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