Gaussian pulse splitting from zero initial velocity ($y(x,t)=\frac{1}{2}[g(x-ct)+g(x+ct)]$)

For the classical wave equation $y_{tt}=c^2{y}_{xx}$ on an effectively infinite string, put $g(x)=A{e}^{-a{x}^2/2}$, $a>0$. Initial data $y(x,0)=g(x)$ and $y_t(x,0)=-c{g}^{\prime }(x)=caxg(x)$ 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.