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5.4 Light

Light is an electromagnetic field and as such is described by Maxwell’s equations. Each
component of the electric and magnetic field satisfies the wave equation. It is through
the boundary conditions that the various components are related to each other. (They
are ”coupled.”) Imagine, for example, light reflecting off a ball with a mirrored surface.
This is a scattering problem. The domain D where the light is propagating is the exterior
of the ball. Certain boundary conditions then are satisfied by the electromagnetic field
components. When polarization effects are not being studied, some scientists use the wave
equation with homogeneous Dirichlet or Neumann conditions as a considerably simplified
model of such a situation.

5.5 Sound

Our ears detect small disturbances in the air. The disturbances are described by the
equations of gas dynamics, which form a system of nonlinear equations with velocity v
and density ρ as the unknowns. But small disturbances are described quite well by the
so-called linearized equations, which are a lot simpler; namely,

∂v c 2 0
+ grad v = 0 (5.4)
∂t ρ 0
∂ρ
+ ρ 0 div v = 0 (5.5)
∂t
(four scalar equations altogether). Here ρ 0 is the density and c 0 is the speed of sound in
still air.
Assume now that the curl of v is zero; this means that there are no sound ”eddies”
and the velocity v is irrotational. It follows that ρ and all three components of v satisfy
the wave equation:
2
2
∂ v ∂ ρ
2
2
= c ∆v and = c ρ. (5.6)
∂t 2 0 ∂t 2 0
Now if we are describing sound propagation in a closed, sound-insulated room D with
rigid walls; Say a concert hall, then the air molecules at the wall can only move parallel
to the boundary, so that no sound can travel in a normal direction to the boundary. So
v · n = 0 on boundary D. Since curl v = 0, there is a standard fact in vector calculus
which says that there is a ”Potential” function ψ such that v = − grad ψ. The potential
2
2
2
also satisfies the wave equation ∂ ψ/∂t = c ∆ψ, and the boundary condition for it is
0
−v · n = n · grad ψ = 0 or Neumann’s condition for ψ.
At an open window of the room D, the atmospheric pressure is a constant and there is
no difference of pressure across the window. The pressure p is proportional to the density
ρ, for small disturbances of the air. Thus ρ is a constant at the window, which means
that ρ satisfies the Dirichlet boundary condition ρ = ρ 0 .
At a soft wall, such as an elastic membrane covering an open window, the pressure
difference p−p 0 across the membrane is proportional to the normal velocity v ·n, namely

p − p 0 = Zv · n,

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