= Fixed-end reflection of a triangular pulse
{title2=$y/H=T(x/L-ct/L)-T(-x/L-ct/L)$}
For a <mechanical string> on $x\le0$ 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>.
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