Geoscience Reference
In-Depth Information
where S = k (0) exp ik (0) x .If k A x
1 ,
( k A x ) 2 .
The contribution from a TEM -mode to the ground field becomes significant
at distances
x km > h
k 0
k A 0 . 5
H 1 ( k A x )
x
0 . 64
i
1 / 2 k 0 h X
ε a
1 / 4
3
10 3 T 1 / 4
×
day,
night.
.
10 3 T 1 / 4
1 . 5
×
For instance, for the nighttime Pi 2 oscillations ( T
50 s) the amplitudes of
G mAS and G 0 AS are comparable at 4000 km.
Expression (8.64) obtained for distances x
d g from a source shows that,
far from the source, the wave decays with a distance of approximately x 2 .
The phase velocity is to be of the order of c A . At larger distances, where the
amplitude distribution is determined by a TEM , the wave amplitude almost
does not depend on distance and the phase velocity grows from c A to the light
velocity c .
Of course, in calculating a TEM-mode it is necessary to take into account
the curvature of the Earth. Because of low attenuation and large wavelength,
the field at the observation point will be the sum of direct and inverse round-
the-world echoes.
Non-Monochromatic Waves
Up to now we have studied transformation of monochromatic oscillations. To
consider signals finite in time we shall perform the inverse Fourier transforma-
tion of (8.64), we assuming a time-dependence in the form of a step-function
θ ( t ). To simplify computation, assume that τ D = k 0 hX
1 for processes
with characteristic times τ
10 s (day) and τ
1 s (night). Then fields on
the ground can be presented as
Y 2
X
h
2 c
g 1
t 3 / 2
Y sin I
X
Q + Y sin I
X
g 1
t 1 / 2
Q
G ( g ) =
,
(8.65)
Y
sin I
h
2 c
g 1
t 3 / 2
g 1
t 1 / 2
where
h
x 2
,
Q ( x, t )= θ ( t 1 )
π
t
t 2 +
h
c 2 A t 2 ( t + t 2 )
c A ,t 2 = t 2
g 1 = ε a
x 2
c 2 A
x
,t 1 = t
,
πh Xc
θ ( t 1 )=1 at t 1
0, and θ ( t 1 )=0 at t 1 < 0 . In deriving (8.65) we used the
fact that phase incursion in a TEM-mode is small ( k (0) x
1).
 
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