32 1. Simple Examples of Propagation
t
x
T
Figure 1.6.1. Reflection in dimension d = 1
An example is sketched in the Figure 1.6.1. In
R1+1
one has an odd
solution of the wave equation. There is a rightward moving wave with
positive profile and a leftward moving wave with negative profile equal to
−1 times the reflection of the first.
Viewed from x 0, there is a wave with positive profile which arrives
at the boundary at time T. At that time a leftward moving wave seems
to emerge from the boundary. It is the reflection of the wave arriving at
the boundary. If the wave arrives at the boundary with amplitude a on an
incoming ray, the reflected wave on the reflected ray has amplitude −a. The
coefficient of reflection is equal to −1 (see Figure 1.6.1).
Example 1.6.2. Suppose that d = 3 and in t 0 one has an outgoing
spherically symmetric wave centered at a point x with x1
0.2
Until it
reaches the boundary, the boundary condition does not play a role. The
reflection is computed by extending the incoming wave to an odd solution
consisting of the given solution and its negative in mirror image. The mo-
ment when the original wave reaches the boundary from the left, its image
arrives from the right.
2The smooth rotationally symmetric solutions u of
1+3
u = 0 centered at the origin are
given by (see [Rauch, 1991])
u(t, x) =
f(t + |x|) f(t |x|)
|x|
, when x = 0, u(t, 0) = 2f (t) ,
where f
C∞(R)
is arbitrary. Equivalently, ru(t, r) is an odd solution of
1+1
(
ru(t, r)
)
= 0.
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