Geoscience Reference
In-Depth Information
Taking advantage of the boundary conditions on the surface and on the bottom, we
obtain:
+
∞
F
osc
(
x
,
z
,
t
)=
η
0
ω
H
exp
{
i
ω
t
}
d
k
2
π
−
∞
)
k
cosh(
kz
)+
2
sinh(
kz
)
(exp
{−
ik
(
x
−
a
)
}−
exp
{−
ik
(
x
+
a
)
}
ω
×
.
(2.88)
k
2
(
2
cosh(
k
)
ω
−
k
sinh(
k
))
Expression (2.88) contains dimensionless variables under the integral sign that
were introduced in accordance with the formulae (the sign '*' has been dropped):
(
x
∗
,
z
∗
,
a
∗
)=(
x
,
z
,
a
)
H
−
1
;
t
∗
=
tg
1
/
2
H
−
1
/
2
;
ω
∗
=
g
−
1
/
2
H
1
/
2
;
k
∗
=
kH
,
ω
but the multiplier before the integral and the velocity potential itself are dimensional
quantities.
To calculate the integral (2.88) it suffices to know the value of an integral of
the following form:
α
}
k
cosh(
kz
)+
2
sinh(
kz
)
+
∞
d
k
exp
{−
ik
ω
,
(2.89)
k
2
(
2
cosh(
k
)
ω
−
k
sinh(
k
))
−
∞
α
±
where the parameter
a
may assume positive, negative and zero values.
Let us continue the integrand function in (2.89) analytically from the real axis
onto the entire complex plane (
=
x
{
Re(k), Im(k)
}
). The integrand has two singular
points on the real axis,
k
=
±
k
0
, and an infinite number of singular points on
the imaginary axis,
k
=
ik
j
. The singular points are poles of the first order, and
their positions are determined from the solutions of the two following transcendental
equations:
±
2
cosh(
k
)
ω
−
k
sinh(
k
)=0
,
(2.90)
2
+
k
sin(
k
)=0
.
cos(
k
)
ω
(2.91)
The integrand function in (2.89) has no other singular points, which is readily
demonstrated with the aid of the theorem on counting the number of zeros of an
analytical function [Sveshnikov, Tikhonov (1999)].
Since the integrand function has poles on the real axis, the integral (2.89) must
be understood in the sense of its principal value (v.p.), according to Cauchy. For its
calculation the theorem of residues was applied. The ultimate expression, determin-
ing the velocity potential of a liquid flow in the case of established oscillations of
a part of the bottom, has the following form:
