34 1. Simple Examples of Propagation
The sign yields the interesting solution
ei(xξ−t|ξ|)

ei(x˜−t|˜|)
ξ ξ
,
which is twice the odd part of
ei(xξ−t|ξ|).
The textbook interpretation of the solution is that
ei(xξ−t|ξ|)
is a plane
wave approaching the boundary x1 = 0, and
ei(x˜−t|˜|)
ξ ξ moves away from the
boundary. The first is an incident wave and the second is a reflected wave.
The factor A = −1 is the reflection coefficient. The direction of motions are
computed using the group velocity computed from the dispersion relation.
Both waves are of infinite extent and of modulus one everywhere in
space-time. They have finite energy density but infinite energy. They both
meet the boundary at all times. It is questionable to think of either one as
incoming or reflected. The next subsection shows that there are localized
waves which are clearly incoming and reflected waves with the property that
when they interact with the boundary, the local behavior resembles the sum
of plane waves just constructed.
To analyze reflections for more general mixed initial boundary value
problems, wave forms more general than plane waves need to be included.
All solutions of the form
ei(xξ+tτ)
with ξ , τ real and Im ξ1 0 must be
considered. When Im ξ1 0, the associated waves are localized near the
boundary. The Rayleigh waves in elasticity are a classic example. They
carry the devastating energy of earthquakes. Waves of this sort are needed
to analyze total reflection described at the end of §1.7. The reader is referred
to [Benzoni-Gavage and Serre, 2007], [Chazarain and Piriou, 1982], [Taylor,
1981], [H¨ ormander, 1983, v.II], and [Sakamoto, 1982] for more information.
1.6.3. Reflected high frequency wave packets. Consider functions
that for small time are equal to high frequency solutions from §1.3,
(1.6.4) u =
ei(xξ−t|ξ|)/
a(, t, x) , a(, t, x) a0(t, x) + a1(t, x) + · · · ,
with
ξ = (ξ1,ξ2,...,ξd) , ξ1 0 .
Then a0(t, x) = h(x tξ/|ξ|) is constant on the rays x + tξ/|ξ|. If the
Cauchy data are supported in a set O {x1 0}, then the amplitudes aj
are supported in the tube of rays
(1.6.5) T := (t, x) : x = x + tξ/|ξ|, x O .
Finite speed shows that the wave as well as the geometric optics approxi-
mation stays strictly to the left of the boundary for small t 0.
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