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.
