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

vtt − Δ v := r in x1 0 , vtt −

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.