= Solution
Spatial Fourier transformation gives
$$
\widetilde u_{tt}+c^2k^2\widetilde u=0,\qquad
\widetilde u(k,0)=\sqrt\pi e^{-k^2/4},\qquad
\widetilde u_t(k,0)=0.
$$
Hence
$$
\widetilde u(k,t)=\sqrt\pi e^{-k^2/4}\cos(ckt).
$$
Inverting and splitting the cosine into exponentials, or applying <D'Alembert formula>, yields
$$
u(x,t)=\frac12\left(e^{-(x-ct)^2}+e^{-(x+ct)^2}\right).
$$
\b[Thus the initial Gaussian separates into two half-amplitude Gaussian pulses travelling without distortion at speeds $c$ and $-c$.]
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