75 lines
2.5 KiB
TeX
75 lines
2.5 KiB
TeX
\begin{frame}
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\frametitle{Introduction}
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\small
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\begin{block}{\bf Setting}
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\begin{itemize}
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\item Outgoing solutions to the ``radial wave equation"
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\item Transparent radiation boundary conditions on spherical domain
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\item \alert{Dirichlet-to-Neumann map}, \alert{Macdonald function}
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\end{itemize}
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\end{block}
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\begin{block}{\bf Outline}
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\begin{itemize}
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\item Radial wave equation and the Macdonald function
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\item Non-reflecting boundary conditions
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\item Identities for roots of the Macdonald function
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\item Other work
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\end{itemize}
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\end{block}
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\end{frame}
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\begin{frame}
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\begin{center}
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\Huge Radial Wave Equation
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\end{center}
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\end{frame}
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\begin{frame}
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\frametitle{Radial Wave Equation}
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\scriptsize
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\begin{block}{\bf Overview}
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\begin{itemize}
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\item
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Ordinary wave equation on $\color{ec}\mathbb{R}^{3}$, ($\color{ec}c=1$ here):
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\hspace{2em}
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$\color{ec} \Delta\psi - \frac{1}{c^{2}}\frac{\partial^{2}\psi}{\partial t^{2}} = 0$
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\item
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Laplace operator in spherical coordinates:
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$$\color{ec}
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\Delta\psi = \frac{1}{r^{2}}\frac{\partial}{\partial r} r^{2} \frac{\partial}{\partial r} \psi + \frac {1}{r^{2}}\Delta_{S^2} \psi, \hspace{2em} \Delta_{S^2} = \frac{1}{\sin \theta}\frac{\partial}{\partial\theta}\sin\theta\frac{\partial}{\partial\theta} + \frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial \phi^{2}}
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$$
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\item
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Generic solution in terms of $\color{ec} Y_{\ell m}(\theta,\phi)$ of $\color{ec} \Delta_{S^2}$, where
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$$\color{ec} \Delta_{S^2}Y_{\ell m}(\theta,\phi) = -\ell(\ell+1)Y_{\ell m}(\theta,\phi)$$
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is:
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$$\color{ec}
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\psi = \sum_{\ell =0}^\infty \sum_{m = - \ell}^\ell \psi_{\ell m}(t,r) Y_{\ell m}(\theta,\phi) = \sum_{\ell =0}^\infty \sum_{m = - \ell}^\ell \frac{\Psi_{\ell m}(t,r)}{r}Y_{\ell m}(\theta,\phi).
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$$
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\end{itemize}
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\end{block}
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\end{frame}
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\begin{frame}
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\frametitle{Radial Wave Equation}
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\scriptsize
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\begin{block}{\bf Overview}
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\begin{itemize}
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\item "multipole" solution $\color{ec}
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\psi(t,r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta)$:
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$$\color{ec}
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\psi_{\ell m}(t,r)Y_{\ell m}(\theta,\phi) = r^{-1}\Psi_{\ell m}(t,r) Y_{\ell m}(\theta,\phi)
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$$
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\item Eliminating angular dependence, multipole coefficients $\color{ec} \psi_{\ell m}$ and $\color{ec} \Psi_{\ell m}$ satisfy:
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$$\color{ec}
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- \partial^2_t \psi_{\ell m}+ \partial^2_r \psi_{\ell m}
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+ \frac{2}{r}\partial_r \psi_{\ell m}- \frac{\ell(\ell+1)}{r^2}\psi_{\ell m} = 0,
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$$
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$$\color{ec}
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-\partial^2_t\Psi_{\ell m} + \partial^2_r\Psi_{\ell m} - \frac{\ell(\ell+1)}{r^2}\Psi_{\ell m} = 0.
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$$
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\end{itemize}
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\end{block}
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\end{frame}
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