Famous PDEs, two dimension

In two dimension we have $u=u(x,y,t)$. We will, unless otherwise stated, simply use $u$ in our discussion below. Familiar situations which this equation can model are waves on a string, pressure waves, waves on the surface of a fjord or a lake, electromagnetic waves and sound waves to mention a few. For e.g., electromagnetic waves we have the constant $A=c^2$, with $c$ the speed of light. It is rather straightforward to extend this equation to two or three dimension. In two dimensions we have $$\begin{equation*} \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=A\frac{\partial^2 u}{\partial t^2}, \end{equation*}$$

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