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but opposite in sign, perturbation forms of the water surface—an induced wave.
Moreover, there arise two free waves, travelling in opposite directions, and the one
propagating in the same direction as the atmospheric perturbation has a larger ampli-
tude, while its polarity is opposite to the polarity of the induced wave. When
V
1,
the amplitude and polarity of the induced wave are in accordance with the values
determined by the inverse barometer law. Practically, all atmospheric processes in
open ocean serve as natural prototypes for slowly propagating atmospheric pertur-
bations. Thus, for instance, the propagation velocity of a tropical cyclone usually
amounts to
V
5-10 m/s, whic
h is
significantly inferior to the propagation velocity
of long waves at large depths
√
g
H
∼
200 m/s (for
H
= 4,000 m).
In the case of resonance (
V
= 1) only two waves are observed on the water sur-
face. The induced wave follows the atmospheric perturbation, linearly increasing its
amplitude with time. The second wave is free. It travels in the opposite direction, and
its amplitude is small. We underline that within the framework of the model prob-
lem considered the amplitude of the induced wave grows without limit. Fulfilment of
the resonance conditions is possible, f
or e
xample, in shallow water (
H
∼
∼
10-100 m),
where the velocity of long waves (
√
g
H
10-30 m/s) may turn out to be close to
the typical propagation velocity of atmospheric perturbations.
If the velocity
V
exceeds the critical velocity (in Fig. 4.10 the case of
V
= 1
.
25
is presented), then the induced wave turns out to be similar in shape and sign to
the perturbation of atmospheric pressure. The polarity of the free wave, travelling in
the same direction as the atmospheric perturbation, differs in polarity from the in-
duced wave. The free wave, travelling in the opposite direction, here, like in all other
cases, repeats the polarity of the atmospheric perturbation. From formula (4.23) it
is seen that at high velocities
V
the amplitude of the free surface response tends
asymptotically toward zero. Note that the similar dependence (2.85) for waves gen-
erated by a running displacement of the ocean bottom exhibits a somewhat different
character: at high velocities
V
the surface displacement tends towards a constant,
instead of zero. In reality, the pressure perturbations, corresponding to the velocity
range
V
∼
1, may be related, for example, to acoustic waves in the atmosphere or
to the propagation of atmospheric internal waves above shallow-water areas.
Figure 4.11 illustrates the character of variation of the maximum amplitude of
waves on the water surface versus the distance covered by the atmospheric pertur-
bation,
L
=
Vt
. The maximum range is calculated by the formula
(
x
,
t
)
(
x
,
t
)
.
A
max
(
x
)=max
t
ξ
−
min
t
ξ
(4.26)
From the figure it is seen that for a noticeable increase in the amplitude it is
necessary that the resonance condition be fulfilled along a path the length of which
holds several horizontal extensions of the atmospheric perturbation. If the velocity
V
= 1, then the growth of the amplitude is limited. At any rate, at the initial stage
of wave formation, when
V
1, the growth rate of the amplitude does not differ
strongly from the resonance case. Therefore, if the velocity of the atmospheric per-
turbation,
V
, varies within limits
≈
±
10% of the resonance velocity, then a tenfold
