42 1. Simple Examples of Propagation
η
η’
ξ’
ξ
Figure 1.7.1.
On a neighborhood (t,x) {x1 = 0} that is small compared to the scale
on which a, b, c vary and large compared to , the solution resembles three
interacting plane waves. In science texts one usually computes for which
such triples the transmission condition is satisfied in order to find Snell’s
law. The asymptotic solutions of geometric optics show how to overcome
the criticism that the plane waves have modulus independent of (t, x) so
cannot reasonably be viewed as either incoming or outgoing.
For a more complete discussion of reflection and refraction, see [Taylor,
1981] and [Benzoni-Gavage and Serre, 2007]. In particular these treat the
phenomenon of total reflection that can be anticipated as follows. From
Snell’s law one sees that sin θr 1/c and approaches 1/c as θi approaches
π/2. The refracted rays lie in the cone θr arcsin(1/c). Reversing time
shows that light rays from below approaching the surface at angles smaller
than this critical angle traverse the surface tracing backward the old incident
rays. For angles larger than arcsin(1/c) there is no possible continuation as
a ray above the surface. One can show by constructing infinitely accurate
approximate solutions that there is total reflection. Below the surface there
is a reflected ray with the usual law of reflection. The role of a third wave
is played by a boundary layer of thickness above which the solution is
O(
∞).
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