= Solution
Assume $v\ne0$ so that the change of variables is invertible. The <chain rule> gives $\partial_x=\partial_r+\partial_s$ and $\partial_t=v(\partial_r-\partial_s)$. Commuting the <mixed partial derivatives> for a twice continuously differentiable solution gives
$$
\partial_x^2-\frac1{v^2}\partial_t^2
=(\partial_r+\partial_s)^2-(\partial_r-\partial_s)^2=4\partial_r\partial_s.
$$
Thus the <wave equation> is $F_{rs}=0$. Integrating once says that $F_r$ depends only on $r$; integrating again gives
$$
\boxed{F(x,t)=f(x+vt)+g(x-vt).}
$$
The two arbitrary twice differentiable functions are waves moving in opposite directions at speed $|v|$. This is <D'Alembert's formula>, on a product domain in $(r,s)$ or globally on the usual full line.
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