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

\[\frac{T^{\prime\prime}}{c^2T} =\frac1R\frac1r(rR^{\prime})^{\prime} +\frac1{r^2}\frac{\Phi^{\prime\prime}}{\Phi}.\]

Set $T^{\prime\prime}/(c^2T)=-k^2$. After multiplying the remaining equation by $r^2$,

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

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

\[\Phi^{\prime\prime}+m^2\Phi=0.\]

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

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

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. A mode $(m,n)$ has $m$ nodal diameters and $n-1$ interior nodal circles; the fixed rim is not counted as an interior circle. The frequency has the correct units because $\alpha_{mn}$ is dimensionless and $[c/a]=\mathrm{time}^{-1}$.

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

Solved Problems

1. The axisymmetric fundamental mode

A circular membrane has radius $a=0.25\,\mathrm{m}$ and wave speed $c=90\,\mathrm{m\,s^{-1}}$. Find its fundamental frequency. The first zero of $J_0$ is

\[\alpha_{01}=2.4048255577.\]

For the axisymmetric fundamental,

\[\omega_{01}=\frac{c\alpha_{01}}a =865.737\,\mathrm{rad\,s^{-1}},\]

and therefore

\[\boxed{ f_{01}=\frac{\omega_{01}}{2\pi} =\frac{90(2.4048255577)}{2\pi(0.25)} =137.786\,\mathrm{Hz}}.\]

Here $m=0$ gives no nodal diameter and $n=1$ gives no interior nodal circle. The displacement is finite at $r=0$, vanishes at $r=a$, and $c/a$ supplies the required inverse-time unit.

2. A non-axisymmetric mode and its nodal circle

For $a=0.40\,\mathrm{m}$ and $c=100\,\mathrm{m\,s^{-1}}$, consider the $(m,n)=(1,2)$ mode. Use

\[\alpha_{11}=3.8317059702, \qquad \alpha_{12}=7.0155866698.\]

Its frequency is

\[\boxed{ f_{12}=\frac{c\alpha_{12}}{2\pi a} =279.141\,\mathrm{Hz}}.\]

The angular factor has one nodal diameter. An interior nodal circle occurs when the radial argument reaches the preceding zero $\alpha_{11}$:

\[\alpha_{12}\frac{r_1}{a}=\alpha_{11}.\]

Thus

\[\boxed{ r_1=a\frac{\alpha_{11}}{\alpha_{12}} =0.218468\,\mathrm{m}}.\]

There is one interior nodal circle, as required by $n-1=1$. Substitution of $r=a$ gives $J_1(\alpha_{12})=0$, so the rim condition is also satisfied.

Descriptive Questions

1. Starting from the given polar wave equation, derive the angular and radial separated equations.
2. Explain how single-valuedness quantizes the angular index and why the singular radial solution is excluded at the centre.
3. Derive the fixed-rim frequency condition in terms of positive zeros of $J_m$.
4. Explain the rotational degeneracy of the sine and cosine angular factors for $m>0$.

Numerical Problems

1. Using $\alpha_{01}=2.4048255577$ and $\alpha_{02}=5.5200781103$, find $f_{02}/f_{01}$. Final answer: $f_{02}/f_{01}=\alpha_{02}/\alpha_{01}=2.29542$.
2. State the interior nodal set and number of angular sectors for a $(3,1)$ mode. Final answer: three nodal diameters, no interior nodal circle, and six angular sectors.
3. For a membrane of radius $a=0.20\,\mathrm{m}$, normalize the fundamental radial shape $R(r)=A J_0(\alpha_{01}r/a)$ by requiring $\int_0^a R^2r\,dr=1$. Use $J_1(\alpha_{01})=0.5191474973$ and $\int_0^a rJ_0^2(\alpha_{01}r/a)\,dr=a^2J_1^2(\alpha_{01})/2$. Final answer: $A=\sqrt2/[a\lvert J_1(\alpha_{01})\rvert]=13.6205\,\mathrm{m^{-1}}$.
4. For $m=2$, rewrite $\sqrt3\cos2\varphi+\sin2\varphi$ as one shifted cosine and find the nodal-diameter orientations in $0\le\varphi<\pi$. Final answer: $2\cos(2\varphi-\pi/6)$; $\varphi=\pi/3$ and $5\pi/6$.

The Bessel-zero ratios, frequencies, nodal radii, radial normalization, and angular identities are verified in the Unit II problem-check worksheet.

References

1. Vibration of a circular membrane — Wikipedia
2. Richard Haberman, Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 5th ed., Chapter 7, §7.7, “Vibrating Circular Membrane and Bessel Functions,” Pearson.
3. George B. Arfken, Hans J. Weber, and Frank E. Harris, Mathematical Methods for Physicists, 7th ed., chapter “Bessel Functions,” Academic Press.
4. Mary L. Boas, Mathematical Methods in the Physical Sciences, 3rd ed., chapters “Special Functions” and “Partial Differential Equations,” Wiley.