Wave Equation in two Dimensions

The $1+1$-dimensional wave equation reads $$\begin{equation*} \frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 u}{\partial t^2}, \end{equation*}$$ with $u=u(x,t)$ and we have assumed that we operate with dimensionless variables. Possible boundary and initial conditions with $L=1$ are $$\begin{equation*} \begin{array}{cc} u_{xx} = u_{tt}& x\in(0,1), t>0 \\ u(x,0) = g(x)& x\in (0,1) \\ u(0,t)=u(1,t)=0 & t > 0\\ \partial u/\partial t|_{t=0}=0 & x\in (0,1)\\ \end{array} . \end{equation*}$$

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