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.
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2012 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.