Geography Reference
In-Depth Information
The mean zonal velocity here plays only a role of
Doppler shifting
the sound
wave so that the frequency relative to the ground corresponding to a given wave
number k is
c
s
)
Thus, in the presence of a wind, the frequency as heard b
y a
fixed observer depends
on the location of the observer relative to the source. If u>0 the frequency of a
stationary source
wi
ll appear to be higher for an observer to the east (downstream)
of th
e
source (c
ν
=
kc
=
k (u
±
=
u
+
c
s
) than for an observer to the west (upstream) of the source
c
=
u
−
c
s
.
7.3.2
Shallow Water Gravity Waves
As a second example of pure wave motion we consider the horizontally propagating
oscillations known as shallow water waves. Shallow water gravity waves can exist
only if the fluid has a free surface or an internal density discontinuity. As shown
in the previous subsection, in acoustic waves the restoring force is parallel to the
direction of propagation of the wave. In shallow water gravity waves, however,
the restoring force is in the vertical so that it is transverse to the direction of
propagation.
The mechanism for propagation of gravity waves can be understood by consider-
ing water in a channel extending in the x direction with an oscillating paddle at the
origin. The back-and-forth oscillations of the paddle generate alternating upward
and downward perturbations in the free surface height, which produce alternating
positive and negative accelerations. These, in turn, lead to alternating patterns of
fluid convergence and divergence. The net result is a sinusoidal disturbance of the
free surface height, which moves toward the right, and has perturbation velocity
and free surface height exactly in phase as shown in Fig. 7.6. A similar sort of
disturbance could be set up moving toward the left, but in that case the velocity
and free surface height perturbations would be exactly 180˚ out of phase.
As a specific example we consider a fluid system consisting of two homoge-
neous incompressible layers of differing density as shown in Fig. 7.7. Waves may
propagate along the interface between the two layers. The assumption of incom-
pressibility is sufficient to exclude sound waves, and we can thus isolate the gravity
waves. If the density of the lower layer ρ
1
is greater than the density of the upper
layer ρ
2
, the system is stably stratified. Because both ρ
1
and ρ
2
are constants, the
horizontal pressure gradient in each layer is independent of height if the pressure is
hydrostatic. This may be verified by differentiating the hydrostatic approximation
with respect to x:
∂p
∂x
∂
∂z
∂ρ
∂x
g
=−
=
0
For simplicity, we assume that there is no horizontal pressure gradient in the
upper layer. The pressure gradient in the lower layer can be obtained by vertical
