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

∂A

∂t

+

∂A

∂x1

= 0,

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,

ei(xξ+τt)

=

ei(xξ−|ξ|t),

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

eixξ/,

then the approximate

solution would have been the sum of two wave packets with group velocities

±ξ/|ξ|,

1

2

ei(xξ−t|ξ|)/

γ x − t

ξ

|ξ|

+

ei(xξ+t|ξ|)/

γ x + t

ξ

|ξ|

.

The approximate solution (1.4.6) is a function H(x − te1) with H(x) =

eix1/

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

equation.

Proposition 1.4.1. If d 1, s ∈ R, K ∈

Hs(Rd)

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 ∈

Hs(Rd)

satisfies 0 =

∑d

k

∂2w/∂2xj,

then

w = 0.