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.