= Solution
Rigid walls require zero normal <velocity>, so $\phi_x=0$ at $x=0,L$ and $\phi_y=0$ at $y=0,b$. <Separation of variables> gives spatial modes $\cos(m\pi x/L)\cos(n\pi y/b)$, for nonnegative integers $m,n$. The <wave equation> therefore gives
$$
\boxed{\omega_{m,n}^2=\pi^2c_0^2(m^2/L^2+n^2/b^2).}
$$
The $(0,0)$ spatially constant mode has zero frequency and is not an ordinary positive-frequency <acoustic wave>[acoustic oscillation].
A spatially uniform boundary forcing excites the $n=0$ transverse mode. Put $\phi=X(x)\cos\omega t$. Then $X''+(\omega/c_0)^2X=0$ and $X'(L)=0$, hence $X=C\cos[\omega(L-x)/c_0]$. The condition $X'(0)=a$ fixes $C$ and gives
$$
\boxed{\phi=\frac{ac_0\cos[\omega(L-x)/c_0]}{\omega\sin(\omega L/c_0)}\cos\omega t.}
$$
Arbitrary free oscillations can be superposed. The denominator vanishes at longitudinal <resonances>. Near them the lossless steady harmonic response ceases to be a uniformly small-amplitude approximation; <damping> from <viscosity>, heat transfer or wall losses regularizes the response, and large amplitudes can require nonlinear <equations of motion> or a more accurate moving-boundary model. At exact <resonance> a switched-on ideal forcing produces secular growth rather than a bounded harmonic particular solution.
For the semi-infinite duct expand $F(y)=\sum_{n\geq0}F_n\cos(n\pi y/b)$. Each axial mode has
$$
\boxed{k_n^2=\omega^2/c_0^2-(n\pi/b)^2.}
$$
For a propagating mode a particular solution is $\phi_n=(F_n/k_n)\cos(n\pi y/b)\cos(k_nx-\omega t)$; differentiating at zero gives the prescribed $F_n\cos(n\pi y/b)\sin\omega t$. A nonzero higher transverse coefficient propagates when $|\omega|>n\pi c_0/b$.
The outgoing <radiation condition> selects $c_0^2k_n/\omega>0$, its positive axial <group velocity>. Thus $\boxed{\omega k_n>0}$. This sign is selected by the absence of waves incident from infinity, not by the differential equation alone. If $F$ has a nonzero mean, its $n=0$ component can propagate even below every higher-mode cutoff.
For $F=\cos(\pi y/b)$ and $0<|\omega|<\pi c_0/b$, put $\kappa=\sqrt{(\pi/b)^2-\omega^2/c_0^2}$. The decaying solution is
$$
\boxed{\phi=-\frac1\kappa e^{-\kappa x}\cos(\pi y/b)\sin\omega t.}
$$
It is evanescent and carries no mean <energy> to infinity. These mode and sign conditions are summarized by the <radiation condition for a waveguide mode>.
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