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(

∞).