Geoscience Reference
In-Depth Information
cumbersome formulae. In this connection, it is much more simple to select a certain
free-surface displacement and, by applying the unambiguous functional relationship
(4.34), to determine the pressure perturbation, which could have been caused by this
displacement.
For definiteness we shall assume the free-surface displacement to be determined
by the function
(
x
,
t
)=
A
max
exp
.
Vt
)
2
a
2
(
x
−
ξ
−
The perturbation of pressure, corresponding to it, which can be calculated by for-
mula (4.34) in the vicinity of the resonance or for large values of the quantity
A
max
,
generally speaking has a form essentially differing from a Gaussian. Therefore, it
has sense to introduce a certain quantity
p
max
, characterizing the intensity of pres-
sure variations. Let
p
max
denote the amplitude of pressure variations, understood
in the sense of formula (4.26). The ratio of quantities
A
max
and
p
max
(in dimen-
sionless form
A
max
ρ
g
/
p
max
) represents the 'amplification coefficient' of the wave
amplitude.
In Fig. 4.12 the quantity
A
max
g
/
p
max
is plotted against the propagation velocity
V
of the atmospheric perturbation. The dotted line in the figure shows the depen-
dence, corresponding to linear theory and calculated with the use of formula (4.35).
The amplification coefficient is seen to depend, in accordance with non-linear the-
ory, not only on the velocity
V
, but also on the sign of the wave produced. For
positive waves the Proudman resonance point turns out to be shifted to the right as
compared with the linear case, while in the case of negative perturbations to the left.
Moreover, the growth of the wave amplitude for any fixed values of velocity
V
turns
out to be limited. This fact is a most important manifestation of non-linearity in
the problem dealt with. Let us briefly dwell upon its physical interpretation. Con-
sider the resonance condition
V
2
/
g
H
= 1 to be fulfilled, starting from a certain
moment of time. Then, at the initial stage, in accordance with linear theory, an in-
crease in the amplitude of the free-surface perturbation will take place. But, as soon
as the quantity
ρ
ξ
reaches sufficiently high values, the actual basin depth, present in
Fig. 4.12 Ratio of free-
surface displacement am-
plitude
A
max
and amplitude
of atmospheric pressure
perturbation,
p
max
, versus
propagation velocity of at-
mospheric perturbation. The
calculation is performed for
positive (
A
max
/
H
= 0
.
25),
negative (
A
max
/
H
=
0
.
25),
and infinitesimal (dotted line)
displacements of the free
surface
−
