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
) 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
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
Together these imply
To have propagation away from the boundary requires ζ1 = −ξ1 so
(1.7.7) R ∼
b(, t, x) , b(, t, x) ∼ b0(t, x)+ b1(t, x)+ · · · .
I + R in x1 0,
T in x1 0.
The continuity required at x1 = 0 forces
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
a + ∂1a −
b + ∂1b =
d + ∂1d .
The first of these relations yields
(1.7.10) aj + bj − dj
= 0, j = 0, 1, 2,....
The second relation in (1.7.9) is expanded in powers of . The coeﬃcients
must match for all all j ≥ −1. The leading order is
a0 − b0 − (η1/ξ1)d0
= 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,
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
cη/|η|). Thus the leading amplitudes are determined throughout the half-
spaces on which they are defined.