Geoscience Reference
In-Depth Information
The boundary conditions on the surface of the liquid and on the bottom remain
the same as in the case of a noncompressible liquid [Yanushkauskas (1981)],
F
tt
=
−
g
F
z
,
z
= 0
,
(3.5)
F
z
=
η
t
,
z
=
−
H
.
(3.6)
If the depth of the basin,
H
, is a function of the horizontal coordinate, then the fol-
lowing formula can be applied as the boundary condition on the bottom:
∂
F
=(U
,
n)
,
(3.7)
∂
n
where n is the normal to the basin bottom plane, U is the deformation velocity vector
of the bottom. From a physical point of view, this condition signifies equality of
the liquid flow velocity component, normal to the bottom, and the motion velocity
of the bottom in the same direction (a no-flow condition).
Displacement of the free surface of the liquid from its equilibrium position,
the dynamical pressure and the flow velocity are calculated via the potential by
the following formulae:
g
−
1
F
t
(
x
,
y
,
0
,
t
)
,
ξ
(
x
,
y
,
t
)=
−
(3.8)
p
(
x
,
y
,
z
,
t
)=
−
ρ
0
F
t
(
x
,
y
,
z
,
t
)
,
(3.9)
v(
x
,
y
,
z
,
t
)=
∇
F
(
x
,
y
,
z
,
t
)
.
(3.10)
Note that the problem for an incompressible liquid, (2.29)-(2.31), is a limit case
of the problems (3.4)-(3.6) in the case of
c
→
∞
. It is readily seen that equation (3.4)
reduces, in the limit, to the Laplace equation (2.29). Therefore, the results, obtained
for an incompressible liquid in Sects. 2.2-2.4, will serve as a reliable benchmark.
Moreover, an answer will be obtained to the question concerning the difference
between the behaviours of compressible and incompressible liquids, and as to when
this difference may be neglected.
We shall restrict ourselves to resolving the two following two-dimensional prob-
lems: the plane problem (in a Cartesian reference frame) and the quasithree-
dimensional problem (exhibiting cylindrical symmetry).
3.1.2.1 Cartesian coordinates
In a Cartesian reference frame, the solution of the problem (3.4)-(3.6) is sought
via the respective Laplace and Fourier transformations relative to the time and space
coordinates in the following form:
s
+
i
∞
+
∞
F
(
x
,
z
,
t
)=
d
p
d
k
Φ
(
z
,
p
,
k
) exp
{
pt
−
ikx
}
.
(3.11)
s
−
i
∞
−
∞
