Geoscience Reference
In-Depth Information
ξ
(
x
,
y
,
t
)
s
+
i
∞
+
∞
+
∞
p
2
exp
1
}
cosh(
kH
)(g
k
tanh(
kH
)+
p
2
)
H
(
p
,
m
,
n
)
,
{
pt
−
imx
−
iny
=
d
p
d
m
d
n
(2.39)
3
i
8
π
s
−
i
∞
−
∞
−
∞
horizontal,
u
(
x
,
y
,
z
,
t
),
v
(
x
,
y
,
z
,
t
) and vertical,
w
(
x
,
y
,
z
,
t
) components of the flow
velocity,
s
+
i
∞
+
∞
+
∞
u
(
x
,
y
,
z
,
t
)=
∂
F
∂
1
=
d
p
d
m
d
n
x
8
π
3
−
∞
−
∞
s
−
i
∞
p
2
tanh(
kz
)
k
cosh(
kH
)(g
k
tanh(
kH
)+
p
2
)
cosh(
kz
)
g
k
mp
exp
{
pt
−
imx
−
iny
}
−
×
H
(
p
,
m
,
n
);
(2.40)
s
+
i
∞
+
∞
+
∞
v
(
x
,
y
,
z
,
t
)=
∂
F
1
=
d
p
d
m
d
n
3
∂
y
8
π
s
−
i
∞
−
∞
−
∞
p
2
tanh(
kz
)
k
cosh(
kH
)(g
k
tanh(
kH
)+
p
2
)
cosh(
kz
)
g
k
np
exp
{
pt
−
imx
−
iny
}
−
×
H
(
p
,
m
,
n
)
(2.41)
s
+
i
∞
+
∞
+
∞
w
(
x
,
y
,
z
,
t
)=
∂
F
∂
1
=
−
d
p
d
m
d
n
3
i
z
8
π
s
−
i
∞
−
∞
−
∞
cosh(
kz
)
g
k
tanh(
kz
)
p
2
p
exp
{
pt
−
imx
−
iny
}
−
×
H
(
p
,
m
,
n
)
(2.42)
cosh(
kH
)(g
k
tanh(
kH
)+
p
2
)
.
In principle, expressions (2.39)-(2.42) provide an exhaustive solution of prob-
lem (2.29)-(2.31), but obtaining concrete results requires the calculation of sixfold
integrals, which represents quite a realistic, but extremely labour-consuming (from
the point of view of the volume of calculations) and irrational task. To be able to
perform part of the calculations analytically it is necessary to set the concrete form
of function
η
(
x
,
y
,
t
).
2.2.2 Cylindrical Coordinates
In a number of cases, when the model displacement of the basin floor exhibits ap-
propriate symmetry, it may turn out to be convenient to apply a cylindrical reference
system, which we shall introduce in a standard manner with respect to the Cartesian
system, described in Sect. 2.2.1. In this case the Laplace equation (2.29) assumes
the following form:
