Geology Reference
In-Depth Information
The corresponding pattern of displacement for
these components is illustrated in Fig.
10.10
.
The importance of these solutions in seismology
arises from the idea that
a distribution of equiva-
lent double couples produces a displacement field
that is indistinguishable from the displacements
around a fault plane after an earthquake
.This
in one of the most fundamental principles of
seismology, which underlies the construction of
equivalent systems of forces in the modeling of
real seismic events, both in the elasto-static and
in the elasto-dynamic cases.
Now we are ready to consider the elasto-
dynamic solutions of the momentum equation. In
the general non-static case this equation can be
written, in vector notation, as follows:
D
A
s
n
, gives the following inhomogeneous wave
equations:
(
r
@
2
A
p
@t
2
D
g.t/
4 r.œ
2
A
p
1
'
2
C
2/
(10.46)
“
2
@
2
A
s
g.t/
4 r
2
A
s
1
r
@t
2
D
Finding a solution to these equations is rather
complicate and involves Fourier transform tech-
niques. The interested reader is referred to the
book of Lay and Wallace (
1995
) for a detailed
treatment of this subject. Therefore, we will give
without proof the classic
Stokes solution
for the
displacement associated with a point force at the
origin in the direction
x
j
:
r=“
Z
4 ¡r
3
3
x
i
x
j
r
2
•
ij
g
t
t
0
t
0
dt
0
1
u
i
.r;t/
D
f
C
.œ
C
2/
r
.
r
u
/
rr
u
D
¡
@
2
u
@t
2
(10.43)
r='
r
2
g
t
1
4 ¡'
2
r
x
i
x
j
r
'
C
This time we shall assume a time-dependent
body force of the form:
f
(
r
,
t
)
D
g
(
t
)•(
r
)
n
,where
g
(
t
) represents the time history. For example,
g
(
t
)
could be a delta function •(
t
) or a step function
H
(
t
). Therefore, (
10.18
) can be rewritten as fol-
lows:
r
2
•
ij
g
t
x
i
x
j
1
4
¡
“
2
r
r
“
(10.47)
We note that the first term of this solution
behaves like 1/
r
2
, thereby it is usually referred
to as the
near
-
field
term, while the other terms
behave like 1/
r
. Consequently, they are called the
far
-
field
terms. The first of them corresponds to a
P
wave that propagates with velocity '. Its contri-
bution to the displacement is a radial component.
The second far-field term is associated with an
S
wave propagating with velocity “. It is easy
to prove that the scalar product with
r
is zero,
so that its contribution consists of a tangential
displacement. Both far-field terms are propor-
tional to the magnitude of the applied force.
Figure
10.11
illustrates the radiation pattern for
a single force applied in the
x
1
direction. Finally,
to obtain the displacement field associated with
force couples and double couples we can apply
the same procedure described in the elasto-static
context.
For a double couple in the
x
j
x
k
plane, oriented
as the coordinate axes, the far-field
P
and
S
wave
radiation patterns are given by:
2
n
4 r
f .r/
D
g.t/•.r/ n
D
g.r/
r
n
4 r
r
n
4 r
D
g.t/
r
Crr
(10.44)
Again, we assume that two vector potentials
exist,
A
p
and
A
s
, such that the displacement
u
has the form (
10.21
). In this instance, the wave
equation splits into two distinct equations, one for
each potential.
We obtain:
(
r
'
2
@
2
A
p
@t
2
D
g.t/n
4 r.œ
2
A
p
1
C
2/
(10.45)
“
2
@
2
1
A
s
g.t/
n
4 r
2
r
A
s
@t
2
D
where ' and “ are the
P
and
S
wave velocities
before that the potentials
A
p
and
A
s
have the
same direction of
n
,sothat
A
p
D
A
p
n
and
A
s
