Geoscience Reference
In-Depth Information
•
For
x
−
a
∞
j
=1
Q
exp
{
k
j
x
}
sinh(
k
j
a
)
2
ξ
osc
(
x
,
t
)=
η
0
exp
{
i
ω
t
}
2
ω
2
P
(exp
+
η
0
ω
{
i
(
ω
t
+
k
0
(
x
−
a
))
}−
exp
{
i
(
ω
t
+
k
0
(
x
+
a
))
}
)
.
(2.97)
•
For
|
x
|
a
j
=1
Qk
j
exp
{−
k
j
a
}
sinh(
k
j
x
)
,
∞
u
(
x
,
z
,
t
)=
η
0
i
ω
exp
{
i
ω
t
}
−
2
(2.98)
1
cosh(
k
j
x
)
,
∞
j
=1
∂
Q
w
(
x
,
z
,
t
)=
η
0
i
ω
exp
{
i
ω
t
}
−
2
exp
{−
k
j
a
}
(2.99)
∂
z
•
For
|
x
|
a
u
(
x
,
z
,
t
)
j
=1
Qk
j
exp
{−
k
j
|
x
|}
sinh(
k
j
a
)
,
∞
=
η
0
i
ω
exp
{
i
ω
t
}
−
2sign(
x
)
(2.100)
2
sinh(
k
j
a
)
,
∞
j
=1
∂
Q
w
(
x
,
z
,
t
)=
η
0
i
ω
exp
{
i
ω
t
}
exp
{−
k
j
|
x
|}
(2.101)
∂
z
Note that, owing to the discontinuity exhibited by the function, describing
the space distribution of oscillations of the basin bottom, expressions (2.98)-(2.101)
do not yield adequate values of the flow velocity at the points with coordinates
{
. To obtain the exact velocity values in the immediate vicinity of
the points indicated it is necessary to take into account quite a large number of terms
of the expansion in
j
.
From the structure of the obtained formulae it is seen that perturbation of a liquid
consists of induced oscillations, occurring in the immediate vicinity of the source
(and exponentially dying away with the distance from it), and a series of progressive
waves starting at points
x
=
x
=
±
a
,
z
=
−
1
}
a
.
From the point of view of tsunami generation precisely the amplitude of pro-
gressive waves is important. From formulae (2.96) and (2.97) it is seen that this
±
amplitude is largely determined by the quantity
ω
2
P
, the dependence of which
upon the cyclic frequency is shown in Fig. 2.26. A most important peculiarity of
the response of the liquid to oscillations of a part of the ocean bottom consists in
the existence of a certain boundary frequency, which, when surpassed, the efficiency
of wave emission drops drastically. Thus, at high frequencies all the motions of
the liquid is concentrated exclusively in the vicinity of the source and represents
induced oscillations.
