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
η1
2
=
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).
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