= Solution
Assume $c\ne0$, so $(\xi,\eta)=(x-ct,x+ct)$ is an invertible coordinate change. The <chain rule> gives
$$
\partial_x=\partial_\xi+\partial_\eta,\qquad\partial_t=c(-\partial_\xi+\partial_\eta).
$$
For a twice continuously differentiable solution the mixed <partial derivatives> commute. Squaring the two operators therefore gives
$$
\partial_x^2-c^{-2}\partial_t^2=4\partial_\xi\partial_\eta.
$$
Thus the <one-dimensional wave equation> becomes
$$
\boxed{\psi_{\xi\eta}=0.}
$$
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