Geoscience Reference
In-Depth Information
Substituting formula (3.11) into expression (3.4), we obtain an equation for deter-
mining function
Φ
( z , p , k ):
2
Φ zz α
Φ
= 0 ,
(3.12)
2 = k 2 + p 2 c 2 .
The solution of equation (3.12) is well known and can be written in the form
where
α
Φ
( z , p , k )= A cosh (
α
z )+ B sinh (
α
z ) ,
where A and B are arbitrary numerical coefficients.
Applying the boundary condition for a free surface, (3.5), we find the relationship
between the coefficients:
Ap 2 (g
) 1 .
B =
α
With the aid of the boundary condition for the basin bottom, (3.6), we determine
the coefficient A :
p
Ψ
( p , k )
A =
H ) ,
H )+ p 2 g 1 cosh (
α
sinh (
α
α
where function
( p , k ) represents the Laplace and Fourier transforms of the space-
time law of motion of the bottom
Ψ
η
( x , t ):
+
1
Ψ
( p , k )=
d t
d x
η
( x , t ) exp (
pt + ikx ) .
(3.13)
2 i
4
π
0
Φ
( z , p , k ) in the following form:
Thus, we have function
Φ
( z , p , k )
p 2 (g
z ) .
p
Ψ
( p , k )
) 1 sinh (
=
α
α
z )
cosh (
α
(3.14)
H )+ p 2 g 1 cosh (
α
sinh (
α
α
H )
Substitution of expression (3.14) into formula (3.11) yields the final expression
for calculation of the potential corresponding to an arbitrary space-time law deter-
mining the displacement of the bottom,
( x , t ). In investigating analytical models,
it is of interest to consider the behaviour of the free surface of the liquid (as the most
illustrative characteristic), the displacement of which relative to its unperturbed level
is expressed via the potential in accordance with formula (3.8). As a result we obtain
the following expression:
η
ξ
( x , t )
s + i
p 2
Ψ
( p , k )
= g 1
d p
d k
H ) exp
{
pt
ikx
}
.
(3.15)
α
α
H )+ p 2 g 1 cosh (
α
sinh (
s
i
 
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