= Solution
For a nonzero-frequency <Fourier mode> proportional to $e^{i(kx+mz-\omega t)}$, the time-independent source has no oscillatory part. The pressure equation gives
$$
\boxed{\omega^2=\frac{N^2k^2+f^2m^2}{k^2+m^2}.}
$$
Take $k,m\ne0$ and $\omega^2\ne f^2,N^2$ for the following pressure-amplitude formulas. With $P_0$ the amplitude of $p/\rho_0$, the momentum and buoyancy equations imply
$$
\hat u=\frac{k\omega}{\omega^2-f^2}P_0,\quad
\hat v=-i\frac{kf}{\omega^2-f^2}P_0,\quad
\hat w=\frac{m\omega}{\omega^2-N^2}P_0,\quad
\hat b=i\frac{mN^2}{\omega^2-N^2}P_0.
$$
Consequently
$$
\hat q=ik\hat v-\frac f{N^2}im\hat b
=fP_0\left[\frac{k^2}{\omega^2-f^2}+\frac{m^2}{\omega^2-N^2}\right]=0,
$$
because the numerator is $(k^2+m^2)\omega^2-N^2k^2-f^2m^2$. \b[Propagating linear <inertia-gravity waves> carry zero perturbation <potential vorticity>.] In degenerate directions the division formulas must be replaced by the original linear equations, or interpreted by a regular limit; the zero-PV property of the nonzero-frequency wave component still follows from $q_t=0$. A steady balanced mode need not have zero <potential vorticity>.
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