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.