Environmental Engineering Reference
In-Depth Information
ofthewavefrequencyonitsamplitude,where
E
is a (small) field amplitude and
ω
0
is the frequency of this wave. Inserting expansions (5.103) and (5.104) into (5.102),
we obtain
D
X
k
a
(
x
,
t
)
a
(
k
)
exp
i
(
k
E
2
(
x
)
t
k
0
)
2
@
v
g
@
k
0
)(
x
x
0
)
ik
0
x
0
i
(
k
k
t
i
α
(5.105)
for the wave packet amplitude. From (5.105) it follows that nonlinear wave interac-
tions can lead to modulation of the wave packet. With certain types of modulation
the wave packet may decay into separate bunches, or it may be compressed into a
solitary wave - a soliton. This phenomenon is known as modulation instability.
Formula (5.105) shows that the compression of a wave packet or its transforma-
tion into separate bunches can take place only if the last two terms in the exponent
in this equation have opposite signs. Only in this case can a nonlinear interaction
compensate for the usual divergence of the wave packet. Therefore, the modulation
instability can only occur if the following criterion holds true:
α
@
v
g
@
<
0 .
(5.106)
k
This condition is called the Lighthill criterion [66].
5.4.2
The Korteweg-de Vries Equation
The above analysis shows that wave dispersion leads to divergence of the wave
packet. If this divergence is weak, a weak nonlinearity is able to change its character.
We now consider an example of such behavior when a wave is characterized by low
dispersion and nonlinearity. Consider the propagation of long-wavelength waves in
a medium in which the dispersion relation has the form
r
0
k
2
),
r
0
k
ω
D
v
g
k
(1
1 .
(5.107)
This formdescribes a variety of oscillations, with sound and ion sound as examples.
We shall obtain below a nonlinear equation that describes such waves.
We take as the starting point the Euler equation (4.6) for the velocity of particles
in a longitudinal wave that has the form
@
v
@
C
v
@
v
@
F
m
D
x
0 .
(5.108)
t
Here
v
(
x
,
t
) is the particle (gas atom or gas molecule) velocity in a longitudinal
wave which propagates along the
x
-axis,
F
is the force per atomic particle, and
m
is
the mass of the atomic particle. Within the framework of a linear approximation,
