 Solution

Take the constant speed parameter $v\ne 0$, so $r=x+vt$ and $s=x-vt$ are independent characteristic coordinates. The chain rule givesApplying these operators twice to a twice differentiable function yieldsConsequently the wave equation reduces to $4F_{rs}=0$. Integrate first with respect to $s$, obtaining $F_r=A(r)$, and then with respect to $r$, allowing a separate integration function of $s$. The general local solution iswith arbitrary sufficiently differentiable functions $G$ and $H$. For $v>0$, the first profile travels towards negative $x$ and the second towards positive $x$. Substitution directly checks the equation. Initial or boundary conditions select the two profiles; none are specified here. This is D'Alembert's formula in characteristic form.