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.
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