1.7. Snell’s law of refraction 41 Once these leading terms are known, the 0 term from the second equa- tion in (1.7.9) shows that on x1 = 0, a1 − b1 − d1 = known . Since a1 is known, this together with the case j = 2 from (1.7.10) suﬃces to determine b1,d1 on x1 = 0. Each satisfies a transport equation along rays that are the analogue of (1.4.12). Thus from the initial values just computed on x1 = 0, they are determined everywhere. The higher order correctors are determined analogously. Once the bj,dj are determined, one can choose b, c as functions of with the known Taylor expansions at x = 0. They can be chosen to have supports in the appropriate tubes of rays and to satisfy the transmission conditions (1.7.9) exactly. The function u is then an infinitely accurate approximate solution in the sense that it satisfies the transmission and initial conditions exactly while the residuals v tt − Δ v := r in x1 0 , v tt − c2 Δ v := ρ satisfy for all N, s, T there is a C so that r Hs([−T,T ]×{x1 0}) + ρ Hs([−T,T ]×{x1 0}) ≤ C N . From the analysis of the transmission problem, it follows that with new constants, u − v Hs([−T,T ]×{x1 0}) ≤ C N . The proposed problem of describing the family of solutions u is solved. The angles of incidence and refraction, θi and θr, are computed from the directions of propagation of the incident and transmitted waves. From Figure 1.7.1 one finds sin θi = |ξ | |ξ| , and sin θr = |η | |η| = |ξ | |ξ|/c . Therefore, sin θi sin θr = 1 c , is independent of θi. The high frequency asymptotic solutions explain Snell’s law. This is the last of the three basic laws of geometric optics. The deriva- tion of Snell’s law only uses the phases of the incoming and transmitted waves. The phases are determined by the requirement that the restriction of the phases to x1 = 0 are equal. They do not depend on the precise trans- mission condition that we chose. It is for this reason that the conclusion is the same for the correct transmission problem for Maxwell’s equations.

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