1.4. Fourier synthesis and rectilinear propagation 23
The approximation is a wave packet with envelope A and wavelength . The
wave packet translates rigidly with velocity equal to e1. The waveform γ is
arbitrary. The approximate solution resembles the columnated light from a
flashlight. If the support of γ is small, the approximate solution resembles
a light ray.
The amplitude A satisfies the transport equation
so it is constant on the rays x = x + te1. The construction of a family
of short wavelength approximate solutions of D’Alembert’s wave equations
requires only the solution of a simple transport equation.
The dispersion relation of the family of plane waves,
is τ = −|ξ|. The velocity of transport, v = (1, 0,..., 0), is the group velocity
v = −∇ξτ(ξ) = ξ/|ξ| at ξ = (1, 0,..., 0). For the opposite choice of sign,
the dispersion relation is τ = |ξ|, the group velocity is −e1, and the rays are
the lines x = x − te1.
Had we taken data with oscillatory factor
then the approximate
solution would have been the sum of two wave packets with group velocities
γ x − t
γ x + t
The approximate solution (1.4.6) is a function H(x − te1) with H(x) =
h(x). When h has compact support, or more generally tends to zero
as |x| → ∞, the approximate solution is localized and has velocity equal
to e1. The next result shows that when d 1, no exact solution can have
this form. In particular the distribution δ(x −e1t) that is the most intuitive
notion of a light ray is not a solution of the wave equation or Maxwell’s
Proposition 1.4.1. If d 1, s ∈ R, K ∈
and u = K(x − e1t)
satisfies u = 0, then K = 0.
Exercise 1.4.1. Prove Proposition 1.4.1. Hint. Prove the following lemma.
Lemma. If k ≤ d, s ∈ R, and w ∈
satisfies 0 =
w = 0.