Geography Reference
In-Depth Information
c
ν/k, the phase speed also depends on the wave number except in the special
case where ν
=
k. For waves in which the phase speed varies with k, the various
sinusoidal components of a disturbance originating at a given location are at a
later time found in different places, that is, they are dispersed. Such waves are
referred to as
dispersive
, and the formula that relates ν and k is called a
dispersion
relationship
. Some types of waves, such as acoustic waves, have phase speeds
that are independent of the wave number. In such
nondispersive
waves a spatially
localized disturbance consisting of a number of Fourier wave components (a
wave
group
) will preserve its shape as it propagates in space at the phase speed of the
wave.
For dispersive waves, however, the shape of a wave group will not remain con-
stant as the group propagates. Because the individual Fourier components of a
wave group may either reinforce or cancel each other, depending on the relative
phases of the components, the energy of the group will be concentrated in limited
regions as illustrated in Fig. 7.3. Furthermore, the group generally broadens in the
course of time, that is, the energy is
dispersed
.
When waves are dispersive, the speed of the wave group is generally different
from the average phase speed of the individual Fourier components. Hence, as
shown in Fig. 7.4, individual wave components may move either more rapidly or
more slowly than the wave group as the group propagates along. Surface waves
in deep water (such as a ship wake) are characterized by dispersion in which
individual wave crests move twice as fast as the wave group. In synoptic-scale
atmospheric disturbances, however, the group velocity exceeds the phase velocity.
The resulting downstream development of new disturbances will be discussed later.
An expression for the
group velocity
, which is the velocity at which the observ-
able disturbance (and hence the energy) propagates, can be derived as follows:
We consider the superposition of two horizontally propagating waves of equal
amplitude but slightly different wavelengths with wave numbers and frequencies
differing by 2δk and 2δν, respectively. The total disturbance is thus
∝
(x, t)
=
exp
{
i [(k
+
δk) x
−
(ν
+
δν) t ]
} +
exp
{
i [(k
−
δk) x
−
(ν
−
δν) t ]
}
Fig. 7.3
Wave groups formed from two sinusoidal components of slightly different wavelengths. For
nondispersive waves, the pattern in the lower part of the diagram propagates without change
of shape. For dispersive waves, the shape of the pattern changes in time.
