 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 isThese terms travel in opposite directions without changing profile, as in D'Alembert's formula.