 Solution

Apply the identities from part (ii) twice. The second time derivative becomes ${\partial }_t^2\tilde{u}$ and the second spatial derivative becomes $(ik{)}^2\tilde{u}=-k^2\tilde{u}$. The Fourier transform method for the wave equation therefore givesFor 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.