= Wave reflection by odd extension
{title2=$u(x,t)=\tfrac12[f_{\rm odd}(x-ct)+f_{\rm odd}(x+ct)]$}
For the half-line <wave equation> $u_{tt}=c^2u_{xx}$ 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 $x\geq0$ 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.
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