円盤領域における波動方程式

$\displaystyle \frac{1}{c^2}u_{tt}$	$\displaystyle =$	$\displaystyle \Laplacian u$ $\displaystyle \mbox{(in $\Omega\times(0,\infty)$)}$ $\displaystyle ,$
$\displaystyle u(x,t)$	$\displaystyle =$	0 $\displaystyle \mbox{(on $\rd\Omega\times(0,\infty)$)}$ $\displaystyle ,$
$\displaystyle u(x,0)$	$\displaystyle =$	$\displaystyle \phi(x)$ $\displaystyle \mbox{($x\in\overline\Omega$)}$ $\displaystyle ,$
$\displaystyle u_t(x,0)$	$\displaystyle =$	$\displaystyle \psi(x)$ $\displaystyle \mbox{($x\in\overline\Omega$)}$

	$\displaystyle u(r,\theta,t)$	$\displaystyle =$	$\displaystyle \frac{1}{2}\sum_{m=1}^\infty A_{0m}J_0(\mu_{0m}r)\cos(\mu_{0m}t) ... ..._n(\mu_{nm}r)\cos(\mu_{mn}t) \left(A_{nm}\cos n\theta+B_{nm}\sin n\theta\right)$
			$\displaystyle + \frac{1}{2}\sum_{m=1}^\infty C_{0m}J_0(\mu_{0m}r)\frac{\sin(\mu... ...{\sin(\mu_{nm}t)}{\mu_{nm}} \left(C_{nm}\cos n\theta+D_{nm}\sin n\theta\right).$