 Wave reflection by odd extension

For the half-line wave equation $u_{tt}=c^2{u}_{xx}$ with a zero Dirichlet boundary condition at $x=0$, extend both the initial displacement and initial velocity as odd functions to the full line. D'Alembert's formula then gives an odd solution in $x$, so $u(0,t)=0$. Restriction to $x\ge 0$ solves the desired initial-boundary value problem. With zero initial velocity, the displayed formula shows reflection at the fixed endpoint with reversal of the displacement sign. Continuous piecewise smooth displacement data give a weak solution; a classical solution requires sufficient additional regularity. In particular, a kink in the initial displacement can prevent the pointwise time derivative at that kink from taking the assigned initial velocity, even though the initial velocity is attained in the $L^2$ sense.