= Reflected-step solution of the wave equation
{title2=$y(x,t)$}
For $y_{tt}=c^2y_{xx}$ on $0<x<L$, initially at rest with zero displacement, apply $y(0,t)=a$ for $t>0$ and keep $y(L,t)=0$. A <geometric series> in the <Laplace transform> gives
$$
y(x,t)=a\sum_{m\geq0}\left[H\!\left(t-\frac{2mL+x}{c}\right)-H\!\left(t-\frac{(2m+2)L-x}{c}\right)\right].
$$
Here $H$ is the <Heaviside step function>, $L,c>0$, and only finitely many terms contribute at any finite time. The second step in each pair is a sign-reversed reflection from the fixed endpoint. The formula solves the <wave equation> classically away from the fronts and distributionally across them; the midpoint is a square wave of period $2L/c$.
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