Geoscience Reference
In-Depth Information
Let the propagating perturbation of atmospheric pressure be described by the
formula
p
atm
(
x
,
t
)=
p
(
x
−
Vt
)
θ
(
t
)
,
(4.22)
where
p
is an arbitrary function determining the space distribution of the pressure,
θ
is the steplike Heaviside function and
V
is the propagation velocity of the pertur-
bation. The dynamics of the atmospheric process, described by formula (4.22), is
such that at time moment
t
= 0 the atmospheric pressure perturbation is 'switched
on' and starts movement unlimited in time with constant velocity
V
in the positive
direction of axis 0
x
. In the case considered the integrals in expression (4.21) are
calculated analytically, and the solution of the problem is given by the formula
−
−
(
x
,
t
)=
p
(
x
Vt
)
p
(
x
t
)
p
(
x
+
t
)
2(
V
+ 1)
.
ξ
1
−
1)
+
(4.23)
V
2
−
2(
V
−
From formula (4.23) it follows that the wave perturbation on the water surface has
three components. One of them propagates with the velocity
V
, following the area
of altered pressure. The other two components correspond to free waves travelling
along axis 0
x
in the positive and negative directions, respectively, with the veloc-
ity of long waves. The amplitude of waves on the water surface depends strongly
on the propagation velocity of the atmospheric perturbation. Here, the amplitude
of waves travelling in the same direction as the atmospheric perturbation may un-
dergo a sharp increase, when
V
1. When the equality
V
= 1 is satisfied exactly,
the growth of amplitude is without limit, within the model considered. This effect is
known as the 'Proudman resonance'. The amplitude of waves travelling in the nega-
tive direction of axis 0
x
exhibit no such peculiarities. Always remaining a relatively
small quantity, it monotonously decreases as the velocity
V
increases.
It is possible to determine the behaviour of a wave perturbation on a water surface
in resonance conditions by calculating the limit of expression (4.23), when
V
≈
.
The resonance effects involve the first two terms of expression (4.23). In the case
of resonance, each of these terms tends towards infinity, but their sum has a finite
limit. We shall, now, expand function
p
(
x
→
∞
−
Vt
) in a Taylor series at point
z
0
=
x
−
t
with an accuracy up to the linear term,
p
(
z
0
)+
p
(
z
0
)(
z
≈
−
z
0
)
.
p
(
z
0
)
Upon performing elementary transformations we obtain an expression, describing
the free-surface displacement in the case of resonance,
(
x
,
t
)
=
p
(
x
−
t
)
t
p
(
x
−
t
)
+
p
(
x
+
t
)
4
ξ
res
(
x
,
t
)= lim
V
ξ
−
−
.
(4.24)
2
4
→
1
From formula (4.24) it follows that, when resonance conditions are fulfilled,
the wave perturbation comprises three components. The first component represents
a wave of amplitude, increasing linearly with time, and the growth rate of the am-
plitude is proportional to the derivative of the distribution of pressure in space. The
other two components describe waves of insignificant and fixed amplitudes.
