Table Of ContentOTHER TITLES IN THE SERIES IN
ELECTROMAGNETIC WAVES
Vol. 1 FOCK-Electromagnetic Diffraction and Propagation Problems
Vol. 2 SMITH and MATSUSHITA - Ionospheric Sporadic-Is
Vol. 3 WAIT-Electromagnetic Waves in Stratified Media
Vol. 4 BECKMANN and SPIZZICHINO - The Scattering of Electromagnetic Waves from Rough Surfaces
Vol. 5 KERKER - Electromagnetic Scattering
Vol. 6 JORDAN - Electromagnetic Theory and Antennas
Vol. 7 GINZBURG - The Propagation of Electromagnetic Waves in Plasmas
Vol. 8 Du CASTEL-Tropospheric Radiowave Propagation beyond the Horizon
Vol. 9 BANOS-Dipole Radiation in the Presence of a Conducting Half-space
Vol. 10 KELLER and FRISCHKNECHT - Electrical Methods in Geophysical Prospecting
Vol. 11 BROWN-Electromagnetic Wave Theory
Vol. 12 CLEMMOW-The Plane Wave Spectrum Representation of Electromagnetic Fields
Vol. 13 KERNS and BEATTY-Basic Theory of Waveguide Junctions and Introductory Microwave
Network Analysis
Vol. 14 WATT-VLF Radio Engineering
Vol. 15 GALEJS: Antennas in In homogeneous Media
TERRESTRIAL PROPAGATION OF
LONG ELECTROMAGNETIC WAVES
BY
JANIS GALEJS
Naval Research Laboratory,
Communication Sciences Division, Washington
PERGAMON PRESS
OXFORD · NEW YORK · TORONTO
SYDNEY · BRAUNSCHWEIG
Pergamon Press Ltd., Headington Hill Hall, Oxford
Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford,
New York 10523
Pergamon of Canada Ltd., 207 Queen's Quay West, Toronto 1
Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street,
Rushcutters Bay, N.S.W. 2011, Australia
Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig
Copyright © 1972 J. Galejs
All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photocopying,
recording or otherwise, without the prior permission of
Pergamon Press Ltd.
First edition 1972
Library of Congress Catalog Card No. 70-175513
Printed in Hungary
08 0167101
ACKNOWLEDGMENTS
THIS monograph reflects to a large extent the author's work at the former Applied Research
Laboratory of Sylvania Electronic Systems. The author is particularly indebted to Dr. J. E. Storer,
Director of the Laboratory, for his support. This work was also made possible by a long-range
interest of the U.S. Navy. Dr. A. Shostak of the Office of Naval Research, Mr. J. Tennyson
and Mr. J. Merrill of the Navy Underwater Sound Laboratory, and Mr. J. E. Don Carlos of the
Naval Electronics Systems Command have encouraged the investigation of several topics.
Professor K. G. Budden of the University of Cambridge and Dr. J. R. Wait of the Environ
mental Science Services Administration provided a number of helpful comments. The author did
benefit from several discussions with Professor T. R. Madden of Massachusetts Institute of
Technology and Dr. E. T. Pierce of Stanford Research Institute. Also acknowledged are
comments by Professor K. G. Booker of University of California at San Diego.
LIST OF PRINCIPAL SYMBOLS
a = radius of the earth = 20000/π (km)
[a ] = solution matrix in (92) of Ch. 5
n
A, B = amplitudes of T.M. waves
e e
A, B = amplitudes of T.E. waves
h h
A = amplitudes of magneto-ionic modes in (181) to (184) of Ch. 5 or (102) of Ch. 6
jn
Ai{f\ Bi(i) = Airy functions in (164) of Ch. 4
Bo = induction of the geomagnetic field
c = free space velocity of E.M. waves = 3X10 m/sec
8
[c ] = solution matrix in (102) of Ch. 6
n
C = received signal due to a median return stroke in Section 3.3.3
C = cos Θ = y/\ — S = cosine of the wave incidence angle
2
Cj = normalized radial wave numbers defined by (154) or (179) of Ch. 4
C = characteristic function in Section 3.4
k
[CJ = column matrix of coefficients defined by (91) or (94) o fCh. 5
[d„] = matrix product defined by (98) of Ch. 5 or (105) of Ch. 6
D = αθ = distance along the surface of the earth
D = α(π— Θ) = ab = distance from antipode
a
D = interference distance of modes i and j in (2) of Ch. 8
i}
Φ = differential operator defined by (59) of Ch. 4
Dj — down-coming waves in Section 1.5
e = charge of a particle
k
Ej = y-component of electric field
E.W. = east to west
/ = frequency in c/s (or Hz)
f = resonance frequency
n
F{w) = attenuation function in (34) of Ch. 9
F F = functions defined by (105), (112) of Ch. 4
n9 m
F , F = functions defined by (208), (209), (219) of Ch. 4
/M im
g(ico), G{ia>) = power spectrum
G (y) or G (z) = height-gain function defined in Section 1.5
q
GJ {y) = height-gain function defined by (272) or (275) of Ch. 5
q
h(t) = impulse response in Section 3.3.3
h = height of the ionospheric boundary
h = upper boundary of ionospheric stratifications
u
he, hm (superscripts) = horizontal electric, magnetic
xii List of Principal Symbols
A(,"°(w) = spherical Bessel function defined by (15) of Ch. 4
hev(u\ h\(u) = radial functions of T.M., T.E. modes defined by (31), (40) of Ch. 4
Hj = /-component of the magnetic field
Hlm\y) or H^\z) = Hankel function of kind m and order v
/, I = current
I = dip angle in Section 7.3.3
Ids = electric dipole moment (Am)
Im S = imaginary part of S
/*, Ihv = integrals defined by (36) of Ch. 4
ie = density of electric current (A/m2)
im = density of magnetic current (V/m2)
i^ = unit vector in x direction
Jn(x) = Bessel functions of the first kind of order n
Jse = electric surface current (A/m)
Jsm = magnetic surface current (V/m)
j (superscript or subscript) = e or h
Kn(z) = modified Bessel function of second kind and order n
k = wave number
k0 = ω Λ/μοεο = wave number of free space
kr = k/k0
K = Boltzmann's constant (1.38X10-16 erg per °K); radial wave number of the exponential
or thin-shell approximations in Sections 4.3 and 5.3; magnetic current (V); aja in Ch. 9
Kds = magnetic dipole moment (V m)
m, M = electron and ion mass
m = index of T.E. modes
M.W. = molecular weight
n = index of T.M. modes
n = unit vector normal to an interface
n = ^/i = refractive index
Nj — particle density per cm3
[pn] = matrix product in (106) of Ch. 6
P = probability
Pj = power flow in the/-direction
Pn(cos Θ) = Legendre polynomial
P^iz) = Legendre function of first kind, degree v, and order μ
q (subscript) = n or m
4j> VgJ ~ normalized impedance or admittance defined by (176), (188) and (189) of Ch. 4 or
following (279) of Ch. 5
q9 = matrix elements in (181) to (184) of Ch. 5 or (109) to (112) of Ch. 6
Q = quality factors defined in Section 7.4.2
r, 0, φ = spherical coordinates
r, φ, z = cylindrical coordinates
Re = „i?,, = reflection coefficient of T.M. waves
R = reflection coefficient of ground surface
List of Principal Symbols xiii
R = R = reflection coefficient of T.E. waves
h ± ±
R = reflection coefficient of ionosphere boundary
(
„i? , i?„ = reflection coefficients of Section 6.8.1
± ±
Re S = real part of S
S = sin Θ = -y/1 — C = sine of the wave incidence angle
2
S = propagation parameter; Re S — c/v Im (5) ~ a
ph9
5' = (a/r)S
[S (r)] = column matrix of tangential field components defined by (90) or (93) o fCh. 5
n
t ~ argument of Airy functions defined by (161) of Ch. 4
t t = t for r = a, a+h
g9 {
T.E. = transverse electric (subscript h)
T.M. = transverse magnetic (subscript e)
T = temperature in degrees (Kelvin)
T = period, time interval or integration time
u = k r
0
u = k a u = k (a+h)
g 0 9 t 0
u(t), v(f) = Airy functions in (164) of Ch. 4
Uj = upgoing waves in Section 1.5
Up Vj = scalar functions in Sections in 7.4.4 and 7.5
U {u\ V (u) = spherical functions defined by (43) of Ch. 7
n n
v = velocity
ve, vm (superscripts) = vertical electric, magnetic
ph phase velocity
v =
V = voltage or potential difference
w = Sommerfeld's numerical distance in (37) of Ch. 9
w(r), W(r) = probability densities
Ί,2(0 = Airy functions in (162) to (164) of Ch. 4
Μ
W(u v) = Wronskian defined by (151) of Ch. 4
9
W.E. = west to east
y y = height of the observation point above the ground surface in a cylindrical shell
9 r
y = height of the source above the ground surface in a cylindrical shell
s
Y = admittance
z, z = height of the observation point above the ground surface in a spherical shell
r
z = height of the source above the ground surface in a spherical shell
s
Z = pressure in atmospheres in Section 2.2.2
Z Z = surface impedance of T.M., T.E. modes at the ionospheric boundary
e9 h
Z = coupling impedance in (284), (217), and (218) of Ch. 5
eh
Z = surface impedance of ground
g
Z £u\ Z (u) = cylinder functions defined by (29) and (47) of Ch. 5
e hv
a = attenuation rate in db/1000 km
Γ(χ) = gamma function
Aj = normalized impedance (ΖΙ^\/μ /ε ).
] 0 0
A {y) or A (z) = normalized height-dependent impedance defined in Section 1.5
q q
A (y) = normalized height-dependent impedance defined by (273) or (276) of Ch. 5
Jq
xiv List of Principal Symbols
b=7t-6
b{x) = delta function
ε = complex relative permittivity
et = elements of permittivity tensor in Section 2.2.3
ε0 = (1/36TZ)X10~9 = permittivity of free space
er = real relative permittivity
Aq = excitation factor defined by (91) or (94) of Ch. 4
AJq = excitation factor defined by (274) or (277) of Ch. 5
λ = 27t/ko — free space wavelength
μο = 4ΤΪΧ 10~7 = permeability of free space
v = koa S (in cyl. coord.); k0a S—0.5 (in spher. coord.)
vjk = collision frequency between particles j and k
a — conductivity (mho/m)
x = time constant
φ = phase
ω = angular frequency
(DC — cyclotron or gyro frequency
ωρ — plasma frequency
ων = conductivity parameter in Section 2.2.3
e~l0)t = implied harmonic time dependence
CHAPTER 1
INTRODUCTION
1.1 Scope and Limitations of the Treatment
The energy of propagating radio waves is confined principally to the shell between the earth
and the ionosphere, and this space is frequently denoted as the terrestrial waveguide. For long
waves the height of this guide becomes comparable to a wavelength, and characteristics of
wave propagation are determined jointly by the properties of the two guide boundaries. There
are a number of propagating modes with distinct cutoff frequencies similar to those present in
microwave problems. But unlike the highly conducting guides of the microwave range, the
upper boundary of the terrestrial guide is diffuse and a poor conductor; the finite conductivity
of the ground surface may be also important. The frequency range to be discussed encompasses
the extremely-low-frequencies (E.L.F.) from 6 to 3000 c/s, the very-low-frequency (V.L.F.)
range from 3 to 30 kc/s, and a brief mention will be made also of low frequencies (L.F.) in the
range from 30 to 300 kc/s. The effective waveguide height h is less than the free-space wavelength
λ in the E.L.F. range, and only one waveguide mode propagates. For V.L.F. h exceeds λ, and
there are several propagating modes. In the L.F. range the number of significant propagating
modes may exceed 10.
Wave propagation in this frequency range has been discussed also in several published text
and review papers. Sommerfeld [1949] and Bremmer [1949] treat dipole radiation above a
plane or curved finitely conducting earth; Bremmer [1949] develops ionospheric reflection
characteristics which are used in solutions of geometric optics. Waves in a planar waveguide
are treated exhaustively in the monograph by Budden [1961b]. Wait [1962] emphasizes wave
propagation in a sharply bounded waveguide; several approximations are developed for curved
waveguides in the V.L.F. range; E.L.F. propagation and the earth-to-ionosphere cavity
resonances [Schumann, 1952a, b] are also treated. E.L.F. propagation and earth-to-ionosphere
cavity resonances are analyzed in more detail by Galejs [1964a] and Madden and Thompson
[1965]. In a monograph on long-wave propagation, Volland [1968] develops reflection character
istics and propagation parameters for a flat waveguide, but the curved waveguide is analyzed
using ray theory.
The magneto-ionic theory of waves in a generally anisotropic ionosphere and the detailed
reflection properties have been discussed by Ratcliffe [1959], Budden [1961a], or Ginzburg
[1964]. Watt [1967] and Wait [1968] treat a weakly anisotropic ionosphere. For more general
conditions [Baybulatov and Krasnushkin, 1967; Galejs, 1967, 1968, and 1969; Pappert, 1968;
Snyder and Pappert, 1969] the effective phase velocity and attenuation rates change for dif-
2 Terrestrial Propagation of Long Electromagnetic Waves
ferent directions of propagation, and there is a coupling between vertically and horizontally
polarized waves.
In this text, wave propagation will be characterized almost exclusively by mode theory. After
developing the fundamental concepts of wave propagation in a planar and curved isotropic
waveguide, a number of examples will illustrate the effects of an anisotropic ionosphere. The
mathematics are developed only for sources at the ground surface or within the waveguide.
This includes artificial sources as well as lightning discharges. Emissions generated at iono
spheric or higher altitudes by incident extraterrestrial particles and whistlers guided in ionized
columns along the geomagnetic field lines [Helliwell, 1965] can be classified as separate topics
and will not be treated. The discussion is limited to steady-state solutions in a waveguide that
is uniform in the direction of propagation; problems of mode conversion caused by changing
boundary properties [Wait, 1968] are not considered.
The remaining part of this chapter summarizes the basic equations and treats plane-wave
reflection from a dielectric interface. A planar waveguide is analyzed using a superposition of
two obliquely incident plane waves. The properties of waveguide boundaries are implicitly
represented by Fresnel reflection coefficients. The propagation parameters of waveguide modes,
the field expressions and concepts of height-gain functions and height-dependent impedances
are then developed.
Chapter 2 discusses boundaries of the terrestrial guide. The ground surface is frequently
approximated as a homogeneous isotropic conductor, but the detailed ground structure is
important in studies of horizontal field components. Stratifications cause an effective an-
isotropy: the conductivity parallel to the interface will differ from the conductivity perpendicu
lar to the interface. The properties of the lower ionosphere are emphasized in the survey of
ionosphere characteristics, but some data of the E- and F-layers are also included. Components
of the ionospheric permittivity or conductivity tensor are calculated from the magneto-ionic
theory.
Lightning discharges are regarded as a natural source of E.L.F. and V.L.F. radiation in
Chapter 3. Apart from return strokes, the more frequent lower amplitude ^-changes also
contribute significantly to the radiated energy. The received noise is the result from integrating
several atmospherics during the response time of the receiver. The amplitude distribution of the
observed radio noise can be related to the statistics of the lightning discharges by considering
return strokes from several thunderstorm centers or by return strokes from a single center in the
presence of the more frequent ^-changes.
The mode theory for waves in an isotropic spherical shell is developed in Chapter 4. The
vertically and horizontally polarized modes excited by various dipole sources are characterized
by wave numbers, excitation factors, height-gain functions, and height-dependent normalized
impedances. An exponential approximation of the radial functions can be interpreted as a
modification of the planar solution. These approximations are less accurate than Airy function
representations for frequencies in the V.L.F. range. Nonpropagating waveguide modes must be
included in the mode summation near the source. Also, antipodal standing wave patterns are
illustrated in several examples.
The terrestrial waveguide is closely represented in a spherical geometry, but a cylindrical
geometry may be required for the analysis of curved anisotropic waveguide boundaries follow
ing Chapter 5. In the Airy function approximation the radial functions are the same in spherical
