Mathematical Methods for Physics and Engineering : A Comprehensive Guide

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES

and is not to be confused with a spherical harmonic (these are written with a

superscript as well as a subscript).

Putting the two parts of the solution together we have

F(ρ, φ)=[Acosmφ+Bsinmφ][CJm(kρ)+DYm(kρ)]. (21.55)

Clearly, for solutions of Helmholtz’s equation that are required to be finite at the

origin, we must setD=0.

Find the four lowest frequency modes of oscillation of a circular drumskin of radiusa

whose circumference is held fixed in a plane.

The transverse displacementu(r,t) of the drumskin satisfies the two-dimensional wave
equation

∇^2 u=

1

c^2

∂^2 u

∂t^2

,

withc^2 =T/σ,whereTis the tension of the drumskin andσis its mass per unit area.
From (21.54) and (21.55) a separated solution of this equation, in plane polar coordinates,
that is finite at the origin is

u(ρ, φ, t)=Jm(kρ)(Acosmφ+Bsinmφ)exp(±iωt),

whereω=kc. Since we require the solution to be single-valued we must havemas an
integer. Furthermore, if the drumskin is clamped at its outer edgeρ=athen we also
requireu(a, φ, t) = 0. Thus we need

Jm(ka)=0,

which in turn restricts the allowed values ofk. The zeros of Bessel functions can be
obtained from most books of tables; the first few are

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

J 1 (x)=0 forx≈ 3. 83 , 7. 02 , 10. 17 ,...,

J 2 (x)=0 forx≈ 5. 14 , 8. 42 , 11. 62 ....

The smallest value ofxfor which any of the Bessel functions is zero isx≈ 2 .40, which
occurs forJ 0 (x). Thus the lowest-frequency mode hask=2. 40 /aand angular frequency
ω=2. 40 c/a.Sincem= 0 for this mode, the shape of the drumskin is

u∝J 0

(

2. 40

ρ

a

)

;

this is illustrated in figure 21.8.
Continuing in the same way, the next three modes are given by

ω=3. 83

c

a

,u∝J 1

(

3. 83

ρ

a

)

cosφ, J 1

(

3. 83

ρ

a

)

sinφ;

ω=5. 14

c

a

,u∝J 2

(

5. 14

ρ

a

)

cos 2φ, J 2

(

5. 14

ρ

a

)

sin 2φ;

ω=5. 52

c

a

,u∝J 0

(

5. 52

ρ

a

)

.

These modes are also shown in figure 21.8. We note that the second and third frequencies
havetwocorresponding modes of oscillation; these frequencies are therefore two-fold
degenerate.