Geography Reference
In-Depth Information
Eliminating u
between (7.20) and (7.21) yields
∂
∂t
+
2
∂
2
h
∂x
2
∂
∂x
gHδρ
ρ
1
h
−
u
=
0
(7.22)
which is a wave equation similar in form to (7.14). It is easily verified by direct
substitution that (7.22) has a solution of the form
h
=
A exp
[
ik(x
−
ct)
]
where the phase speed c satisfies the relationship
±
gHδρ
ρ
1
1
/
2
c
=
u
(7.23)
If the upper and lower layers are air and water, respectively, then δρ
≈
ρ
1
and
the phase speed formula simplifies to
gH
The quantity
√
gH is called the
shallow water
wave speed. It is a valid approx-
imation only for waves whose wavelengths are much greater than the depth of
the fluid. This restriction is necessary in order that the vertical velocities be small
enough so that the hydrostatic approximation is valid. For an ocean depth of 4 km,
the shallow water gravity wave speed is
c
=
u
±
200ms
−
1
. Thus, long waves on the
ocean surface travel very rapidly. It should be emphasized again that this theory
applies only to waves of wavelength much greater than H . Such long waves are not
ordinarily excited by the wind stresses, but may be produced by very large-scale
disturbances such as earthquakes.
3
Shallow water gravity waves may also occur at interfaces within the ocean where
there is a very sharp density gradient (diffusion will always prevent formation of
a true density discontinuity). In particular, the surface water is separated from the
deep water by a narrow region of sharp density contrast called the
thermocline
.If
the horizontal pressure gradient vanishes in the layer above the thermocline, then
(7.22) governs the displacement, h
, of the thermocline from its mean height H .
If the density changes by an amount δρ /ρ
1
≈
≈
0.01, across the thermocline, then
from (7.23) the wave speed for waves traveling along the thermocline will be only
one-tenth of the surface wave speed for a fluid of the same depth.
4
3
Long waves excited by underwater earthquakes or volcanic eruptions are called
tsunamis.
4
Gravity waves propagating along an internal density discontinuity are sometimes referred to as
internal
waves. We will, however, reserve that terminology for the vertically propagating waves con-
sidered in Section 7.4.
