Wave equation in two dimensions

The wave equation in two dimension can be written as

\begin{equation} c \frac{\partial f}{\partial x} - \frac{\partial f}{\partial t} = 0 \end{equation}

The most common way to solve this equation is by change of variables, using the null coordinates

\begin{eqnarray} u &=& \frac{x + ct}{2}\\ v &=& \frac{x - ct}{2} \end{eqnarray}

This is easily shown to be invertible as we have

\begin{eqnarray} x &=& u + v\\ t &=&u - v \end{eqnarray}

The corresponding change becomes