= Gaussian pulse splitting from zero initial velocity
{c}
{title2=$y(x,t)=\tfrac12[g(x-ct)+g(x+ct)]$}
For the classical <wave equation> $y_{tt}=c^2y_{xx}$ on an effectively infinite string, put $g(x)=Ae^{-ax^2/2}$, $a>0$. Initial data $y(x,0)=g(x)$ and $y_t(x,0)=-cg'(x)=caxg(x)$ give the single right-moving pulse $y=g(x-ct)$. The same displacement with zero initial velocity instead gives $y=[g(x-ct)+g(x+ct)]/2$, two oppositely moving half-amplitude pulses. Both satisfy their initial data by direct differentiation. Their total energies differ because their initial kinetic energies differ; the split solution initially stores only elastic energy.
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