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).