= Solution
Take the constant speed parameter $v\ne0$, so $r=x+vt$ and $s=x-vt$ are independent <characteristic coordinates>. The <chain rule> gives
$$
\partial_x=\partial_r+\partial_s,\qquad\partial_t=v(\partial_r-\partial_s).
$$
Applying these operators twice to a twice differentiable <function> yields
$$
F_{xx}=F_{rr}+2F_{rs}+F_{ss},\qquad F_{tt}=v^2(F_{rr}-2F_{rs}+F_{ss}).
$$
Consequently 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 is
$$
\boxed{F(x,t)=G(x+vt)+H(x-vt)},
$$
with 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.
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