= Odd-extension solution for a semi-infinite fixed string
{title2=$y(x,t)=\tfrac12[H(x-ct)+H(x+ct)]$}
For zero initial <velocity> and initial <displacement> $h(x)$ on $x\geq0$ with $h(0)=0$, let $H$ be its <odd extension> to the real line. The displayed <D'Alembert's formula> solves the <wave equation> with $y(0,t)=0$, since $H(-ct)+H(ct)=0$. A pulse travelling toward the endpoint reflects with reversed <displacement>. For piecewise linear data the equation holds classically between moving corners and in the sense of <weak derivatives> globally.
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