 Independent-edge radiation from an open planar duct

For the even acoustic waveguide mode $\cos (n\pi y/h)e^{-ikx}$, set $\sin \alpha =n\pi /(k_{0}h)$ and $k=k_0\cos \alpha >0$. At leading independent-edge order for large $k_{0}h$, the two incident travelling constituents each have amplitude $(-1{)}^n/2$ at the relevant wall endpoint. Reflecting the upper-edge coordinates reverses the edge-wave sign. Their path phases are $e^{\pm i{k}_{0}h\sin \theta }$. Thus Sommerfeld half-plane diffraction gives the displayed sum, where $D(r)=\sqrt{2/(\pi k_{0}r)}{e}^{-i{k}_{0}r-i\pi /4}$. For $n>0$, the apparent poles at $\theta =\pm \alpha$ are removable in the combined expression, with limit $ikhD(r)/2$. Single-edge nonuniform far-field poles are not physical infinities. This combined far-field limit assumes a radiation distance beyond the aperture's near field and neglects further edge interactions; it is not uniform at duct cutoff.