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extend over hundreds of kilometers, exhibiting a small angle to the horizontal plane.
Therefore, a displacement of the basin bottom, as a rule, has a component that can
be represented as a perturbation propagating in the horizontal direction. In the litera-
ture, such perturbations of the ocean bottom are conventionally termed 'running dis-
placements' [Novikova, Ostrovsky (1979); Vasilieva (1981); Marchuk et al. (1983)].
Let us name several other natural prototypes of the running displacement. This
role may be assumed by a non-simultaneous (sequential) displacement of blocks
of the bottom [Lobkovsky, Baranov (1982)], a crack propagating over the basin bot-
tom [Bobrovich (1988)], surface seismic waves [Belokon' et al. (1986)], the motion
of an underwater landslide [Garder et al. (1993); Kulikov et al. (1998)], [Watts et al.
(2001)]. Similar effects may be observed, also, in the case of wave generation by
a moving area of low or elevated pressure [Pelinovsky et al. (2001)].
The interest in running displacements arose, because when the propagation ve-
locity of a displacement coincides (even approximately) with the velocity of long
waves, (g H ) 1 / 2 , a resonance pumping is realized of energy into the tsunami wave.
Like in the preceding section, we shall first turn to the linear theory of long waves.
We shall take advantage of the one-dimensional wave equation (2.75), describing
perturbation of the surface,
ξ
( x , t ), that arises with deformation of the basin bot-
( x , t ). Assume a deformation of the basin bottom, the shape of which is set
by a certain function f , to propagate in the positive direction of the 0 x axis with
a constant velocity v :
η
tom,
vt ). Consider the motion to be established,
therefore the solution of equation (2.75) will also have the form of a perturbation
ξ
η
( x , t )= f ( x
vt ) running over the surface, where A 0 is a constant. Substituting
the form of the solution,
( x , t )= A 0 f ( x
( x , t ) into the wave equation, we find
the dependence of the constant A 0 upon the velocity of long waves and the propa-
gation velocity of the perturbation,
ξ
( x , t ), and function
η
v 2
ξ
( x , t )=
f ( x
vt ) .
(2.85)
v 2
g H
From formula (2.85) it is seen that over a deformation of the basin bottom,
travelling horizontally, there always exists a similar in shape perturbation of the wa-
ter surface. Given the condition v < (g H ) 1 / 2 , the perturbations of the surface and
of the basin bottom exhibit different polarities, while, when v > (g H ) 1 / 2 , their
polarities coincide. The velocities of the perturbation and of long waves being
close to each other result in a sharp enhancement of the amplitude of the surface
perturbation.
If the problem of an established running displacement is considered within
the framework of linear potential theory, then the main conclusion concerning res-
onance pumping of energy into the wave, when v
(g H ) 1 / 2 , does not change. At
velocities v > (g H ) 1 / 2 there will exist over the displacement a perturbation of sim-
ilar polarity. But for velocities v < (g H ) 1 / 2 , besides the perturbation of opposite
polarity located over the displacement, there also exists behind it a periodic in space
and stationary in time perturbation with a wavelength determined by the velocity v .
Standing waves, similar in nature, form when underwater obstacles are bypassed by
the flow [Sretensky (1977)].
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