Geoscience Reference
In-Depth Information
Approximation for Short Acoustic Wavelength
For simplicity, the radius of seismic wave front r
l
is assumed to be much greater
than the seismic source radius R
0
. This implies that the region of integration in
Eq. (
7.92
) is restricted, in fact, by the radii within interval 0<r<C
l
t.Themass
velocity V of the medium reaches a peak value in the acoustic wave zone, which
is located at the distance r much greater than the acoustic wavelength. In this zone
the last term on the right-hand side of Eq. (
7.40
) can be neglected. In this approach,
substituting Eq. (
7.40
) for the mass velocity in Eq. (
7.92
), and taking into account
that
R
0
C
l
@
t
f .t
1
/
D
C
l
R
0
@
r
f .t
1
/;
r@
r
V
C
2V
D
(7.108)
we obtain
C
l
t
0
Z
Z
t
2C
l
R
0
1=2
r
dt
0
r
0
g
3
r;r
0
;r
0
@
r
0
f
t
1
dr
0
;
g
1
.r;t/
D
(7.109)
0
0
where t
1
D
t
0
r
0
=C
l
and the functions g
3
and r
0
are defined by Eq. (
7.93
)
and (
7.94
). Suppose that the profile of mass velocity is described by a continuous
smoothing function so that the first and second derivatives of the function f
t
1
are zero at the wave front, i.e., at r
0
D
C
l
t
0
or at t
1
D
0. Then we can transform
Eq. (
7.109
) integrating with respect to r
0
by parts twice. Taking into account that
g
3
D
0 at r
0
D
0, we get
Z
Z
C
l
t
0
t
2C
l
R
0
1=2
r
dt
0
r
0
@
r
0
g
3
r;r
0
;r
0
@
r
0
f
t
1
dr
0
:
g
1
.r;t/
D
(7.110)
0
0
Recall that the normalized potential f .t
1
/ can be expressed through the radial
displacement
u
.t/ at the source surface. Suppose that the radial displacement at
the source surface increases gradually and tends to a constant value
u
0
as t
!1
.
In other words,
u
.t/ is an increasing or weakly oscillating function, which tends
to the value of static displacement
u
0
as t
!1
. As it follows from Eq. (
7.106
),
the characteristic time of the displacement and potential variations depend on the
parameters t
r
;R
0
=C
l
and R
0
=C
t
which define the stress relaxation time in the
source and the typical periods of the seismic vibrations and relaxation. At far
distances all of these parameters are much smaller than the time t
D
r=C
l
of
seismic wave arrival to the observation point. In such a case we may simplify
Eq. (
7.41
) for the function f .t/ by replacing
u
.t
0
/ under the integral sign by the
constant value
u
0
. Calculating this integral we find that f .t/
!
u
0
=R
0
as t
!1
in accordance with the plot shown in Fig.
7.6
. It follows from this fact that the
derivative of the normalized potential under the integral sign in Eq. (
7.110
) changes
considerably within a short interval near the point r
0
D
C
l
t
0
and it tends to zero
Search WWH ::
Custom Search