ABSTRACT
This chapter is devoted to the investigation of waves whose dispersion relations are nonlinear in
nature.
Consider a one-dimensional wave pulse,
ψ(x, t) =
C(k) cos(ω t − k x) dk, (9.1)
made up of a linear superposition of cosine waves, with a range of different wavenumbers, all
traveling in the positive x-direction. The angular frequency, ω, of each of these waves is related to
its wavenumber, k, via the so-called dispersion relation, which can be written schematically as
ω = ω(k). (9.2)
In general, this relation is derivable from the wave disturbance’s equation of motion. Up to now, we
have only considered sinusoidal waves that have linear dispersion relations of the form
ω = k v, (9.3)
where v is a constant. The previous expression immediately implies that such waves have the same
phase velocity,
vp = ω
k = v, (9.4)
irrespective of their frequencies. Substituting Equation (9.3) into Equation (9.1), we obtain
ψ(x, t) =
C(k) cos[k (v t − x)] dk, (9.5)
which is the equation of a wave pulse that propagates in the positive x-direction, at the fixed speed
v, without changing shape. (See Chapter 8.) The previous analysis seems to suggest that arbitrarily-
shaped wave pulses generally propagate at the same speed as sinusoidal waves, and do so without
dispersing or, otherwise, changing shape. In fact, these statements are only true of pulses made up of
superpositions of sinusoidal waves with linear dispersion relations. There are, however, many types
166 Oscillations and Waves: An
of sinusoidal wave whose dispersion relations are nonlinear. For instance, the dispersion relation of
sinusoidal electromagnetic waves propagating through an unmagnetized plasma is (see Section 9.3)
ω =
√ k 2 c 2 + ω 2p e, (9.6)
where c is the speed of light in vacuum, and ωp e is a constant, known as the (electron) plasma
frequency, that depends on the properties of the plasma. [See Equation (9.28).] Moreover, the dis-
persion relation of sinusoidal surface waves in deep water is
ω =
√ g k +
T
ρ k 3, (9.7)
where g is the acceleration due to gravity, T the surface tension of water, and ρ the mass density.
Sinusoidal waves that satisfy nonlinear dispersion relations, such as (9.6) or (9.7), are known as dis-
persive waves, as opposed to waves that satisfy linear dispersion relations, such as (9.3), which are
called non-dispersive waves. As we saw previously, a wave pulse made up of a linear superposition
of non-dispersive sinusoidal waves, all traveling in the same direction, propagates at the common
phase velocity of these waves, without changing shape. How does a wave pulse made up of a linear
superposition of dispersive sinusoidal waves evolve in time?
