40 1. Simple Examples of Propagation
From section 1.6.3, the leading amplitude d0 must be constant on the
rays t (t , x + c t η/|η|). To determine d0, it suffices to know the values
d0(t,
0+,x
) at the interface. One could choose d0 to guarantee the continuity
of u or of ∂1u, but not both. One cannot construct a good approximate
solution consisting of just an incident and transmitted wave.
Add to the recipe a reflected wave. Seek a reflected wave in x1 0 in
the form
R
ei(xζ+tσ)/
b(, t, x) , b(, t, x) b0(t, x) + b1(t, x) + · · · .
In order that the reflected wave oscillate with the same phase as the incident
wave in the interface x1 = 0, one must have ζ = ξ and σ = −1. To satisfy
the wave equation in x1 0 requires
σ2
=
|ζ|2.
Together these imply
ζ1
2
= ξ1.
2
To have propagation away from the boundary requires ζ1 = −ξ1 so
ζ =
˜.
ξ Therefore,
(1.7.7) R
ei(x˜−t)/
ξ
b(, t, x) , b(, t, x) b0(t, x)+ b1(t, x)+ · · · .
Summarizing, seek
v
=
I + R in x1 0,
T in x1 0.
The continuity required at x1 = 0 forces
(1.7.8)
ei(x ξ −t)/
(
a(, t, 0,x ) + b(, t, 0,x )
)
=
ei(x ξ −t)/
d(, t, 0,x ) .
The continuity of u and ∂1u hold if and only if at x1 = 0 one has
(1.7.9) a + b = d , and
iξ1
a + ∂1a
iξ1
b + ∂1b =
iη1
d + ∂1d .
The first of these relations yields
(1.7.10) aj + bj dj
x1=0
= 0, j = 0, 1, 2,....
The second relation in (1.7.9) is expanded in powers of . The coefficients
of
j
must match for all all j −1. The leading order is
−1
and yields
(1.7.11)
(
a0 b0 (η1/ξ1)d0
)
x1=0
= 0 .
Since a0 is known, the j = 0 equation from (1.7.10) together with (1.7.11)
is a system of two linear equations for the two unknown b0,d0,
−1 1
1 η1/ξ1
b0
d0
=
a0
a0
.
Since the matrix is invertible, this determines the values of b0 and d0 at
x1 = 0.
The amplitude b0 (resp., d0) is constant on rays with velocity
˜
ξ (resp.,
cη/|η|). Thus the leading amplitudes are determined throughout the half-
spaces on which they are defined.
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