5.4 Problems in Polar Coordinates 309
If we consider now the vibrations of a circular membrane or heat conduc-
tion in a circular plate, we shall see common features again. In what follows,
these two problems are given side by side for the region 0<r<a,0<t.
Wave Heat
∇^2 v=1
c^2∂^2 v
∂t^2 ∇(^2) v=^1
k
∂v
∂t
v(a,θ,t)=f(θ ) v(a,θ,t)=f(θ )
v(r,θ, 0 )=g(r,θ) v(r,θ, 0 )=g(r,θ)
∂v
∂t(r,θ,^0 )=h(r,θ)
In both problems we require thatvbe periodic inθwith period 2π:
v(r,θ,t)=v(r,θ+ 2 π,t),
as in Section 4.5.
Although the interpretation of the functionvis different in the two cases,
we see that the solution of the problem
∇^2 v= 0 ,v(a,θ)=f(θ )
is the rest-state or steady-state solution for both problems, and it will be
needed in both problems to make the boundary condition atr=ahomo-
geneous. Let us suppose that the time-independent solution has been found
and subtracted; that is, we will replacef(θ )by zero. Then we have
∇^2 v=^1
c^2
∂^2 v
∂t^2
∇^2 v=^1
k
∂v
∂t
v(a,θ,t)= 0 v(a,θ,t)= 0
plus the appropriate initial conditions. If we attempt to solve by separation of
variables, settingv(r,θ,t)=φ(r,θ)T(t),inbothcaseswewillfindthatφ(r,θ)
must satisfy
1
r
∂
∂r(
r∂φ∂r)
+r^12 ∂(^2) φ
∂θ^2 =−λ
(^2) φ, (1)
φ(a,θ)= 0 , (2)
φ(r,θ+ 2 π)=φ(r,π), (3)
φbounded asr→ 0. (4)
Now we shall concentrate on the solution of this two-dimensional eigen-
value problem. We can separate variables again by assuming thatφ(r,θ)=