Geoscience Reference
In-Depth Information
The main equation of linear potential wave theory, (2.29), will remain without
changes. But in the boundary condition on the free surface there will now appear
a time-dependent coefficient,
g+
a
(
t
)
∂
2
F
F
∂
−
∂
=
t
2
,
z
= 0
.
(7.14)
∂
z
At the lower boundary in the new noninertial reference system it is necessary to
set the vertical velocity component equal to zero,
∂
F
= 0
,
z
=
−
H
(or
z
→−
∞
)
.
(7.15)
∂
z
The general solution of the problems (2.29), (7.14) and (7.15) is given by the fol-
lowing formula (analogue of expression (2.34)):
+
∞
+
∞
F
(
x
,
y
,
z
,
t
)=
d
m
d
n
exp(
imx
−
iny
)
−
∞
−
∞
A
(
t
,
m
,
n
) cosh(
kz
)+
B
(
t
,
m
,
n
) sinh(
kz
)
.
×
(7.16)
Substituting formula (7.16) into the boundary conditions (7.14) and (7.15), we
obtain that the potential is determined by the expression
+
∞
+
∞
F
(
x
,
y
,
z
,
t
)=
d
m
d
n
exp(
imx
−
iny
)
−
∞
−
∞
A
(
t
,
m
,
n
)
cosh(
kz
)+tanh(
kH
) sinh(
kz
)
,
×
(7.17)
and the coefficient
A
(
t
,
m
,
n
) can be found from the solution of the known Mathieu
equation
+ g
k
tanh(
kH
)
1 +
η
0
ω
t
)
A
= 0
.
2
A
∂
2
∂
cos (
ω
(7.18)
t
2
g
The properties of the solution of this equation are such, that, when the following
equality is satisfied:
g
k
tanh(
kH
)=
n
2
2
,
n
= 1
,
2
,
3
...,
(7.19)
in the system there arises a parametric resonance, in the case of which the solution
increases exponentially with time. The most rapid growth takes place, when
n
= 1.
The growth of oscillations is possible not only when relationship (7.19) is satisfied
exactly, but also within certain finite intervals of the pumping frequency values,
termed instability zones. The widths of these zones increase with the coefficient
η
0
ω
2
/
g in equation (7.18).
