ABSTRACT
In general, this relation is derivable from the wave disturbance’s equation of motion. Up to now, we have considered only sinusoidal waves that have linear dispersion relations of the form
ω = k v, (9.3)
where v is a constant. The above 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 above 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 of sinusoidal waves whose dispersion relations are nonlinear. For instance, the dispersion relation of sinusoidal electromagnetic waves propagating through an unmagnetized plasma is (see Section 9.2)
ω = √
k2 c2 + ω 2p , (9.6)
and An
where c is the speed of light in vacuum, and ωp is a constant, known as the plasma frequency, that depends on the properties of the plasma. [See Equation (9.28).] Moreover, the dispersion relation of sinusoidal surface waves in deep water is (see Section 9.7)
ω =
√
g k + T ρ
k3, (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 dispersive waves, as opposed to waves that satisfy linear dispersion relations, such as (9.3), which are called nondispersive waves. As we saw previously, a wave pulse made up of a linear superposition of nondispersive 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?
