= 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>:
$$
\cos(\lambda y)e^{-ikx}=\frac12e^{-ikx-i\lambda y}+\frac12e^{-ikx+i\lambda y}.
$$
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 $D(r)=\sqrt{2/(\pi k_0r)}e^{-ik_0r-i\pi/4}$. 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 gives
$$
\begin{aligned}
\phi_{d,+}&\sim-\frac{(-1)^n}{2}D(r)\frac{\cos(\alpha/2)\sin(\theta/2)}{\cos\theta-\cos\alpha}e^{+ik_0h\sin\theta},\\
\phi_{d,-}&\sim+\frac{(-1)^n}{2}D(r)\frac{\cos(\alpha/2)\sin(\theta/2)}{\cos\theta-\cos\alpha}e^{-ik_0h\sin\theta}.
\end{aligned}
$$
The sign difference is the reversal of the upper-edge transverse coordinate. Their sum is the leading <independent-edge radiation from an open planar duct>:
$$
\boxed{\phi_d\sim-i(-1)^nD(r)\cos(\alpha/2)\sin(\theta/2)\frac{\sin(k_0h\sin\theta)}{\cos\theta-\cos\alpha}.}
$$
Restore $e^{i\omega t}$ if a time-dependent potential is desired. This is the leading large-$k_0h$ approximation, neglecting further diffraction between edges, and assumes a far-field distance $r\gg k_0h^2$ 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.
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