Fixed-end reflection of a triangular pulse ($y/H=T(x/L-ct/L)-T(-x/L-ct/L)$)

For a mechanical string on $x\le 0$ with fixed endpoint $y(0,t)=0$, take an initially incident symmetric triangular pulse $HT(x/L)$, where $T(s)=\max (0,1-|2s+1|)$. The superposition principle gives the title formula. Both terms satisfy the wave equation away from the corners and satisfy it weakly everywhere; their values cancel at the endpoint. At $ct=L/2$ their displacements cancel on the whole half-line, but the velocity is nonzero on $-L/2<x<0$. Hence a flat instantaneous shape need not mean zero wave energy.