29 lines
736 B
TeX
29 lines
736 B
TeX
%--------------------------------------------------------------
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\begin{frame}
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\frametitle{Radial Wave Equation}
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\small
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\begin{itemize}
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\item
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Radial Wave Equation:
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$$
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\color{eqncolor}
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- \partial^2_t \Psi_\ell+ \partial^2_r \Psi_\ell
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- \frac{\ell(\ell+1)}{r^2}\Psi_\ell = 0
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$$
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\item
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For $\color{eqncolor} \ell=0$, generic solutions from factoring wave equation:
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$$\color{eqncolor}
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\underrightarrow{f(t-r)}, \hspace{2em} \underleftarrow{g(t+r)}
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$$
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\end{itemize}
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\begin{block}{\bf Lemma}
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For $\color{eqncolor} \ell>0$, generic outgoing solution:
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$$
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\color{eqncolor}
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\Psi_{\ell}(t,r)=\sum_{k=0}^\ell \frac{c_{\ell k}}{r^k}f^{(\ell-k)}(t-r)
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\hspace{2em}
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c_{\ell k} = \frac{1}{2^k k!}\frac{(\ell + k)!}{(\ell - k)!}
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$$
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\end{block}
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\end{frame}
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