Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 82 / 2 / b / i / Solution

Take a cut-on mode with $k>0$, put $\lambda =n\pi /h$ and $\alpha ={\tan }^{-1}(\lambda /k)$, so $k=k_0\cos \alpha$, $\lambda =k_0\sin \alpha$, and initially take $n>0$. The acoustic waveguide mode splits into two acoustic plane waves:At the upper edge $y=h$, the upward-going constituent is incident from below. Use the reflected local coordinate $Y_{+}=h-y$; its incident potential becomes $(-1{)}^n{e}^{-ikx+i\lambda Y_{+}}/2$, with local incidence angle ${\theta }_0=\pi -\alpha$. At the lower edge use $Y_{-}=y+h$ and the downward-going constituent; it has the same local incident amplitude and incidence angle. At each wall the other constituent is the geometrical optics reflection of this incident half-amplitude, so it must not be counted again as an independent incident wave at the same edge.

Let $D(r)=\sqrt{2/(\pi k_{0}r)}{e}^{-i{k}_{0}r-i\pi /4}$. For a far observer, the upper and lower local observation angles are $-\theta$ and $\theta$, and their distances are $r_{+}=r-h\sin \theta +O(h^2/r)$ and $r_{-}=r+h\sin \theta +O(h^2/r)$. Applying the Wiener-Hopf solution of rigid half-plane diffraction separately givesThe sign difference is the reversal of the upper-edge transverse coordinate. Their sum is the leading independent-edge radiation from an open planar duct:Restore $e^{i\omega t}$ if a time-dependent potential is desired. This is the leading large-$k_{0}h$ approximation, neglecting further diffraction between edges, and assumes a far-field distance $r\gg k_0{h}^2$ as well as $r\gg h$. It is not a uniform approximation for a mode approaching cutoff. The zero mode follows as the grazing-incidence limit $\alpha =0$; its beam-direction limit is treated separately below.