Mathematical Methods for Physics and Engineering : A Comprehensive Guide

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES

As in the two-dimensional case, single-valuedness ofurequires thatmis an

integer. However, in the particular casem= 0 the solution is

Φ(φ)=Cφ+D.

This form is appropriate to a solution with axial symmetry (C= 0) or one that is

multivalued, but manageably so, such as the magnetic scalar potential associated

with a currentI(in which caseC=I/(2π)andDis arbitrary).

Finally, theρ-equation (21.34) may be transformed into Bessel’s equation of

ordermby writingμ=kρ. This has the solution

P(ρ)=AJm(kρ)+BYm(kρ).

The properties of these functions were investigated in chapter 16 and will not

be pursued here. We merely note thatYm(kρ) is singular atρ= 0, and so, when

seeking solutions to Laplace’s equation in cylindrical coordinates within some

region containing theρ= 0 axis, we requireB=0.

The complete separated-variable solution in cylindrical polars of Laplace’s

equation∇^2 u= 0 is thus given by

u(ρ, φ, z)=[AJm(kρ)+BYm(kρ)][Ccosmφ+Dsinmφ][Eexp(−kz)+Fexpkz].

(21.35)

Of course we may use the principle of superposition to build up more general

solutions by adding together solutions of the form (21.35) for all allowed values

of the separation constantskandm.

A semi-infinite solid cylinder of radiusahas its curved surface held at 0 ◦Cand its base

held at a temperatureT 0. Find the steady-state temperature distribution in the cylinder.

The physical situation is shown in figure 21.5. The steady-state temperature distribution
u(ρ, φ, z) must satisfy Laplace’s equation subject to the imposed boundary conditions. Let
us take the cylinder to have its base in thez= 0 plane and to extend along the positive
z-axis. From (21.35), in order thatuis finite everywhere in the cylinder we immediately
requireB=0andF= 0. Furthermore, since the boundary conditions, and hence the
temperature distribution, are axially symmetric, we requirem=0,andsothegeneral
solution must be a superposition of solutions of the formJ 0 (kρ)exp(−kz) for all allowed
values of the separation constantk.
The boundary conditionu(a, φ, z) = 0 restricts the allowed values ofk,sincewemust
haveJ 0 (ka) = 0. The zeros of Bessel functions are given in most books of mathematical
tables, and we find that, to two decimal places,

J 0 (x)=0 forx=2. 40 , 5. 52 , 8. 65 ,....

Writing the allowed values ofkasknforn=1, 2 , 3 ,...(so, for example,k 1 =2. 40 /a), the
required solution takes the form

u(ρ, φ, z)=

∑∞

n=1

AnJ 0 (knρ)exp(−knz).