Mathematical Methods for Physics and Engineering : A Comprehensive Guide

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES

and the coefficientsAnmay be expressed as

An=

2 T 0

a^2 J^21 (kna)

[

aJ 1 (kna)

kn

]

=

2 T 0

knaJ 1 (kna)

.

The steady-state temperature in the cylinder is then given by

u(ρ, φ, z)=

∑∞

n=1

2 T 0

knaJ 1 (kna)

J 0 (knρ)exp(−knz).

We note that if, in the above example, the base of the cylinder were not kept at

a uniform temperatureT 0 , but instead had some fixed temperature distribution

T(ρ, φ), then the solution of the problem would become more complicated. In

such a case, the required temperature distributionu(ρ, φ, z)isingeneralnotaxially

symmetric, and so the separation constantmis not restricted to be zero but may

take any integer value. The solution will then take the form

u(ρ, φ, z)=

∑∞

m=0

∑∞

n=1

Jm(knmρ)(Cnmcosmφ+Dnmsinmφ) exp(−knmz),

where the separation constantsknmare such thatJm(knma) = 0, i.e.knmais thenth

zero of themth-order Bessel function. At the base of the cylinder we would then

require

u(ρ, φ,0) =

∑∞

m=0

∑∞

n=1

Jm(knmρ)(Cnmcosmφ+Dnmsinmφ)=T(ρ, φ).

(21.37)

The coefficientsCnmcould be found by multiplying (21.37) byJq(krqρ)cosqφ,

integrating with respect toρandφover the base of the cylinder and exploiting

the orthogonality of the Bessel functions and of the trigonometric functions. The

Dnmcould be found in a similar way by multiplying (21.37) byJq(krqρ)sinqφ.

Laplace’s equation in spherical polars

We now come to an equation that is very widely applicable in physical science,

namely∇^2 u= 0 in spherical polar coordinates:

1

r^2

∂

∂r

(

r^2

∂u

∂r

)

+

1

r^2 sinθ

∂

∂θ

(

sinθ

∂u

∂θ

)

+

1

r^2 sin^2 θ

∂^2 u

∂φ^2

=0. (21.38)

Our method of procedure will be as before; we try a solution of the form

u(r, θ, φ)=R(r)Θ(θ)Φ(φ).

Substituting this in (21.38), dividing through byu=RΘΦ and multiplying byr^2 ,

we obtain

1

R

d

dr

(

r^2

dR

dr

)

+

1

Θsinθ

d

dθ

(

sinθ

dΘ

dθ

)

+

1

Φsin^2 θ

d^2 Φ

dφ^2

=0. (21.39)