Geology Reference
In-Depth Information
r
2
@
@r
sin ™
@
@™
where
k
is the wave vector. In the case of a
monochromatic plane wave
that is propagating in
a unique direction, the dilatation must also be a
function of (
t
—/'). Therefore we have that,
¨t
C
ª
D
¨
t
1
r
2
@
@r
1
r
2
sin ™
@
@™
C
@
2
@¥
2
@
2
@t
2
D
0
1
r
2
sin
2
™
1
'
2
C
(8.36)
—
'
In this equation, the spherical coordinates
(™,¥) do not represent global coordinates of
colatitude and longitude; they are simply
local
angular coordinates. Assuming spherical
symmetry implies that does not depend from
these variables. Therefore, the wave equation for
the dilatation can be rewritten as follows:
¨t
k—
(8.33)
If
n
is the versor in the direction of propaga-
tion and œ
D
'/ is the wavelength, then the wave
vector is given by:
¨
'
n
D
2
'
n
D
2
œ
n
k
D
(8.34)
r
2
@
@r
@
2
@t
2
D
0
1
r
2
@
@r
1
'
2
(8.37)
The magnitude
k
of this vector represents the
number of oscillations in a segment of length
2 (wavenumber). Using a complex notation, the
general plane wave monochromatic solution to
(
8.12
) will have the following form:
To solve this equation, we set:
.r;t/
D
Ÿ.r;t/=r
(8.38)
Then, substituting into (
8.37
):
.r;t/
D
Ae
i.kr¨t/
(8.35)
@
2
@r
2
@
2
Ÿ
@t
2
1
r
1
'
2
D
0
(8.39)
where
A
is the amplitude. A similar formula
can be written for the transverse waves. Of
course, only the real part of (
8.35
) has physical
significance. In the crust, where '
5.5 km s
1
,
a
P
wave that is propagating with period
T
D
1/
D
5 s will have wavelength œ
27.5 km.
In general, seismic waves associated with
earthquakes have periods between 1 and 10 s,
that is, frequency 0.1 Hz
1 Hz. Therefore,
typical crustal wavelengths of
P
waves range
between 5.5 and 55 km. Other kinds of seismic
waves have different periods. For example, in
the case surface waves 10 s
T
100 s, while
free oscillations of the Earth occur with periods
100 s
T
1,000 s. Conversely, artificial waves
generated in exploration geophysics have very
short periods between 10
4
s and 10
3
s.
More realistic solutions to the wave Eqs.
(
8.12
)and(
8.27
) can be found assuming a
spherical rather than planar symmetry. In this
instance, it is necessary to represent the equations
reference frame having the origin at the seismic
source. In this frame, (
8.12
) assumes the form:
For
r
¤
0 this reduces to a classic plane waves
Eq. (
8.16
). Therefore, the solution for has the
form:
f.t
˙
r='/
r
.r;t/
D
(8.40)
This is the spherically symmetric solution to
the wave equation for the dilatation. A similar
solution can be written for the components of
the vector field . The solution describes wave
fronts that are spherical surfaces centered about
the origin
r
D
0 and having amplitude that is in-
versely proportional to the distance from the ori-
gin. For
r
D
0, (
8.40
)is
not
a solution to the wave
equation. However, it is possible to show that it
is a solution of the following non-homogeneous
wave equation (e.g., Aki and Richards
2002
):
@
2
@t
2
D
4 •.r/f.t/
1
'
2
2
r
(8.41)
where •(
r
) is the Dirac delta function centered
at the origin. The term at the right-hand side of
