Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

ω=2. 40 c/a ω=3.^83 c/a

ω=5. 14 c/a ω=5.^52 c/a

a

Figure 21.8 The modes of oscillation with the four lowest frequencies for a

circular drumskin of radiusa. The dashed lines indicate the nodes, where the

displacement of the drumskin is always zero.

Helmholtz’s equation in cylindrical polars

Generalising the above method to three-dimensional cylindrical polars is straight-

forward, and following a similar procedure to that used for Laplace’s equation

we find the separated solution of Helmholtz’s equation takes the form

F(ρ, φ, z)=

[

AJm

(√

k^2 −α^2 ρ

)

+BYm

(√

k^2 −α^2 ρ

)]

×(Ccosmφ+Dsinmφ)[Eexp(iαz)+Fexp(−iαz)],

whereαandmare separation constants. We note that the angular part of the

solution is the same as for Laplace’s equation in cylindrical polars.

Helmholtz’s equation in spherical polars

In spherical polars, we find again that the angular parts of the solution Θ(θ)Φ(φ)

are identical to those of Laplace’s equation in this coordinate system, i.e. they are

the spherical harmonicsYm(θ, φ), and so we shall not discuss them further.

The radial equation in this case is given by

r^2 R′′+2rR′+[k^2 r^2 −(+1)]R=0, (21.56)

which has an additional termk^2 r^2 Rcompared with the radial equation for the

Laplace solution. The equation (21.56) looks very much like Bessel’s equation.

In fact, by writingR(r)=r−^1 /^2 S(r) and making the change of variableμ=kr,

it can be reduced to Bessel’s equation of order+^12 , which has as its solutions

S(μ)=J+1/ 2 (μ)andY+1/ 2 (μ) (see section 18.6). The separated solution to