= Independent-edge radiation from an open planar duct
{title2=$\phi_d\sim-i(-1)^n D(r)\cos(\alpha/2)\sin(\theta/2)\sin(k_0h\sin\theta)/(\cos\theta-\cos\alpha)$}
For the even <acoustic waveguide> mode $\cos(n\pi y/h)e^{-ikx}$, set $\sin\alpha=n\pi/(k_0h)$ and $k=k_0\cos\alpha>0$. At leading independent-edge order for large $k_0h$, 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 $e^{\pm ik_0h\sin\theta}$. Thus <Sommerfeld half-plane diffraction> gives the displayed sum, where $D(r)=\sqrt{2/(\pi k_0r)}e^{-ik_0r-i\pi/4}$. 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.
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