Table Of ContentDiese Mitteilungen sefzen eine von Erich Regener begrundete Reihe fort, deren Hefte
am Ende dieser Arbeif genannf sind.
Bis Heft 19 wurden die Mitteilungen herausgegeben von J. Bartels und W. Dieminger.
Von Heft 20 an zeichnen W. Dieminger, A. Ehmert und G. Pfotzer als Herausgeber.
Das Max-Pianck-lnsfitut fUr Aeronomie vereinigt zwei Institute, das lnstitut fur Strato
sphiirenphysik und das lnsfitut fur lonosphiirenphysik.
Ein (S) oder (I) beim Tile I deutet an, a us welchem lnsfitut die Arbeif stammt.
Anschrift der beiden Institute:
3411 Lindau
APPLICATION OF THE
GEOMETRICAL THEORY OF DIFFRACTION TO
TERRESTRIAL LF RADIO WAVE PROPAGATION
by
R. MICHAEL JONES
ISBN 978-3-662-34408-8 ISBN 978-3-662-34679-2 (eBook)
DOI 10.1007/978-3-662-34679-2
- 3 -
Contents
1. Introduction ........ . .......................... Seite 5
2. The path of a surface diffracted ray. 5
3. Excitation of the groundwave ... 8
4. Radiation from the groundwave. 12
5. Application of the GTD to a concentric, isotropic ionosphere 13
6. Discussion. 19
7. Conclusions 19
Summary, Zusammenfassung 20
Appendix A : The rigorous solution for the field of a vertical
Hertzian dipole over a homogeneous, isotropic, sphe-
rical earth.... . . . . . . . . . . . . . . . . 21
Appendix B Rigorous solution for the field of a vertical Hertzian
dipole over a homogeneous, isotropic, spherical earth
below a homogeneous, isotropic, concentric, sharply-
bounded ionosphere. . . . . . . . . . . . . . . . 23
Appendix C Calculation of the effect of one hop for shedding from
a surface diffracted ray and reflecting from a concen-
tric ionosphere ........... . 25
References ............................ . 26
- 4 -
Nomenclature
c formula given by equation (51)
mth hop sky-wave field
H)1
l(x),
H)2 Hankel functions
)(x)
L transmitter normalizing factor (see equation (A-2))
T ionospheric reflection coefficient
T ionospheric reflection coefficient including the phase integral from the surface
diffracted ray up to the ionosphere and back
a radius of the earth
b distance from the transmitter to the center of the earth
ds differential path length
g radius of the lower boundary of a concentric, sharply-bounded ionosphere
h height
m number of hops (reflections from the ionosphere)
n complex phase refractive index
r distance from the receiver to the center of the earth
a. great circle angle traveled by the groundwave
a nearly the same as a. (see equation (B-10))
e
central earth angle between the transmitter and the receiver
Tt ratio of the circumference to the diameter of a circle
w angular wave frequency
Other variables not defined directly in the text are defined in Table 1 or in the figures.
- 5 - 1.. 2.
1. Introduction
Until now, a practical method for calculating the propagation of LF radio waves for a homogeneous,
isotropic, spherical earth, but arbitrarily varying ionosphere did not exist: 1) Analytical fullwave solu
tions exist only for particular geometries. The solution for a spherical geometry (i. e. , a homogeneous,
isotropic, concentric, sharply-bounded ionosphere) has been particularly well developed [WATSON,1919;
BREMMER, 1949; WAIT, 1960, 1961; JOHLER and BERRY, 1962, 1964; BERRY, 1964; BERRY and
CHRISMAN, 1965 a, b ] . 2) Numerical full-wave solutions are not practical for an ionosphere that varies
arbitrarily in 2 or 3 dimensions. 3) Waveguide solutions have been developed for only special shapes of
ionospheric tilts or bumps in the direction of propagation [WAIT, 1962, 1964a, b, c; RUGG, 1967).
4) Ray tracing in complex space [ JONES, 1968 ) can calculate propagation of LF radio waves in an
ionosphere that varies arbitrarily in three dimensions, but does not take into account diffraction by the
earth.
KELLER's [ 1962 ] geometrical theory of diffraction (GTD) can be used to calculate diffraction of
LF radio waves by the earth. Since the GTD and ray tracing in complex space are both ray-type methods,
the two can be combined to calculate propagation of LF radio waves (including diffraction by the earth)
in the presence of an ionosphere that varies arbitrarily in three dimensions.
This report represents a groundwave mode by a surface-diffracted ray at a complex height dependent
on the propagation constant of the groundwave mode, calculates groundwave excitation and shedding coef
ficients for this representation, and applies the results to LF radio wave propagation below a concentric,
isotropic ionosphere.
2. The path of a surface diffracted ray
LEVY and KELLER [ 1959 ] apply the geometrical theory of diffraction (GTD) to diffraction by a
sphere. Using the GTD postulate that diffracted rays obey Fermat's principle, they obtain a path of the
type SPQO in figure 1, i.e., the path of a string stretched between the transmitter and receiver in con
tact with the sphere. They also obtain paths that wind around the sphere many times. The straight line
segments are geometrical optical rays, and the rays following the surface of the sphere are called surface
diffracted rays. LEVY and KELLER [ 1959 ] infer from this picture of diffraction that a geometrical
optical ray tangent to the sphere will excite a surface diffracted ray that will in turn shed rays tangential
ly as it follows the surface of the sphere.
The GTD can be used to calculate diffraction of radio waves by the earth (at least above 30 kHz). It
is clear that a surface diffracted ray represents one of the infinite number of groundwave modes, each of
which travels along the ground with its own characteristic propagation constant v . (That is,
-iva.
e (1)
Figure Representing a ground
wave mode by a surface
diffracted ray on the
ground
2. - 6 -
gives the change in field strength of the groundwave mode in a central earth angle <1 , and v is complex
to give both amplitude and phase). Thus, the GTD solution must include a sum over the groundwave modes.
The ray paths of figure 1 do not, however, quite satisfy Fermat's principle because the propagation
constant of the surface diffracted rays is slightly different from that of free space. Thus, the optical
length of the total path is not proportional to the geometrical length, and the stretched string analogy breaks
down. The path of figure 2 is a generalization of that in figure 1 in that the surface diffracted ray travels
at a height r -a (where a is the radius of the earth) instead of on the ground. To find a height r -a that
0 0
satisfies Fermat's principle, we take the derivative of the phase path with respect to r and set it equal
0
to 0. The total (complex) phase path length from figure 2 is
1-
C!l..
.c..
0
~0
Transmitter
Figure 2.: Representing a groundwave mode by a surface-diffracted ray at a height of r 0 - a
- 7 - 2 .
p k .J. '~ b--r~ + II ( 9 - cos -1 0ro - cos -1 rr 0 ) + k .J. '/ r2--2r~ (2)
Applying Fermat's principle gives
dP k r 0 k r 0 0 (3)
~
Receiver
II
enl .
c
E.
-
?I-><
....
Transmitter
Figure 3.: Representing a groundwave mode by a surface-diffracted ray at a height of 11 /k-a
3. - 8 -
which gives
v
(4)
k
Equation 4 shows several things. First, the height at which the surface diffracted ray travels depends on
the propagation constant of the groundwave mode it represents. Second, the height is complex because v is
complex. The real part of this height is about 20 km for the dominant groundwave mode for typical ground
properties at 30kHz. Third, this height is such that the effective propagation constant of the surface dif
fracted ray ( v /r ) exactly equals the propagation constant of free space, k. In other words, Fermat's
0
principle chooses a height at which the propagation constant is the same all along the path, and thus the
phase path length equals k times the total geometrical length of the ray. This phase path length is
v' v'
P = (kb) 2 - v 2 + (kr) 2 - v 2 + v ( 9 - cos -1 kvD - cos -1 kvr ) (5)
as shown in figure 3 .
3. Excitation of the groundwave
The groundwave is excited at the point at which a geometrical optical ray is tangent to that surface
diffracted ray which represents the groundwave mode. At that point, a fraction of the incident field is con
verted to a groundwave mode, and the rest continues. Analagous to reflection and transmission coeffi
cients, an excitation coefficient gives the fraction of the incident field that is converted to the groundwave
mode.
I define the excitation coefficient Dv as the ratio of the transverse electric field of the groundwave
at the point of excitation to that of the radiation field of the source at the same point.
To evaluate Dv , it is necessary to compare the GTD solution with a rigorous solution. For the com
parison I use a transmitter with a small, vertical dipole antenna at a height b-a above the earth, as
shown in figure 4. The GTD then gives the vertical electric field of groundwave mode v at a height
V /k-a above the earth at a central earth angle 9 from the transmitter as
(6)
where the four terms are, respectively, (1) the incident field from the transmitter at the point of exci
tation of the groundwave mode, (2) the groundwave excitation coefficient, (3) the phase integral contribu
tion of the surface diffracted ray, and (4) a convergence factor due to azimuthal focusing. I neglect paths
that circle the earth many times because they have negligible effect above 30 kHz, where the groundwave
attenuation coefficient is large. With the geometry of figure 4, (6) becomes
-i ,/(kb) 2 - v 2 -i v ( e - cos -1 J:.)
iLV Dv e V e Ku
(7)
For r v /k, the rigorous solution from (A- 10) for one groundwave mode is
