# 6.6: Spherically Symmetric Vibrations

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Another application of spherical harmonics is a vibrating spherical membrane, such as a balloon. Just as for the two-dimensional membranes encountered earlier, we let $$u(\theta, \phi, t)$$ represent the vibrations of the surface about a fixed radius obeying the wave equation, $$u_{t t}=c^{2} \nabla^{2} u$$, and satisfying the initial conditions

$u(\theta, \phi, 0)=f(\theta, \phi), \quad u_{t}(\theta, \phi, 0)=g(\theta, \phi) .\nonumber$

In spherical coordinates, we have (for $$\rho=r=$$ constant.)

$u_{t t}=\frac{c^{2}}{r^{2}}\left(\frac{1}{\sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial u}{\partial \theta}\right)+\frac{1}{\sin ^{2} \theta} \frac{\partial^{2} u}{\partial \phi^{2}}\right),\label{eq:1}$

where $$u=u(\theta, \phi, t)$$.

The boundary conditions are given by the periodic boundary conditions

$u(\theta, 0, t)=u(\theta, 2 \pi, t), \quad u_{\phi}(\theta, 0, t)=u_{\phi}(\theta, 2 \pi, t),\nonumber$

where $$0<t$$, and $$0<\theta<\pi$$, and that $$u=u(\theta, \phi, t)$$ should remain bounded.

Noting that the wave equation takes the form

$u_{t t}=\frac{c^{2}}{r^{2}} L u, \quad \text { where } \quad L Y_{\ell m}=-\ell(\ell+1) Y_{\ell m}\nonumber$

for the spherical harmonics $$Y_{\ell m}(\theta, \phi)=P_{\ell}^{m}(\cos \theta) e^{i m \phi}$$, then we can seek product solutions of the form

$u_{\ell m}(\theta, \phi, t)=T(t) Y_{\ell m}(\theta, \phi) .\nonumber$

Inserting this form into the wave equation in spherical coordinates, we find

$T^{\prime \prime} Y_{\ell m}=-\frac{c^{2}}{r^{2}} T(t) \ell(\ell+1) Y_{\ell m},\nonumber$

or

$T^{\prime \prime}+\ell(\ell+1) \frac{c^{2}}{r^{2}} T(t)\nonumber$

The solutions of this equation are easily found as

$T(t)=A \cos \omega_{\ell} t+B \sin \omega_{\ell} t, \quad \omega_{\ell}=\sqrt{\ell(\ell+1)} \frac{c}{r} .\nonumber$

Therefore, the product solutions are given by

$u_{\ell m}(\theta, \phi, t)=\left[A \cos \omega_{\ell} t+B \sin \omega_{\ell} t\right] Y_{\ell m}(\theta, \phi)\nonumber$

for $$\ell=0,1, \ldots, m=-\ell,-\ell+1, \ldots, \ell$$.

The general solution is found as

$u(\theta ,\phi ,t)=\sum\limits_{\ell=0}^\infty\sum\limits_{m=-\ell}^{\ell}[A_{\ell m}\cos\omega_{\ell}t+B_{\ell m}\sin\omega_{\ell}t]Y_{\ell m}(\theta ,\phi ).\nonumber$

An interesting problem is to consider hitting the balloon with a velocity impulse while at rest. An example of such a solution is shown in Figure $$\PageIndex{3}$$. In this images several modes are excited after the impulse.

This page titled 6.6: Spherically Symmetric Vibrations is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Russell Herman via source content that was edited to the style and standards of the LibreTexts platform.