Mathematical Methods for Physics and Engineering : A Comprehensive Guide

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES

whereA, B, C, Dare arbitrary constants andnis any integer.

We have not yet, however, considered the solution whenn=0.Inthiscase,

the solutions of the separated ordinary equations (21.28) and (21.27), respectively,

are easily shown to be

Φ(φ)=Aφ+B,

P(ρ)=Clnρ+D.

But, in order thatu=PΦ is single-valued, we requireA= 0, and so the solution

forn= 0 is simply (absorbingBintoCandD)

u(ρ, φ)=Clnρ+D.

Superposing the solutions for the different allowed values ofn, we can write

the general solution to Laplace’s equation in plane polars as

u(ρ, φ)=(C 0 lnρ+D 0 )+

∑∞

n=1

(Ancosnφ+Bnsinnφ)(Cnρn+Dnρ−n),

(21.31)

wherencan take only integer values. Negative values ofnhave been omitted

from the sum since they are already included in the terms obtained for positive

n. We note that, since lnρis singular atρ= 0, whenever we solve Laplace’s

equation in a region containing the origin,C 0 must be identically zero.

A circular drumskin has a supporting rim atρ=a. If the rim is twisted so that it

is displaced vertically by a small amount(sinφ+2sin2φ),whereφis the azimuthal

angle with respect to a given radius,find the resulting displacementu(ρ, φ)over the entire

drumskin.

The transverse displacement of a circular drumskin is usually described by the two-
dimensional wave equation. In this case, however, there is no time dependence and so
u(ρ, φ) solves the two-dimensional Laplace equation, subject to the imposed boundary
condition.
Referring to (21.31), since we wish to find a solution that is finite everywhere inside
ρ=a,werequireC 0 =0andDn=0foralln>0. Now the boundary condition at the
rim requires

u(a, φ)=D 0 +

∑∞

n=1

Cnan(Ancosnφ+Bnsinnφ)=(sinφ+2sin2φ).

Firstly we see that we requireD 0 =0andAn=0foralln. Furthermore, we must
haveC 1 B 1 a=,C 2 B 2 a^2 =2andBn=0forn>2. Hence the appropriate shape for the
drumskin (valid over the whole skin, not just the rim) is

u(ρ, φ)=

ρ

a

sinφ+

2 ρ^2

a^2

sin 2φ=

ρ

a

(

sinφ+

2 ρ

a

sin 2φ

)

.