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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