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10.7 Circles, wedges, and annuli

The technique of separating variables in polar Coordinates works for domains whose bound-
aries are made up of concentric circles and rays. The purpose of this section is to present
several examples of this type. In each case we get the expansion as an infinite series.
(But summing the series to get a Poisson-type formula is more difficult and works only in
special cases.) The geometries we treat here are

• A wedge: (0 < θ < θ 0 , 0 < r < a)

• An annulus: (0 < a < r < b)
• The exterior of a circle: (a < r < ∞).

We could do Dirichlet, Neumann, or Robin boundary conditions. This leaves us with a
lot of possible examples!

Example 10.4 The Wedge
Let us take the wedge with three sides θ = 0, θ = β, and r = a and solve the
Laplace equation with the homogeneous Dirichlet condition on the straight sides and
the inhomogeneous Neumann condition on the curved side (see Figure 10.7). That
is, using the notation u = u(r, θ), the BCs are
∂u
u(r, θ) = 0 = u(r, β), (a, θ) = h(θ).
∂r
The separation-of-variables technique works just as for the circle, namely,

2
0
θ” + λθ = 0, r R” + rR − λR = 0.

0
= u = h(θ)
u r

β

u = 0

Figure 10.7: The wedge

So the homogeneous conditions lead to

Θ” + λΘ = 0, Θ(0) = Θ(β) = 0.
This is our standard eigenvalue problem, which has the solutions

2
πn πnθ
λ = , Θ(θ) = sin .
β β

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