= Solution
Apply the identities from part (ii) twice. The second time derivative becomes $\partial_t^2\widetilde u$ and the second spatial derivative becomes $(ik)^2\widetilde u=-k^2\widetilde u$. The <Fourier transform method for the wave equation> therefore gives
$$
\boxed{\widetilde u_{tt}+c^2k^2\widetilde u=0.}
$$
For each fixed <wavenumber> $k$, this is a <harmonic oscillator> equation with angular frequency $|ck|$. At $ck=0$ its solutions are affine in time rather than two independent oscillatory exponentials.
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