Geology Reference
In-Depth Information
Substituting (
8.9
) into the momentum equa-
tion gives a differential equation in terms of
displacements only:
region to its neighboring areas. Clearly, this ap-
proach is effective when one takes into account of
the presence of discontinuities in the mechanical
properties at some boundaries, for example at
the Moho, the CMB, etc. We shall explore this
technique, which is referred to as the
seismic ray
method
, in the next chapter. We will prove that the
method can be used to describe the propagation
of high-frequency seismic waves, thereby the
Earth's free oscillations (with frequencies of the
order of one mHz or less) are excluded from a
description in terms of seismic rays.
¡
@
2
u
i
@x
i
C
.œ
C
/
@
@œ
@
@x
j
@t
2
D
@x
i
C
@
u
i
@x
j
C
@
u
j
@x
i
C
@
2
u
i
@x
j
(8.10)
This second-order partial derivatives equation
is the
seismic wave equation
. It can be solved
numerically when the three scalar fields ¡
D
¡(
r
),
œ
D
œ(
r
), and
D
(
r
) are known. Sometimes
this technique is used to create computer simu-
lations of seismic wave propagation, following a
hypothetical earthquake with assigned source pa-
rameters. The corresponding theoretical ground
motion that would be observed at a station is
called a
synthetic seismogram
, and it is also pos-
sible to predict the associated damage to human
structures. The terms that include gradients of the
Lamé parameters at the right-hand side of (
8.10
)
are zero in the case of a homogeneous material.
This is often a useful approximation in the study
of seismic wave propagation within small quasi-
homogeneous regions. In this instance, Eq. (
8.10
)
reduces to:
8.3
Seismic Waves
Although the seismic wave Eq. (
8.10
) can be
solved numerically or assuming specific distribu-
tions of the elastic parameters, it is possible to
gain some insight about the physics of seismic
waves propagation from the more simple homo-
geneous version (
8.11
). Taking the divergence of
this equation gives:
¡
@
2
@t
2
@x
i
D
.œ
C
/
@
2
@
2
u
i
@x
j
@
u
i
@x
i
C
@
@x
i
Therefore,
¡
@
2
u
i
@x
i
C
@
2
u
i
@t
2
D
.œ
C
/
@
(8.11)
@
2
@t
2
D
0
¡
œ
C
2
@x
j
2
r
(8.12)
There are two ways for using this equation
in appropriate manner. First, it is possible to
assume that the Earth is composed by a se-
quence of quasi-homogeneous layers with vari-
able thickness, so that (
8.11
) is applied indepen-
dently within each layer and the local solutions
are linked together a posteriori. For example, the
propagation of seismic waves through the oceanic
crust is often modelled this way. In general, this
approach assumes that the lateral variations of the
mechanical parameters are negligible. Second,
when both and œ vary smoothly, it is possible to
find an approximate solution to (
8.10
) consider-
ing the material as formed by small homogeneous
regions where (
8.11
) holds, and assuming smooth
variations of the parameters and œ from each
This
is
a
standard
wave
equation
(or
D
'
Alambert
'
s
equation
).
It
implies
that
in
isotropic
homogeneous
media
a
volume
perturbation propagates with velocity:
s
œ
C
2
¡
'
(8.13)
To prove this, let us assume that at any time
t
the dilatation
D
(
r
,
t
) is constant along a
plane having distance — from the origin, as shown
in Fig.
8.1
.
Although this assumption is a good approxi-
mation of reality only at great distance from the
source, it is useful to understand the physics of
