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 byu(ρ, φ, z)=∑∞
n=12 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 ata 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=1Jm(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=1Jm(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 polarsWe 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 formu(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
Rd
dr(
r^2dR
dr)
+1
Θsinθd
dθ(
sinθdΘ
dθ)
+1
Φsin^2 θd^2 Φ
dφ^2=0. (21.39)