Wave Equation 2D

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The wave equation in two dimensions is given by

\begin{align} u_{tt} = c^2 (u_{xx} + u_{yy}) \end{align}

An application of the 2D wave equation is the modeling of vibrations of a drumhead. In this article we reduce the Partial Differential Equation to Ordinary Differential Equations by Separation of Variables. In polar coordinates the PDE becomes \begin{align} u_{tt} = c^2 \left(u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta \theta} \right) \end{align} We make the ansatz $u(r,\theta,t)=T(t)R(r)\Theta (\theta)$ \begin{align} T''R\Theta =c^2 \left(T R'' \Theta+\frac{1}{r}TR'\Theta+\frac{1}{r^2}TR\Theta '' \right) \end{align} and devide by $c^2u$ \begin{align} -\lambda^2=\frac{T''}{c^2T} = \frac{R''}{R}+\frac{1}{r}\frac{R'}{R}+\frac{1}{r^2}\frac{\Theta''}{\Theta} \end{align} we find the first ODE \begin{align} T''+\lambda^2 c^2 T=0 \end{align} with the solution \begin{align} T(t)=a_1 \, cos \left( \lambda c t \right) + a_2 \, sin \left( \lambda c t \right) \end{align} Now we mulitply the equation \begin{align} \mu^2=-\frac{1}{r^2}\frac{\Theta''}{\Theta} =\frac{R''}{R}+\frac{1}{r}\frac{R'}{R}+\lambda^2 \end{align} by $r^2$ and get \begin{align} \mu^2=-\frac{\Theta''}{\Theta} =r^2\frac{R''}{R}+r\frac{R'}{R}+\lambda^2 r^2 \end{align} we find the second ODE \begin{align} \Theta''+\mu \Theta=0 \end{align} with the solution \begin{align} \Theta(\theta)=c_{\mu} \, cos \left( {\mu} \theta \right)+ d_{\mu} \, sin \left( {\mu} \theta \right) \end{align} and the third ODE (Bessel's Differential Equation) \begin{align} r^2R''+r R'+ (\lambda^2 r^2 -\mu^2) R=0 \end{align} which can be transformed into standard form. The solutions of Bessel's equation are the Bessel functions \begin{align} R(r)=c_{\pm} J_{\pm \mu}\left( \lambda r \right) \end{align} or \begin{align} R(r)=c_1 J_{\mu}\left( \lambda r \right) + c_2 Y_{\mu}\left( \lambda r \right) \end{align} The solution to a specific boundary value problem can be found in the article on vibrations of a drumhead.