Geoscience Reference
In-Depth Information
water particles can be neglected. To describe the waves we take advantage of linear
potential theory. In this problem it is essential to take into account phase dispersion,
so application of the long-wave approximation is not permitted. Note that owing to
the large wave amplitudes (comparable to the depth) the application of linear the-
ory is also not quite correct. But for approximate estimation of the properties of
a cosmogenic tsunami such an approach is quite justified.
In the case of an ocean of constant depth, evolution of the initial free-surface
perturbation, exhibiting radial symmetry, is described by the following expression
(see general theory in Sect. 2.2.2):
∞
k
d
kA
(
k
)
J
0
(
kr
) cos
ω
(
k
)
t
,
ξ
(
r
,
t
)=
(4.47)
0
A
(
k
)=
∞
r
d
r
ξ
0
(
r
)
J
0
(
kr
)
,
(4.48)
0
where
ξ
0
(
r
) is the function describing the form of the initial perturbation,
J
0
is
the Bessel function of the first kind of 0th order. The relation between the cyclic
frequency and the wave number is determined by the known dispersion relation for
gravitational waves on water,
2
= g
k
tanh(
kH
). For an initial perturbation, deter-
mined by formula (4.36), the Fourier-Bessel transformation (4.48) yields the fol-
lowing form for the dependence of the amplitude of space harmonics upon the wave
number:
ω
R
D
R
D
−
2
R
D
J
2
(
kR
D
)
R
C
kJ
1
(
kR
D
)
−
A
(
k
)=
D
C
.
(4.49)
R
C
k
2
Figure 4.17 presents the example of a calculation of waves caused by the fall
into an ocean 4 km deep of a celestial body of radius 100 m, density 3,000 kg/m
3
,
moving with a velocity of 20 km/s. It is seen that during the first minutes after
the fall the waves in the immediate vicinity of the incidence point may reach colos-
sal heights of the order of 1 km and more. Figure 4.18 presents in dimensionless
coordinates the wave number dependences of phase and group velocities of surface
gravitational waves on water. The same plot shows the amplitude distribution of
space harmonics over the wave numbers, calculated in accordance with the form
of the initial perturbation (4.36) for an internal cavity radius equal to the ocean
depth. The amplitude distribution is determined by function
, where
A
(
k
) is
given by formula (4.49). The position of the space spectrum on the wave number
axis is related in an evident manner to the cavity radius. Therefore, on the basis
of calculations for
R
C
=
H
, from which the maximum turns out to be located at
the value
kH
|
kA
(
k
)
|
2
.
97, it is possible to write the formula determining the position
of the maximum as function of the cavity radius,
k
max
≈
≈
2
.
97
/
R
C
. Recalculation
for the wave lengths reveals this to correspond to
2
.
12
R
C
, which somewhat
exceeds the cavity diameter. Taking advantage of the dispersion relation, it is not
difficult to determine the period corresponding to the maximum of the spectrum,
λ
max
≈
