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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