28 Jun 2025

Vibrational Modes of a Circular Membrane

Angular periodicity, the radial Bessel equation, fixed-rim roots, and circular-membrane frequencies.

bsc semester-ii mathematical-physics circular-membrane bessel-functions

For a circular membrane of radius $a$, take the polar wave equation as given:

\[\boxed{ u_{tt}=c^2\left[ \frac1r\frac{\partial}{\partial r} \left(r\frac{\partial u}{\partial r}\right) +\frac1{r^2}\frac{\partial^2u}{\partial\varphi^2} \right]}.\]

Use $u(r,\varphi,t)=R(r)\Phi(\varphi)T(t)$. Division by $c^2R\Phi T$ gives

\[\frac{T''}{c^2T} =\frac1R\frac1r(rR')' +\frac1{r^2}\frac{\Phi''}{\Phi}.\]

Set $T’’/(c^2T)=-k^2$. After multiplying the remaining equation by $r^2$,

\[\frac rR(rR')'+k^2r^2=-\frac{\Phi''}{\Phi}.\]

The two sides depend on different variables, so each equals a constant $m^2$. The angular equation is

\[\Phi''+m^2\Phi=0.\]

Single-valuedness requires $\Phi(\varphi+2\pi)=\Phi(\varphi)$, hence $m=0,1,2,\ldots$ and

\[\Phi=A\cos m\varphi+B\sin m\varphi.\]

The radial equation is

\[r^2R''+rR'+(k^2r^2-m^2)R=0.\]

With $s=kr$, this becomes Bessel’s equation,

\[\boxed{s^2R_{ss}+sR_s+(s^2-m^2)R=0}.\]

Its solution finite at the centre is $J_m(s)$. The second independent solution is singular at $r=0$ and is excluded. Therefore

\[R(r)=J_m(kr).\]

The fixed rim requires $R(a)=0$, so

\[J_m(ka)=0.\]

Let $\alpha_{mn}$ be the $n$th positive zero of $J_m$. Then

\[k_{mn}=\frac{\alpha_{mn}}a, \qquad \boxed{\omega_{mn}=\frac{c\alpha_{mn}}a}.\]

A real mode is

\[\boxed{ u_{mn}=J_m\!\left(\alpha_{mn}\frac ra\right) \left(A\cos m\varphi+B\sin m\varphi\right) \cos(\omega_{mn}t+\delta)}.\]

A mode with angular index \(m\) has \(m\) nodal diameters; increasing the radial index introduces interior nodal circles at the corresponding zeros of \(J_m\). The outer circle is the fixed rim.

For $m>0$, the cosine and sine angular factors have the same frequency and differ only by a rotation. The frequency has the correct units because $\alpha_{mn}$ is dimensionless and $[c/a]={\rm time}^{-1}$.

The $m=0$ radial function is checked directly in Bessel’s differential equation in the Unit II Maxima worksheet.