Odd-extension solution for a semi-infinite fixed string ($y(x,t)=\frac{1}{2}[H(x-ct)+H(x+ct)]$)

For zero initial velocity and initial displacement $h(x)$ on $x\ge 0$ 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.