Past exam of the waves and compressible flow course of the University of Oxford / 2023 / 2 / c / i

Use the Fourier transform convention $\hat{W}(k,z,t)={\int }_{\mathbb{R}}W(x,z,t)e^{-ikx}dx$, with inverse factor $1/(2\pi )$. The initial profile occupies only the vertical mode $\sin (\pi z/h)$. Its transform isThe transformed evolution is a harmonic oscillator for each $k$, withTherefore inversion givesUsing the nonnegative even frequency instead gives the same cosine. At $t=0$, ${\int }_{\mathbb{R}}{e}^{-a|k|}{e}^{ikx}dk=2a/(a^2+x^2)$ with $a=ϵh$, recovering the specified initial profile and its zero time derivative.