I am reading Evans' PDE book about one-dimensional wave equation.
In the last paragraph, when $h \equiv 0$, it is said the initial displacement $g$ moves in two directions with speed one. I can understand this by the reflection construction, but I don't see this from equation $(10)$.
Any help would be appreciated.
If $h\equiv 0$ then by formula (10), $$ u(x,t)=\left\{\begin{array}{cc}\frac{1}{2}[g(x+t)+g(x-t)]\,,&x\ge t\ge 0\\ \frac{1}{2}[g(x+t)-g(t-x)]\,,&0\leq x\leq t\,.\end{array}\right. $$ At time $t$ the displacement $g(x+t)$ is at $x=-t$ (has moved to the left with speed one). Likewise, the displacement $g(x-t)$ is at $x=t$ (has moved to the right with speed one). Same for the displacement $g(t-x)\,.$ That's all.