 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 givesTake $k,m\ne 0$ 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 implyConsequentlybecause the numerator is $(k^2+m^2){\omega }^2-N^2{k}^2-f^2{m}^2$. 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.