36 1. Simple Examples of Propagation The analogue of (1.6.3) in this case is (1.6.8) ∀n ≥ 0, ∂2n+1u ∂x2n+1 1 x1=0 = 0 . Exercise 1.6.1. Prove Proposition 1.6.4. Exercise 1.6.2. Prove uniqueness of solutions by the energy method. Hint. Use the local energy identity. Exercise 1.6.3. Verify the assertion concerning the reflection coeﬃcient by following the examples above. That is, consider the case of dimension d = 1, the case of spherical waves with d = 3, and the behavior in the future of a solution which near t = 0 is a high frequency asymptotic solution approaching the boundary. 1.7. Snell’s law of refraction Refraction is the bending of waves as they pass through media whose propa- gation speeds vary from point to point. The simplest situation is when media with different speeds occupy half-spaces, for example x1 0 and x1 0. The classical physical situations are when light passes from air to water or from air to glass. Snell observed that for fixed materials, the ratio of the sines of the angles of incidence and refraction sin θi/ sin θr is independent of the incidence angle. Fermat observed that this would hold if the speed of light were different in the two media and the light chose a path of least time. In that case, the quotient of sines is equal to the ratio of the speeds, ci/cr. In this section we derive this behavior for a model problem quite close to the natural Maxwell equations. The simplified model with the same geometry is (1.7.1) utt−Δ u = 0 in x1 0 , utt−c2 Δ u = 0 in x1 0 , 0 c 1 . In x1 0, the speed is equal to 1 which is greater than the speed c in x 0.3 A transmission condition is required at x1 = 0 to encode the interaction of waves with the interface. In the one-dimensional case, there are waves that approach the boundary from both sides. The waves that move from the boundary into the interior must be determined from the waves that arrive from the interior. There are two arriving waves and two departing waves. One needs two boundary conditions. 3To see that c is the speed of the latter equation, one can (in order of increasing sophistication) factor the one-dimensional operator ∂2 t −c2∂2 x = (∂t +c∂x)(∂t −c∂x), or use the formula for group velocity with dispersion relation τ 2 = c2|ξ|2, or prove finite speed using the differential law of conservation of energy or Fritz John’s Global H¨ olmgren Theorem.

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