= Triangular pulse splitting under the wave equation
{title2=$u(x,t)=\tfrac12(1-|x-ct|)_++\tfrac12(1-|x+ct|)_+$}
For the <triangular function> $f(x)=(1-|x|)_+$ released with zero initial velocity in the whole-line <wave equation>, the <D'Alembert formula> gives the displayed solution. Here $r_+=\max(r,0)$. Two equal half-amplitude triangles translate in opposite directions, overlapping at early times and separating after $|c|t>1$. This defines a <weak solution>, because the initial slope has jumps; away from the translated corner lines it is a classical solution. Its <Fourier transform> is $2(1-\cos k)\cos(ckt)/k^2$, with removable value one at $k=0$.
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