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
˜ x := (−x1,x2,...,xd),
˜
ξ := (−ξ1,ξ2,...,ξd).
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