Geoscience Reference
In-Depth Information
2.3.1 Construction of the General Solution
We shall consider (Fig. 2.9) a layer of ideal incompressible homogeneous liquid, in-
finite along the 0
x
axis, of constant depth
H
, and in the field of gravity. We shall put
the origin of the Cartesian reference system, 0
xz
, on the unperturbed free surface,
the 0
z
will be directed vertically upward. To find the perturbation of the free sur-
face,
(
x
,
t
), and the field of flow velocities v(
x
,
z
,
t
), arising in the layer of liquid,
when the basin floor undergoes motion in accordance with the law
ξ
(
x
,
t
), we shall
resolve the problem with respect to the potential of the flow velocity,
F
(
x
,
z
,
t
):
η
2
F
2
F
∂
∂
+
∂
= 0
,
(2.53)
∂
x
2
z
2
2
F
g
∂
F
∂
−
∂
=
t
2
,
z
= 0
,
(2.54)
z
∂
∂
F
=
∂η
∂
t
,
z
=
−
H
.
(2.55)
∂
z
Without dwelling on the details of resolving the problem (2.53)-(2.55) that were ex-
posed above for the three-dimensional case, we shall present the resultant formulae.
s
+
i
∞
+
∞
1
F
(
x
,
z
,
t
)=
−
d
p
d
k
2
i
4
π
s
−
i
∞
−
∞
p
2
tanh(
kz
)
k
cosh(
kH
)(g
k
tanh(
kH
)+
p
2
)
cosh(
kz
)
g
k
p
exp
{
pt
−
ikx
}
−
×
H
(
p
,
k
)
,
(2.56)
s
+
i
∞
+
∞
p
2
exp
1
}
cosh(
kH
)(g
k
tanh(
kH
)+
p
2
)
H
(
p
,
k
)
,
{
pt
−
ikx
ξ
(
x
,
t
)=
d
p
d
k
(2.57)
4
π
2
i
−
∞
s
−
i
∞
s
+
i
∞
+
∞
u
(
x
,
z
,
t
)=
∂
F
1
=
d
p
d
k
∂
x
4
π
2
−
∞
s
−
i
∞
p
2
tanh(
kz
)
cosh(
kH
)(g
k
tanh(
kH
)+
p
2
)
cosh(
kz
)
g
k
p
exp
{
pt
−
ikx
}
−
×
H
(
p
,
k
)
,
(2.58)
Fig. 2.9 Mathematical
formulation of the 2D
problem
