20 1. Simple Examples of Propagation
The analysis of Exercise 1.3.3 does not apply to the fundamental so-
lution since the latter does not have finite energy. However it belongs to
Cj(R
:
Hs−j(R))
for all s 1/2 and j N. The next result provides a
good replacement for (1.3.4).
Exercise 1.3.4. Suppose that u is the fundamental solution of the Klein–
Gordon equation (1.1.6) and that s 1/2. If 0 χ
C∞(R)
is a plateau
cutoff supported on the positive half line, that is
χ(x) = 0 for x 0 and χ(x) = 1 for x 1 ,
then for all 0 and R 0 there is a δ 0 so that
(1.3.5) lim sup
t−∞
χ(R + |x| (1 δ)t) u(t, x)
Hs(Rx)
.
Hints. Prove that
χ u(t)
Hs(R)
C u(0)
Hs(R)
+ ut(0)
Hs−1(R)
with C independent of t and the initial data. Conclude that it suffices to
prove (1.3.4) with initial data u(0),ut(0) dense in Hs × Hs−1. Take the
dense set to be data with Fourier transform in C0
∞(R).
These examples illustrate the important observation that the propaga-
tion of singularities in solutions and the propagation of the majority of the
energy may be governed by different rules. For the Klein–Gordon equation,
both answers can be determined from considerations of group velocities.
1.4. Fourier synthesis and rectilinear propagation
For equations with constant coefficients, solutions of the initial value prob-
lem are expressed as Fourier integrals. Injecting short wavelength initial
data and performing an asymptotic analysis yields the approximations of
geometric optics. This is how such approximations were first justified in
the nineteenth century. It is also the motivating example for the more gen-
eral theory. The short wavelength approximations explain the rectilinear
propagation of waves in homogeneous media. This is the first of the three
basic physical laws of geometric optics. It explains, among other things, the
formation of shadows. The short wavelength solutions are also the building
blocks in the analysis of the laws of reflection and refraction presented in
§1.6 and §1.7.
Consider the initial value problem
(1.4.1) 0 = u :=
∂2u
∂t2

d
j=1
∂2u
∂xj 2
, u(0,x) = f, ut(0,x) = g.
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