= Solution
If $\psi=f(\xi)+g(\eta)$ with twice differentiable $f,g$, its mixed <partial derivative> vanishes because each summand depends on only one coordinate. Conversely $\psi_{\xi\eta}=0$ means $\psi_\xi$ is independent of $\eta$; integration in $\xi$ supplies $f(\xi)$ and an integration function $g(\eta)$. Hence, on a characteristic rectangle, the general solution is
$$
\boxed{\psi(x,t)=f(x-ct)+g(x+ct).}
$$
These terms travel in opposite directions without changing profile, as in <D'Alembert's formula>.
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