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NEW YORK UNIVERSITY /

WASHINGTON SQUARE COLLEGE OF ARTS AND SCIENCE
MATHEMATICS RESEARCH GROUP
RESEARCH REPORT No. EM-26



NEW YORK UNIVERSITY;
gOURANT INST2TUTE a LIBRARY



ON THE SCATTERED REFLECTION OF
ELECTROMAGNETIC WAVES

by
VICTOR TWERSKY



The research reported in this document has been made
possible through support and sponsorship extended by the
GEOPHYSICAL RESEARCH DIRECTORATE of the Air
Force Cambridge Research Laboratories, AMC, under Con-
tract No. AF-19(122)-42. It is published for technical
information only and does not represent recommendations
or conclusions of the sponsoring agency.

JANUARY, 1951



New York University
Washington Square College of Arts and Science
Mathematics Research Group
Research Report No. BM-26



ON THE SCATTERED REFLECTION OF ELECTROMAGNETIC WAVES

by
Victor Tv/ereky



Written by
Victor Tversky//'




Morris Kline
Project Director



Title page

82 numbered pages

January, 19 $1



CONTENTS

Article Page

Abstract 1

Notation 2

1, Introduction 5

2, Formulation of the Single Boss Problem and Some Preliminary 7
Consideration*

3, The Seraicylindrical Boss 10

A. The Single Boss 10

B. Distributions of Bosses 23

C. Distributions of Cylinders 33
U. The Hemispherical Boss 36

A. The Single Boss 36

B. Distributions of Bosses 51

C. Distributions of Spheres 6l
5« Discussion and Comparisons 65
Appendix: Extensions to Absorbent Booses 81



-1-

ABSTRACT

The non-specular reflection of plane electromagnetic waves of arbi-
trary polarization by certain perfectly conducting surfaces composed of
either semi cylindrical or hemispherical bosses on an infinite plane is
analyzed. Solutions in terms of eigenfunctions for the problem of the single
boss on an infinite plane and a plane wave at an arbitrary angle of incidence
are derived and extended, subject to the 6ingle scattering hypothesis, to ob-
tain the far field solutions for certain small finite patterned distributions
and both small finite and infinite uniform random distributions of bosses
small compared to the wavelength. The results for the various cases are
then compared in the plane of incidence and similarities between the analogous
expressions for the distributions of semicylinders and hemispheres ere noted.
Expressions are obtained for the ratios of the reflected intensities and
radial energy flux polarized parallel and perpendicular to the plane of inci-
dence's well as for the total intensity and radial energy flux for the
case where the incident wave is unpolarized. It is found that for certain
values of the parameters the reflected radiation may consist only of either
the specular or the scattered contributions, while for other values of the
parameters one of the scattered contributions, either parallel or perpendi-
cular component may vanish. The results also indicate the occurrence of an
extremum in the reflected radiation in the vicinity of the specular angle of
reflection which for certain ranges of the parameters for the small finite
distributions may be a minimum rather than a maximum. For these cases there
is also some critical angle of incidence (not necessarily Tf/2 or grazing
incidence) for which the reflection at the 6pecular angle is completely
specular.

The analogous distributions of cylinders and spheres are also con-
sidered.



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NOTITIOH

E, H, - The electric and magnetic field intensities which comprise the electro-
magnetic field .

i - Superscript or subscript character! ?ing the Incident wave,

r - Superscript or subscript character! 7ing the total reflected wave.

p - Superscript or subscript characterizing the specularly reflected rave components,

c, 8 - Superscript or subscript characterizing the scattered components of the total
reflected wave for the (semi) cylindrical and (hemi) spherical cases re-
spectively.

A, B,- Arbitrary constants (determined from proscribed initial conditions) that
specify the polarization parallel and perpendicular to the plane of
incidence.

II', II" - Vector Hertz potentials of the electric and magnetic type respectively.

y^yw _ scalar components of the Hertz potentials,

K = Kn = (27T/71 )n = (ou/cjfi" - Propagation vector (the symbol^ denotes the

unit vector) .

o(,Q , - Angles of incidence,

i,i,k - Unit rectangular coordinate vectors,

R = r(cos 9 i + sin 8 j)+ zk - Position vector in cylindrical coordinates .

A 1 , B* - Constants introduced for brevity.

a - Radius of the boss.

£ n - Neumann's factor: € n - 1 if n-0, £ n = 2ifn>0

J - Bessel functions ,
n

B_ - Hankel functions of the first kind,

Z n - Derivatives of the cylindrical functions with respect to their arguments.

A , A - Cylindrical scattering coefficients,

o - Left superscript denoting the approximate solution (Kr » 1, Ka < 1) for the
single boss at the origin of the reference coordinates.

Q_ (or K) , c i - Abbreviations for the amplitude and phase of the scattered com-
ponents.

F - Either E or H



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a - Summation index ranging from -N to + N ,
?!) - Total number of bosses in e distribution,

/ - Fraunhofer phase factor ,

j - Subscript labeling the superposed distributions.
/?? , - Factor defined so that 'V . - for the particular value associated
with the bosses of largest radii and *y . - 1 for all other values.

M - Density function eoual to the number of bosses per square centimeter •

d - Linear extent of the small finite distributions •

C, D - Constant coordinates of a field point for the infinite distributions,

V - Volumetric departure from the plane equal to the total volume of bosses
per square centimeter of distribution.

r = r (sin 9 cost^ i t sin 6 eir.tf J + cos 9 k)- Position vector in spheri-
cal coordinates.



- Spherical Bessel functions - Morse's notation ,



D , etc. - Partial derivative with respect to r, etc.
r

n nm' G nm» ^ P nm» P nm * Symbols introduced for brevity*

P (cos 9) - Derivatives of the Associated Legendre functions with respect to 9,

X n 5 m , 2£ n^-m - Sums over n and m such that n and m have odd or even parity
respectively^

a , a - Spherical scattering coefficients .
n n

y - Angle made by a line of boeses with the x-axis in the xy-plene: also
used to label a particular line of bosses.




O ,u - Polar coordinates in xy-plane

Angle made by the axes of a recta

the xy-axes: also used to label a particular rectangle.



S - Angle made by the axes of a rectangular distribution of spheres with



-u-



u - "Reflection coefficient" for the infinite distributions,

Y - Phase factor of u .

A - function - A function such as sin P /P or sin OOP /sinP which has a
single principle maximum or a series of such maxima and
which reduces to*?7 at the specular angle of reflection ,

J s Re(E«E ) - Quantity proportional to the intensity



S - Padial component of the complex Poynting vector,

. < 1). Although the functions employed
are tabulated and in principal the initially derived expressions are valid
for all values of Ka, it is only for Ka< 1 that the serieB are sufficiently
convergent to be approximated by their first few terms. The expressions for
the distributions derived subject to this limitation sre therefore only ap-
propriate for the diffraction range of phenomena (or the "Rayleigh scattering"
region) as might occur with surfaces whose irregularities are small compered
to the wavelength. In this region the distributions of bosses on a plane
present features characteristic of either striated or rough surfaces and

should be useful in investigating some of the general aspects of non-specular

2

reflection.

The restriction of Ka< 1 of necessity precludes the discussion of a
wide variety of phenomena. In particular it excludes the optical range
where there is reason for believing that reflection is governed by phenomena

arising because the usual surface irregularities are large compered to wave-

2. As was mentioned in the previous report, the solution for the simple semi-
cylindrical boss had been obtained by Rayleigh, reference 6. It is in-
teresting to note that another approach to such problems as reflection
from a rough surface introduced by Rayleigh, "Theory of Sound", Section
2728, Dover, N.T., (1945) — who considered the problem of a plane wave
incident on a corrugated surface whose cross-section could be repre-
sented by a Fourier series — is being developed by C. T. Tai, "Reflection
and Refraction of a Plane Electromagnetic Wave at a Periodical Surface",
Harvard Tech. Report No. 28, Cruft Lab., (Jan. 194-8) and S. 0. Rice, "Wave
Propagation Over Rough Surfaces", paper delivered nt the Symposium on
the Theory of Electromagnetic Waves, New York University, June, 1950.



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length — essentially the region for which the empirical Lambert '3 Law of
diffuse reflection is considered valid. A certain amount of theoretical
work has been done in this range, particularly in an effort to derive
Lambert's Law which is believed to be a statistical effect of specular re-
flections occurring at the irregularities themselves.- 5 This phase
of the problem will perhaps be investigated by the techniques of geometrical
optics in a future report.

The region Ka


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