Geoscience Reference
In-Depth Information
3.1.4 The Running Displacement
In dealing with the running-displacement problem in Sect. 2.3.3 we noted that move-
ments of the ocean bottom of such types are characterized by high propagation
velocities, at which the theory of incompressible liquids cannot be applied. The
velocity, with which the fault ruptures at the earthquake source, the fissure propagat-
ing along the bottom and surface seismic waves are all phenomena characterized by
velocities exceeding the speed of sound in water. And it is only in the case of under-
water slumps (landslides) that the velocity of a running slide is significantly inferior
to the speed of sound in water. Therefore, the aim of this section is the construction
of a mathematical model for the excitation of waves by a running displacement of
the ocean bottom in a compressible liquid.
Consider a plane problem, the general formulation of which corresponds to (3.4)-
(3.6). We shall choose the model law of motion of the ocean bottom in the case of
a running displacement to be of the form (Fig. 2.10)
0 θ
a ) 1
vt ) ,
η
( x , t )=
η
( x )
θ
( x
θ
( x
(3.32)
θ
where
( z ) is the Heaviside step-function. The residual deformation of the ocean
η
bottom,
0 , is the same over the entire active zone of length a and equals zero outside
this zone. The horizontal propagation velocity of the displacement is v . A similar
problem has been resolved in Sect. 2.3.1 for the case of an incompressible liquid.
We shall apply the general solution of the problem (3.15) and pass to dimen-
sionless variables in accordance with formulae (3.27), which in the case of a run-
ning displacement must be complemented with the expression v = v (g H ) 1 / 2 (we
drop the asterisk '*'). As a result we arrive at the following expression describing
the surface perturbation of a compressible liquid, caused by a running displacement
of the ocean bottom:
+
s + i
( x , t )= η 0 c 2
4
d p p (exp
{
a
γ }−
1) exp
{
pt
ikx
}
ξ
d k
,
(3.33)
π
2 i
γ
cosh (
α
)(
α
tanh (
α
)+ p 2 c 2 )
s i
= ik
pcv 1 ,
2 = k 2 + p 2 .
As a function of the complex parameter p the integrand expression has two or an
infinite number (depending on the sign of
where
γ
α
2 ) of poles located on the axis Im( p )=0.
Since the positions of the poles are determined from the solution of a transcendental
equation, and, besides, they depend on the parameter k , over which external integra-
tion is performed, further analysis of expression (3.33) was carried out numerically.
Formula (2.72), obtained in Sect. 2.3.1, is the analogue of expression (3.33) for
the case of incompressible liquids.
The following parameter values were chosen for calculations: c = 8, a = 10,
which at ocean depths of 4,000 m approximately corresponds to the velocity of
sound in water, 1,500 m/s, and to a horizontal size of the source equal to 40 km. The
propagation velocity of the displacement, v , was varied within limits from 0.125 up
to 32 (from 23 up to 6,000 m/s).
α
 
Search WWH ::




Custom Search