1.6. The law of reflection 33 Figure 1.6.2. Spherical wave arrives at the boundary In Figure 1.6.2 the wave on the left has positive profile and that on the right, a negative profile. Figure 1.6.3. Spherical wave with reflection In Figure 1.6.3. the middle line represents the boundary. Viewed from x1 0, the wave on the left disappears into the boundary and a reflected spherical wave emerges with profile flipped. The profiles of outgoing spherical waves in three-space preserve their thickness and shape. They decrease in amplitude as time increases. 1.6.2. The plane wave derivation. In many texts you will find a deriva- tion that goes as follows. Begin with the plane wave solution ei(xξ−t|ξ|) , ξ ∈ Rd . The solution is everywhere of modulus one, so it cannot satisfy the Dirichlet boundary condition. Seek a solution of the initial boundary value problem which is a sum of two plane waves, ei(xξ−t|ξ|) + A ei(xη+tσ) , A ∈ C . To satisfy the wave equation, one must have σ2 = |η|2. In order that the plane waves sum to zero at x1 = 0, it is necessary and suﬃcient that η = ξ , σ = −|ξ|, and A = −1. Since σ2 = |η|2, it follows that |η| = |ξ|, so η = (±ξ1,ξ2,...,ξd) . The sign + yields the solution u = 0. Denote ˜ := (−x1,x2,...,xd), ˜ := (−ξ1,ξ2,...,ξd).

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