1.7. Snell’s law of refraction 39 folding them to a boundary value problem and using the results of [Rauch and Massey, 1974] and [Sakamoto, 1982]. Next consider the mathematical problem whose solution explains Snell’s law. The idea is to send a wave in x1 0 toward the boundary and ask how it behaves in the future. Suppose ξ Rd, |ξ| = 1 , ξ1 0 , and consider a short wavelength asymptotic solution in {x1 0} as in §1.6.3, (1.7.5) I ei(xξ−t)/ a(, t, x) , a(, t, x) a0(t, x)+ a1(t, x)+ · · · , where for t 0 the support of the aj is contained in a tube of rays with compact cross section and moving with speed ξ. Take a to vanish outside the tube. Since the incoming waves are smooth and initially vanish identically on a neighborhood of the interface {x1 = 0}, the compatibilities are satisfied and there is a family of piecewise smooth solutions u defined on R1+d. We construct an infinitely accurate description of the family of solutions u . Seek an asymptotic solution that at {t = 0} is equal to this incoming wave. A first idea is to find a transmitted wave which continues the incoming wave into {x1} 0. Seek the transmitted wave in x1 0 in the form T ei(xη+tτ)/ d(, t, x) , d(, t, x) d0(t, x) + d1(t, x) + · · · . On the interface, the incoming wave oscillates with phase (x ξ t)/ and the proposed transmitted wave oscillates with phase (x η + tτ)/. In order that there be any chance at all of satisfying the transmission conditions one must take η = ξ , τ = −1, so that the two expressions oscillate together. In order that the transmitted wave be an approximate solution on the right with positive velocity, one must have τ 2 = c2|η|2, η1 0 . The equation τ 2 = c2|η|2 implies η2 1 = τ 2 c2 −|η |2 = 1 c2 −|ξ |2 , so η1 = 1 c2 −|ξ |2 1/2 ξ1 0. Thus, (1.7.6) T ei(xη−t)/ d(, t, x) , η = ( (c−2 |2)1/2 , ξ ) . value problem is u = cos t −A f + sin t −A −A g . For piecewise H∞ data, the sequence of compatibilities is equivalent to the data belonging to j D(Aj).
Previous Page Next Page