2 . (W2 2 ) 2=P2K - -0
22 ( W2 0
o" n
8(y) C n en -neo em
-4-
Eq. (1) becomes
Z) 2 E 1 LE+ Y - K 2 _g0(y E - 0 (2)
The characteristic radius of the scattering region is p0 and the
subacripts 0 and - on ne and W refer to p - 0 and P - , respectively.e p
The dimensionless parameters K and 0 are related to the wave number
k and the parameter O of Part I by
K - kP°
(3)
0=k2 P2 Y=K2 O~-k 2 poK0
The parameter 0 is not a function of the frequency of the radiation.
For monotonic electron-density distributions, it depends upon the
difference between the maximum and minimum electron densities as well
as on the size of the scattering region. It is a more convenient
parameter than of for small K. since it remains finite as K 0 0. The
small-K limit refers to j1 _(W iW)2( Wpic) < 1. Note that this
situation can occur even if Wpo/c is large, provided that W is not
much larger than W • For example, for the ionosphere, W is of the
order of 10+ 8 rad/sec (ne ~ 3 x 10 5/cm 3). Large negative values
of 0 can occur for electron-density "holes" such as the partially
evacuated region behind a hypersonic body moving through the ionosphere
if the body radius is larger than a few meters. For K to be less than
one under these conditions, we require radiation very close in fre-
quency to the ionospheric plasma frequency, i.e., of the order of
20 Mc/sec.
-5-
Expanding E in a Fourier series, we get
CO
E= Z €m cos (mp) *m(y)
6 = 1 m 0 (4)m
-2 m 0
The function 4m(y), which is related to urn(y) of Part I by
( ) Urn(y)my Kmy) satisfies the differential equation
* (y) .+y1(y F2 + [ g(y) - 1 m-(y) - 0 (5)m y m + 2 jm
with the boundary conditions
#m(O) is finite
*ms * m mi " constant xe
as y
The subscripts s and i refer to the scattered and incident wave,
respectively. If the asymptotic form of *m is written as
21/2 eimT/2- i co m'-0, !r' (6)" 2W- -2/4
where qm is the phase shift of the mth partial wave, the scattered
field E afor large y becomes
-6-
e i(Ky'Tn/4) m e 2 1%m e My
Es " 1 IKm -JO Im Cos m(p -e 1) f(C) _/Yo (7)
The differential-scattering cross section If(Cp)I 2 then is given by
2m. o W -21 m )12
If 0 Im Cos (W)(e - (8)
It was shown in Section IV of Part I that the smaller the K,
the fewer the phase shifts which contribute to the scattering. As
K - 0, we can generally expect all the phase shifts except 1o to be
negligible. The resultant scattering will be isotropic, with If(cp)I 2
given by
If( -)I 2_- sin2 qo (9)
As we will see, exceptions can arise when it is necessary to consider
one or more higher phase shifts, even as K - 0. The exceptions occur
when either Iqoj is an integral multiple of TT, causing sin qo to
vanish, or when one of the higher m partial waves is in resonance,
causing a phase shift which is an odd multiple of IT/2.
If K is mall but finite and we do not have a resonance situation
for m > 1, we need consider only qo and q 1" The differential cross
section If(T)12 becomes
If-o- in2 qo + 4 coo2 cp sin2 11 + 4 cos ( (sin2 q 0 sin 2 q,
(10)
+ sin qo cos qo sin N cos q 1)]
-7-
We can see from Eq. (10) that the ratio of the contribution of the
m - 1 term to the m - 0 term to the total cross section
2!
is proportional to sin 2 rI" However, the ratio of the largest angle-
dependent term to the constant term in the differential cross section
is proportional to sin 1"* Thus, the partial wave m - 1 generally
manifests itself in the differential cross section at a smaller value
of K than that at which it becomes significant in the total cross
section. For example, if % M 45 deg and 1 = 4.5 deg, the m - 1
wave contributes only about 2.4 per cent to the total scattering,
while it makes the forward scattering (rf - 0) about twice as large
as the backward scattering (cp - T).
-8-
III. SMALL-K PHASE SHIFTS
FINITE-RADIUS CYLINDER SOLUTIONS
Let us consider functions g(y) which vanish for y > 1. For
such cases, the Eq. (5) solutions inside and outside the cylinder
are matched at the boundary; for the T.M. mode, both m (y) and
*m(y) are continuous at y = 1. The phase shifts are then given
by
KN'(K) - ymN (K)
cot h~,Ym( ( ) - KJ'(K)
m (11)
The functions Jm (K) and Nm (K) are Bessel and Neumann functions, respec-
tively. The Ym's are found from the solutions to Eq. (5) inside the cylin-
der; for each m, the required solution is that which is finite at y - 0.
For K << 1, we have approximately
cot M i l2 2 Ke (12a)0 T o2
C - Euler's constant (12b)
-(m + ')N M(K) + KNm.(K) m (1ccoT, - (Y- " m).,(K) + KJ.() (1c)
cot . (m - ( m (12d)
-9-
We will discuss two cases for which we can obtain analytic expressions
for the YM'S8
Homogeneous Cylinder
Equation (5) can be solved analytically for a step-functioD
electron-density distribution (homogeneous cylinder):
g(y) 1 y 9 1
=0 y>l
The solutions are the Bessel functions Jm(P - 0 y) regular at the
origin. The Ym to are given by
r"7- ' o(4j _,)YrM 2 M (4X (13)
Using the recurrence relatienships of the Bessel functions, we
get for m 2 1
y+m J(4 T)
S(14)
so that an approximate expression for cot for K << 1 and m z 1 is
m!om -_ W, 2)2mCJ-l( .) €cot TIm a !6 1)! (K) M 1 1 ( .i (15)
For a given small K and P < K2 , there are maxima and minima for sin2_o an i iven
(m 2t 1) which occur at the zeroes of tmil
respectively.
-10-
Quadratic Distribution
Another electron-densitl distribution for which we can obtain an
analytic solution to Eq. (5) is the so-called quadratic distribution
8(y) - 1- y2 y S1
M0 y;l
The solution inside the cylinder for this case is
2
e y c --- " (16)
a + i (K2~pa a -2
c m+m
The function 1,(a; b; Z) is the confluent hypergeometric function
(a; b; Z) - 1+,! Z a(a + 1) Z 2b T. +b(b + 1) 3+" (17)
The Ym s can be found from the recursion formulas for the hyper-
geometric functions and are given by
14- + I + 2( _ _ 1)l1(m -+ 41) (18)Y M I n c m ? i1 ( a ; c a; " -
EVALUATIO( TO FIR ARBITRARY EBICTRUFDMnSITY DISTRIBUT(IqS
With certain exceptions to be discussed later, Sq. (9) repre-
sents a fair approximation to the mall-K cross section. It would
-11-
be especially useful, therefore, to develop a technique for calcu-
lating 1 0 for arbitrary functions g(y) (subject to the restriction
that g(y) falls off faster than 1/y2 for large y). We will do this
by analogy with a technique used for obtaining an approximation to
the s-wave phase shift for the scattering from spherically symmetric
scattering regions.
Let us rewrite Eq. (5) for m - 0 as follows
d ( *Kd
( *6 K K2 - g(y)) Y*., . 0 (19)
with boundary conditions
OK (0) finite
coas yy*oK(Y ) Y'_ 4 ) a a y -4c
The subscript K simply refers to the solution for a particular K.
The function * oo(y) would be the solution to Eq. (19) (finite at the
origin) with K - 0.
Let us also consider functions Wk(y) which have the same
asymptotic behavior as *OK(y) and which satisfy Eq. (19) with 0 - 0
d K 2
VK(y) -. *d(y) as y -. - (20)
-12-
Equation (28) can be solved immediately, and we get
WK(y) - B [cot 1oJo(Ky) + N( 0Ky)] for K 0 0
Wo(y) = ny+ (21)
The factor 9 is put in the function W (y) to cause yWo(y) and yW (y)1.r 0
to be equal at y = 0. The parameter y can only be found from thedW
solution of Eq. (19) with K = 0. Since y is equal to L -y at y
0
it is the same as y of Eq. (11) for K = 0 and for those functions
g(y) which vanish for y > 1. The amplitude B is as yet unspecified,
since we have not specified the value of *OK at y = 0. We will
choose B by requiring, for convenience, that *oK (0) - 1. From ther oK
differential Eq. (19), we then find that [*o(y)/y] - 0/2. Obtaining
numerical solutions of Eq. (19) with K - 0 is a straightforward
problem for functions g(y) which fall off faster than -L for large y.2
yFrom the solutions, we obtain the parameters B and Y as functions of
Let us now see how we can obtain cot 10 for arbitrary small K
from the function 00(y). Let us consider the following two equalities:
oKd ( d*O d*0 2y j oK y + Kyo* -0 (22a)
W 7d 6yKy d (y dy )0 oo
o dy I % - do + K.0 (22b)o dyV dy ) Kdy ' ) YoWK (22b
Subtracting Eq. (22b) from Eq. (22a), integrating over y from 0 to m,
and inserting the boundary conditions, we get
-13-
2 [2 2W.K JIWW.- W)] = I id (23) 10
= 2j (.K - 000OK)yy(3
From this exact equation, an approximate expression for cot no
correct to first order in K2 may be obtained by replacing WK by W °
and *oK by * 0 in the integral
oK 00
cot ' o -- n +-+ - K2A
A = [ n y + -' " (]y dy (24)
Equation (24) can be expected to break down as
Y - 0 (1 cot 101- O
For values of 0 which are such that we are near a zero of y, we
cannot replace the wave functions for finite K by those for K - 0,
since small changes in K have large effects on the wave functions.
The zeros of sin I for small K will cause the cross section
to exhibit deep minima. If 0 is negative, the smallest value of
101 for which sin no M 0 will be that which will cause the m - 0
wave to be pulled in by just half a cycle, giving a phase shift
nI of -rT as illustrated in the following sketch.
-14-
(Y (y)(Ky00
N,
y--
The effect is a diffraction of the wave around the cylinder,
in which the wave inside the cylinder is distorted in such a way
that it fits on smoothly to an undistorted wave outside. This par-
ticular phenomenon with K small is associated only with negative 0;
for positive 0, 1 is 17 (m - 0 wave pulled out by half a cycle)
only for fairly large K, so that the higher m phase shifts are not
negligible. Such minima in the cross section are well known in the
analogous situation of the scattering of low-energy particles from
spherically symmetric attractive potentials. In particular, an
extremely low minimum known as the Ramsauer-Townsend effect occurs
in the scattering cross section of low-energy electrons by rare-gas
atoms.
When we are not at such a minimum, we can see from Eq. (24)
that for finite 0, as K - 0
cot -i ) (25)
so that
-15-
2 - n (26)
For small but finite K, it is necessary to use the complete Eq. (24)to evaluate cot 10'
The parameter Y and the ii.tegral A were computed on the 124 70902
for a Gaussian distribution g(y) = e-y for a variety of values of1 2
positive and negative 0. In Fig. 1, we have shown - and - A versusY T
for the Gaussian distribution for positive 0. In Fig. 2, we have1
plotted I versus 0 for negative 0 for the Gaussian distribution,Y
homogeneous cylinder, and quadratic distribution for those values
of 0 for which 1-1 < 2. In Fig. 3, we have plotted versus 0 for
the homogeneous cylinder and Gaussian distribucion in the region in
which y -. 0. The integral A for the Gaussian distribution is shown
as a function of 0 for negative 0 in Fig. 4.
The approximate cot q0 , given by Eq. (24), and the exact cot 10
were calculated for a homogeneous cylinder for -= -4, -14.2, and
-14.6. For 0 = -4(1 = -0.194), it was found that the error involved
in the approximation was less than I per cent for K up to unity, and
the results were not plotted. For 0 - -14.2 and -14.6, lyl is small,
and one can expect the approximation given by Eq. (24) to break down
as K -. 1. In Fig. 5, we have plotted both the approximate and exact
values of cot 1o versus K for - -14.2 and -14.6. For - -14.2(j 5).
the approximation is surprisingly good for K near unity; the error
in cot Il is only 20 per cent for K - 1. For -1~4.6(14- 22.3)
the approximation breaks down badly as K -1 but is., nevertheless,
excellent for small K; for K - 0.5, the error is only 8.5 per cent.
Gaussian distribution
0
0.4 0.6 0,8 1.0 2 4 6 8 10 20
1 2Fig. 1-Curves of-7 and - A versus R3 for 83 positive
-17-
1.6
I I
1.2 _ _ __/
/l ///0.8 -1--
/ .0.4 / . - _ __,*
S ----- Gaussian distriboition0 / '- -- Homogeneous cylinder
0 . Quadratic distribution
-0.4
II,-0.8 /
/00I, /
-1.2I' /I-" __ _ _____/
-2.05 9 13 17 21 25
-Ff
Fig. 2-Curves of versus /0 for negative /3( FI < 2)
-18-
10080
60
40
Homogeneous cylinder
20-I10 /Gaussion distribution
1!
2o--- K
0.8
0.6
0.4 __
0.2
0.19 10 II 12 13 14 15 16
Fig. 3-Curves of 14-Iversus /3 for negative /3 near y --s-0
-19-
1000
600
400
200
100
80s0
40
20
10
6
4
2
0.80.6
0.2
0. _ __ _
0.080.06
0.04____Gaussian distribution
0.02 I
0.010 4 8 12 16 20 24 28
Fig. 4-Curves of - A versus A for negative/3
-20-
23 -
19
13
(CotCo 7'?O)APPRRx
1 ___1 - M T _
Hom oene0 oucyir (co 1 large
-21-
IV. RESONANCE SCATTERING
We can see from Eq. (11) for the finite-radius cylinder that'(K)
when Ym - K 1(KY' cot 'm will be zero. Whenever cot Im - 0, the
th mm partial wave is said to be in resonance.
Small-K resonances for m ; 1 cannot arise if 0 is positive. We
can easily prove this for finite-radius cylinders by showing thatNI(K)
for > 0, m > 1, and K < 1, y is always positive. Since K---- < 0m N m(K)
for small K, a resonance can only occur if Ym is negative.
Let us rewrite Eq. (5) as follows
_ y 'm) Y* m [g~y) + 2-- K (1Sdy[= 2 2](27)
Integrating both sides of Eq. (27) over y from 0 to 1, we get
d~m1 2r]d-dy1 * (y)[ g(y) + !- -K(28)
= ym4 (l)
If 0 > 0, m k 1, and K < 1, the quantity [og(y) + - K2 is positivedm y2
throughout the range of integration, y always has the same sign as
, and Y is positive.
On the other hand, if 0 is negative and sufficiently large in
magnitude, the quantity g(y) + - K2] may be negative for part
of the range of integration, and the solution *m(y) may be of thed*m
periodic type ( *m(y ) and y-y of opposite sign) for y near one, causing
Y to be negative. For a given g(y), m, and small K we can expect an
-22-
infinite sequence of values 0mn of negative 0 for which Ym will have
precisely the right negative value to cause Im - -(2n - 1) 2, wherem2
n is a positive integer. For example, for the homogeneous cylinder,
the 0 's are found from the roots of the equation Jm- O.thmn _ jV mn)'0
Considered as a function of m, the lowest root--K2 ml will increase
with m.
In a small-K resonance situation.for m ! 1, the scattering will
usually be the result of the interference between the wave in resonance
and the m - 0 wave. The differential cross section in the vicinity
of an m 1 i resonance will be approximately
2 2Po 2 22
If (C)2 -;F-[ s i n 1o + 4 cos ,mp sin2 1m
+ 4 cos cK(sin 2 1 sin 2 Im + sin 1 0sin % cos 110 cos 1)] (29)
The angular distribution S(cp) at the peak of the resonance will be
given by
a f(P) 1 2 sin2 T1 + 4 cos 2 W + 4 cos p sin 2 1(2 (30)
s Tj" if(p)l 2 sin q° + 20
Illustrations of resonant scattering are shown in Figs. 6 to 9.
In Fig. 6, we have shown lain T111 and Iam 101 as functions of K near
an m - 1 resonance for a homogeneous cylinder. In Fig. 7, we have
plotted the radar cross section If(TT) I 2 near the resonance in com-2P e 2
parison with the small-K cross section W- sin 10 in the absence of
an m ! 1 resonance. Figure 8 shows the angular distribution S(cp) at
the peak of this particular m - 1 resonance and compares it to 2 cos2 T,
-23-
the angular distribution of a pure m - 1 wave. In Fig. 9, we have
shown the angular distribution at the peak of an m - 1 resonance for
the quadratic distribution in comparison with the 2 cos2 cp distri-
bution. Figures 8 and 9 show the effects of the interference between
the m - 0 and m - 1 waves on the angular distribution. The distor-2
tion of the 2 cos 2p distribution by the m - 0 wave is more pro-
nounced in Fig. 9 than in Fig. 8 simply because Isin 1o f is largerin the former case.
In Fig. 10, we have shown lain .1, lain 1 and Jin 121
near an m - 2 resonance for the homogeneous cylinder. It happens
that this particular resonance occurs near a minimum of Ilin 1o1,
so that at the peak of this m - 2 resonance we have an almost pure
m - 2 wave. This result is not surprising. We can see from Eq. (15)
that for the homogeneous cylinder, cot 2 _ 0 when J1o2- )- 0.
However, from Eq. (12) we see that for small K, I cot 101 when
is very small in magnitude or when is near zero. We have
not plotted the angular distribution at the peak of the particular
m - 2 resonance being considered here, since it is essentially a pure
2 coon2 2cp distribution.
So far we have discussed only resonance maxima for the m 1 1
partial waves. We may also have small-K resonances for m - 0; they
occur in the vicinity of - - - In - (see Eq. (24)).. If K is verysmallin the vi-n0 r oance m l oCsmall, the m - 0 resonance maxima will occur for Y negative and fairly
-24-
small in magnitude; however, we have seen that the minima in sin 2 10
occur for I very small. Therefore, for a given small K, if we
consider sin 2 10 as a function of 0 for negative 0 in a region where
Y passes through zero, we can expect a zero and a maximum in sin2 10
to be fairly close together. We have shown this effect in Fig. 11 by
plotting sin 2 no versus 0 for a homogeneoue cylinder for K - 0.06 and
-0 between 13 and 16. We can see that a minimum occurs for -0 near
14.7 and a maximum for -0 near 15.4. For K smaller, the minimum and
maximum would be still closer together.
-25-
1.01
0.8 __ _ _ __ _ _ _
0.6
0.4 -
Is 7701
0.2jsin 77, 1
0.1
0.08
0.06
0.04
Homogeneous cylinder S=:- 5.76
0.02
0.01 1 __ _ _
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08K
Fig. 6- Curves of m - 0 and m a 1 phase shifts near an m *1resonance
-26-
100
80
60
40--_ _ _ _
20
I(7r32PO
10 Z 0
I0 ____-
8
Homogeneous cylinder
22 sin 8 5.76
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08K
Fig. 7- Effect of an m - 1 resonance on the radar cross section
-27-
2.4
2.2
2.0
1.8
1.6 Homogeneous cylinder
s(O)1.4
2 cos2
1.2
1.0
0.8
0.6
0.4
0.2
000 "r/8 7r4 3"w/B 7'l/2 3/B 3"7r4 7w/8 /
Fig. 8-Angular distribution at the peak of an m a 1 resonance(homogeneous cylinder; /3 - -5.76, K 0 0. 058)
-28-
2.4_ _
2.2
2.0-- - _ _ -_ _ _ _ _
1.6 Quadratic distribution -
1.0
0.8
0.6
0.4
0.2
0-
0 7r/8 7r/4 37r/ 8 7r/ 2 57r/ 8 377/4 7 7r/8 7
Fig. 9-Angular distribution at the peak of an m - 1 resonance(quadratic distribution; )9 -12, K - 0. 25)
-29-
1.0
0.8
0.6
0.4
0.2 Homogeneous cylinderJ B=_14.6
0.1
0.08
0.06 sin___
0.04 _ _-
/. Ji~n 7,1
0.02 "_ _
0.01 10.15 0.17 0.19 0.21 0.23 0.25 0.27 0.29
K
Fig. 10-Phase shifts near an m -2 resonance
-30-
1.0
0.8
0.6
0.4
Homogeneous cylinder
/ = - 14.6
0.2
sin2 00.1
0.08
0.06
0.04
0.02
0.0113 14 15 16
Fig. 11-Curve of sin2 77(o versus / for fixed K in regionwhere y passes through zero
-31-
V. CROSS SECTION FOR I I < 1 AND K < 1
Since 1I1 and K are both proportional to Pop in many situations
in which K is small, 101 will also be less than one. It would be
useful, therefore, to develop a simple method for calculating the
cross section for 101 < 1 and K < 1.
We know that for K finite, we can use the Born approximation to
estimate the cross section or the individual phase shifts for 101 - 0.
For example, the Born approximation for 1o 4 1 ) is given by
2o -2 0 U yg(y)J (Ky) dy (31)0
If we let K go to zero, this gives
0 IT yg(y) dy (32)0
2 1However, we have seen that for 0 finite, if we let K -0 0, cot qo "*-9 n 1,
0 r K'
so that
- (33)
K
Therefore, for K -. 0, we cannot use Eq. (32) for finite 0, even for
101 small. We need a simple approximation to 1 for K < 1 and
101 < 1, which reduces to Eq. (32) for K finite and I -. 0, and to
Eq. (33) for 101 finite and K -. 0.
Since Eq. (24) represents a good approximation to 1o for small
K and all finite values of 101 (except at the cross-sectional minima),
we can obtain the approximation we are seeking by obtaining a power
-32-
series expansion in 0 of 1/y and A. This can be done by expanding
*00 (y) in a power series in 0
00O
(Y) = n Y Y) (34)
From the differential equation for * 00, we get the equations for
the functions Y n(y)
d (dY) o (35)
d (y dYny . -y ) = yg(y)Ynil(y) n 1 1
Yo(0) - 1
Yn(0) = 0 n 0
Y' (0) = 0n
The solutions then are
Y o(y) = 1
0 0
n ' n-n~ 1
We can see from Eq. (36) that for n > 1
Yn(y) - Cn In y + Dn as y- (37)
-33-
where
C.= y'g(y')Yn.l(y') dy'0
D n ' y'g(y') In y' Yn-l(y') dy'0
Since *o(y) - B (In y" + -) as y - =, we have
SB J1 n nCn (38)TT n= n
M2 1+ E n= D
Try n n
1 1+ - n
We see that O/Y - L as 0 -. 0. The next approximation yields
C 1
1 1 +0( L2(39)Y O 1 11 1D C 1)
The integral A in Eq. (24) is proportional to - for small B, and the
leading term is
A 1 2 r y3g(y) dy (40)2o 1
-34-
Substituting Eqs. (39) and (40) into Eq. (24), we get
cot o- nK + I + 0 I C
0 LT Ke l\ R/
+ _2 2 j y3g(y) dy (41)2 1 0 J
If we let 0 remain fixed, then as K - 0, Eq. (41) reduces to Eq. (33).
For K small but finite, as 0 -. 0, Eq. (41) becomes
T10o 1 i C1 _ 2 y3g (y) dy) = 0 yg(y) dy - L y 3 g(y) da 2l 2C1 2 LoT 20 J]
This is the s. ixe as Eq. (31) if we expand J 2(Ky) and keep the first0
two terms.
Thus, we can expect Eq. (41) to represent a good approximation
to 1o for K < l and el < 1 for those functions g(y) which fall off
rapidly enough at large distances to insure convergence of the inte-
grals.
Let us note from Eq. (41) that for negative P we can have an
m = 0 resonance even for I I small. For a resonance to occur for
10P1 < 1, K is given by
=AL + (D - (42)
In Fig. 12, we have shown the correct radar cross section for
K - 0.06 for a homogeneous cylinder for small P and compared it to
the Born approximation. The peak at -0 - 0.63 is due to an m - 0
-35-
resonance, and it can be seen that the Born approximation is very
poor near this resonance. The fact that for the particular case
considered the Born approximation cross section happens to be good
at 0 = -1 is fortuitous.
-36-
100__ _ _
80
60
40-_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
20Homogeneous cylinder
I008
6
2 -10 If(7r
0.8
0.6
0.4
0.2
0.1 _ _ _
0.08
0.06
0.04
0.02
0.0110 0.2 0.4 0.6 0.8 1.0
Fig. 12- A comparison of the Born approximation with the correctradar cross section for small 1,81 and K
-37-
VI. RiSMd
In Part I, we described the Born approximation and the geometrical-
optics approximation to the scattering cross section and discussed
their limits of applicability. We also described methods for approxi-
mating the individual phase shifts for those values of large K for
which the Born or geometrical-optics approximations are not applicable.
In this Memorandum we have confined ourselves to a consideration of
the cross section for K < 1.
The following briefly summarizes the methods one can use to
calculate the cross section for different K values for 101 < 1 and
0 > 1. Since we did not consider specific methods for calculating
the cross section for 8 < -1 and K > 1 in either Part I or Part II
(except in the Born-approximation limit), we uill not include 0 < -1
in this summary.
101 < 1
For small 101 and K < 1, we may use Eq. (24) for the evaluation
of cot 1 and the Born approximation to the higher phase shifts.0
For larger values of K, the Born approximation to the differential
cross section is applicable. Thus, for 101 < 1, we can get a good
estimate of the cross section for all K.
As an eiample, in Fig. 13, we have shown the radar cross section
as a function of K for the Gaussian distribution for - k. The cross
section was computed from Eqs. (9) and (24) for K I , and from the
Born approximation for K . For K - , the two methods differed
by only 10 per cent.
10 -38-
1.0 -'N
Gaussian distribution
z3 =.5
0.1
PO.O
0.01-
0.001 -- - - - -- - - -
0.0001 -IiL
0.01 0.02 0.06 0.1 0.2 0.4 0.6 0.81 2 4 6 8130K
Fig. 13- Radar cross section as a function of Kfor fixed small positive S3
-- 39-
8>1
For 8 > 1, we can use Eq. (24) to calculate cot 1o for K < 1
and neglect the other phase shifts. For K > $, we can use the Born
approximation to the differential cross section. If $ is large, then
the geometrical-optics approximation can be applied for I < K2 < 8.
For all other values of K > 1, we can calculate the individual phase
shifts by the WKB approximation when the phase shifts are large, and
by the small-perturbation (Born) approximation when the phase shifts
are small.
As an example, we have calculated the radar cross section as a
function of K for the Gaussian distribution for 8 - 18. We used
Eqs. (9) and (24) for K < 1 and the WKB method for K : 1. The results
are shown in Fig. 14. In the same diagram, we have also plotted the
geometrical optics radar cross section for S K ! 4. This very useful
approximation is apparently valid for K as small as unity, provided
that 8/K2 is sufficiently large.
-40-
1.0
0.8
0.6
0.4Giussian distribution
0.2 1f
0.1 2 %P0 .0 8 I O N I G
0.06 PO
0.04
0.02
0.01
0.008
0.006
0.004
0.002
0.0010 2 3 4 5
K
Fig. 14-Radar cross section as a function of Kfor fixed large positive /3
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