I am studying the linear oscillation of the spherical droplet of water with azimuthal symmetry. I have written the surface of the droplet as [tex]F=r-R-f(t,\theta)\equiv 0.[/tex](adsbygoogle = window.adsbygoogle || []).push({});

I have boiled the problem down to a Laplace equation for the perturbed pressure, [itex]p_{1}(t,r,\theta)[/itex]. I have also reasoned that the function [tex]f(t,\theta)=\sum_{n=0}^{\infty}\beta_{n}(t)P_{n}(\cos\theta).[/tex]

The pressure is given by the following:

[tex]p_{1}=\sum_{n=0}^{\infty}\frac{\ddot{\beta}_{n}(t)}{(n+2)r^{n+1}}P_{n}(\cos\theta)[/tex]

There is a boundary condition to determine an equation for [itex]\beta_{n}(t)[/itex] which is:

[tex]p_{1}=-\cot\theta\frac{\partial f}{\partial\theta}-\frac{\partial^{2}f}{\partial\theta^{2}}[/tex]

I've tried expanding and using a few identities but have gotten bogged down, I can't seem to eliminate the derivatives of [itex]P_{n}(\cos\theta)[/itex] but get bogged down and I am unsure where to go from here.

Any suggestions?

I have done the following:

[tex]

\begin{eqnarray}

p_{1}\Big|_{r=1} & = & -\cot\theta\frac{\partial f}{\partial\theta}-\frac{\partial^{2}f}{\partial\theta^{2}} \\

& = & -\cot\theta\sum_{n=0}^{\infty}\beta_{n}(t)\frac{\partial}{\partial\theta}P_{n}(\cos\theta)-\sum_{n=0}^{\infty}\beta_{n}(t)\frac{\partial^{2}}{\partial\theta^{2}}P_{n}(\cos\theta) \\

& = & \cot\theta\sum_{n=0}^{\infty}\beta_{n}(t)P_{n}'(\cos\theta)\sin\theta+\sum_{n=0}^{\infty}\beta_{n}(t)\frac{\partial}{\partial\theta}\left[P_{n}'(\cos\theta)\sin\theta\right] \\

& = & \sum_{n=0}^{\infty}\beta_{n}(t)P_{n}'(t)\cos\theta+\sum_{n=0}^{\infty}\beta_{n}(t)\left[P_{n}'(\cos\theta)\cos\theta-P_{n}''(\cos\theta)\sin^{2}\theta\right] \\

& = & \sum_{n=0}^{\infty}\beta_{n}(t)\left[2\cos\theta P_{n}'(\cos\theta)-P_{n}''(\cos\theta)\sin^{2}\theta\right]

\end{eqnarray}

[/tex]

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# A Problems with identities involving Legendre polynomials

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