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A technical report on the radiation due to an oscillating dipole over a lossless semi-infinite moving dielectric medium. It covers topics such as electrodynamics of moving media, reflection and refraction of a plane electromagnetic wave, and oscillating dipole over a moving dielectric medium. The report includes mathematical equations, diagrams, and numerical results. The report was prepared for the National Aeronautics and Space Administration and was administered through the Office of Research Administration at the University of Michigan.
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THEUNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING Radiation L abototory
Rudiution Due to an Oscillating Dipole Over u Lossless Semi-lnfinite Moving Dielectric Medium
b y
r- r[. s
February 1966 -
Grant NGR-23-005-
National Aeronautics and Space Administration Langley Research Center Langley Station Hampton, Virginia 23365
Contract M t h :
O F F I C E O F R E S E A R C H A D M I N I S T R A T I O N A N N A R B O R
b r'^ T H E^ U N I V E R S I T Y7322-2-T^ O F^ , ~ I C H I G A N
I *
. (^) by Vittal P. Pyati
February 1966
Report No. 7322-2-T on
Prepared for NATIONAL AERONAUTICS AND SPACE ADMINISTRATION NASA-LANGLEY RESEARCH CENTER LANGLEY STATION HAMPTON, VIRGINIA
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T H E U N I V E R S I T Y d~ M I C H I G A N 7322-2-T
The author is indebted to Professor Chen-To Tai for suggesting the problem and invaluable guidance and encouragement throughout the inves - tigation. Thanks are also due to the members of the Thesis committee for their helpful suggestions. A portion of this investigation w a s conducted while the author was with the Radio Astronomy Observatory of The Univer- sity of Michigan (NsG-572) and he is grateful to the Director, Professor Fred T. Haddock, for his support. The author wishes to thank Professor
F. White and M i s s Madelyn L. Hudkins for typing the manuscript.
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T H E U N I V E R S I T Y O F M I C H I G A N
A CKNQWLEDGEMENT LIST OF ILLUSTRATmNS CHAPTER I I1 ELECTRODYNAMICS O F MOVING MEDIA
2.1 The Lorentz Transformation 2.2 Maxwell-Minkowski Equations
2. 3 The Method of Potentials for Moving Media III REFLECTION AND REFRACTION OF A PLANE ELECTRO- MAGNETIC WAVE AT THE BOUNDARY OF A MOVING DIELECTRIC MEDIUM 3.1 Geometry of the Problem 3.2 Plane Waves in Moving Media 3. 3 The Modified Snell's Law 3.4 Electric Field Perpendicular to the Plane of Incidence 3.5 Electric Field Parallel to the Plane of Incidence 3.6 Perpendicular Incidence 3.7 Summary OSCILLATING DIPOLE OVER A MOVING DIELECTRIC MEDIUM 4.1 Introduction 4.2 Vertical Dipole
4.2.1 Fourier Integral Method 4.2.2 Method of Weyl 4.2.3 Approximation of the Integrals; Asymptotic Forms 4.2.4 Numerical Results
4. 3.1 Fourier Integral Method 4.3.2 Approximation of the Integrals; Asymptotic Forms 4.3.3 Numerical Results
4.3 Horizontal Dipole in the Direction of the Velocity
Page ii iii V 1 3 3 4 6 9 9 9
40 40 40 40 49 55 77 84 84 87 93 101 103
105 108
iv
J
Fig. No. Page
IN THE AIR IN THE Y Z PLANE FOR A VERTICAL DIPOLE R n = 2 , h=O.5X 80
b A0 97
vi
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r' (^8)
l
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T H E U N I V E R S I T Y O F M I C H I G A N 7322-2-T
CHAPTER I INTRODUCTION AND STATEMENT OF THE PROBLEM
Sayiedl have presented an alternate derivation of Frank and Tamm's2 formula
and one or more moving dielectric media have been considered by Sayied3,
dipole in miform motion, the complementary problem, in which the medium is
dently solved by Tai^6^ and^ Lee^ and^ Papas^7^.^ The present work^ is^ concerned with the following boundary value problem.
4
in whose frame of reference all the fields will be determined. The problem may be regarded as an extension of Sommerfeld's8 dipole problem to moving media. The object of this study is, first to develop techniques of formulation of boundary value problems in moving media, and then to apply these techniques to the above problem to ascertain the extent to which the radiation patterns are
It may be recalled that Weyl' developed a method by which Sommerfeld's solution for a dipole over a flat earth could be interpreted as a b-e of plane waves reflected and refracted by the earth at various angles of incidence. In
s a r y that we extend F r d l ' s results to moving me'dia, nameky:
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i^ #
W '
T H E U N I V E R S I T Y O F M I C H I G A N 7322-2-T
Z'
V
V
Z'
from one system into another are given by
y' = y(y-vt) y = y(y'+vt')
The above is known as the Lorentz or the Lorentz-Minkowski transformation. The various constants appearing above are given by
p, = permeability of free space = 4 ~ x 1 0 - ~henries/m
p = v/c.
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T H E U N I V E R S I T Y O F M I C H I G A N 7322-2-T
according to the theory of relativity, Maxwell equations must have the same form in all inertial frames of reference, that is, they must be covariant under the Lorentz transformation (2.1). Therefore in the unprimed o r laboratory system, we have aB V X E = - --at a s V x H = J + -- - (^) at (2.3)
and by attaching primes, we get Maxwell equations in the primed system, for instance, (2.2) becomes
It may be noted that the divergence equations follow from the curl equations and the equation of continuity; hence do not yield any new information. To formulate a problem completely the constitutive relations must be known. These can be derived in the following manner. In the primed system where an observer is at rest with the medium, we have
of this type of formulation a modified version of Nag and Sayied's method for Cerenkov radiation is presented in Appendix A.
we get
V x - H 1 = - i w E a,. - El+e i w R y J- Now introduce the vector potential A1 such that p q. l l l =^ vx^ a-1 - -^ * A or
(2. 17) (2. 18)
.
. (2.19) (2.20)
(2.22) (2.23)
Substituting in (2.22), we obtain
-1E^ = i w a - l .= A-1^ - V g^ 1 (2.25) where $d1 is the scalar potential. Substituting for (^) - El and H -1 in (2.23), we obtain
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T H E U N I V E R S I T Y O F M I C H I G A N 7322-2-T
dinates thuf
va=x-+y^ n a^ ~a -^ +z- h^ a
Now define the gauge condition V. A = -ik2a -1 w I, so that (2.26) becomes
G which satisfies the equation
where q refers to source point.
Case 1: a > o o r v < c/n The solution can be obtained by dimensional scaling a@ .ika@Rl
where R1, the modified distance is given by 2
~ a s e 2 :a < O o r v > c / n In this case, we have a two-dimensional Klein-Gordon equation and the
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T H E U N I V E R S I T Y O F M I C H I G A N 7322-2-T
3. 1 Geometrv of the (^) ~~ Problem~ A s shown in Fig. 2, the region z < 0 is filled by a medium, with a per- meability p , and a permittivity E, moving uniformly in the y-direction with a velocity v. The region z > 0 is free space bo, eo) and is stationary. A plane electromagnetic wave traveling in free space in an arbitrary direc- tion is incident upon the interface; as a result there will be a reflected wave and a transmitted wave. azimuthal angles being measured from the x-axis.
Let the orientation of the three waves be as in Fig. 2 , the
The Maxwell equations in the absence of sources and
VxE=- i w g ( 3. 1 ) VxH_ = -iwD_ (3.2) V * B = O ,- V * D = O (3.3) Making use of the constitutive relations (2.15) and (2.16) we obtain
The plane wave solutions which we a r e seeking can be represented thus i K K i K K
E = E e H = H e
(3.4) (3.5)
(3.6) (3.7) where the first factor denotes complex amplitude, K the propagation constant and
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T H E U N I V E R S I T Y O F M I C H I G A N 7322-2-T
Reflected
Transmitted
FIG. 2: PLANE WAVE INCIDENCE ON A MOVING MEDIUM
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Eliminating one variable at a time between (3.14) and (3.15), we find that gf and H_" satisfy the modified vector wave equation
(3.17)
find and H_" satisfy
(Va.V) H_"+ak2 HIf - = 0
These equations separate into three scalar equations in rectangular coordinates of the form
[i2+: 5 +-+A? 2 I+!/=o
given by i(klx +$y -k3z) e provided kl, %, k3 (^) rl satisfy the characteristic equation 2 k kl t +^ k i -ak2^ =^0
ns of the above a r e
with either the electric or the magnetic field. Thus setting
we get
b
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T H E U N I V E R S I T Y O F M I C H I G A N 7322-2-T
The amplitudes in (3.23) must satisfy the relation (3.16), so that klE" ox +$E'' OY-k3E" 02 = 0. Comparing (3.24) with (3.9), we get from the phase functions
K COS et = kg
(3.26a) (3.26b) (3. substituting for kl, %, k3 in (3.22) we get the following dispersion relation for K
K 2 sin 2 et (^) , a 1 (K sinet singt+ua) 2 + K 2 cos 2 et-ak2 = o (3.27)
where CY is the angle between the direction of propagation and the velocity of the
page 231, Eq. 61 ). We also note that the amplitudes in (3.9) will have to satis- fy the following relation
K sinOtcOS$tE (^) OX +- a^1 (KshOtsingt+wQ)EOY -KCOSOtE oz = O (3.29)
The magnetic field H- can be obtained from (3.4). Making use of (3.27) and (3.29), one can show that the H- field thus obtained satisfies (3.5). From this it follows that the divergence relations in (3.3) are also satisfied. We conclude this section by summarizing the method of construction of plane wave solutions in moving media.