D'Alembert formula

The Cauchy problem $u_{tt}=c^2{u}_{xx}$ has

A rectangular initial displacement with zero initial velocity splits into two equal half-height copies moving in opposite directions. The D'Alembert formula reads $u(x,t)=[u_0(x+ct)+u_0(x-ct)]/2$. Overlap adds the heights; separated copies each retain the original width. For discontinuous fronts the formula defines a weak solution of the wave equation.

For $u_{tt}-u_{xx}=0$ with $u(0,x)=0$ and $u_t(0,x)=f(x)$,

If $u(x,0)=0$, $u_t(x,0)=g(x)$, and the odd function $g$ is supported on $[-a,a]$, thenThe interval $[x-t,x+t]$ then contains the entire support, whose integral vanishes by oddness.