= Exponential-profile stationary stratified waves
{title2=$\phi''+[N_0^2/U_0^2-M^2-k^2]\phi=0$}
For positive background velocity $U=U_0e^{Mz}$ and <buoyancy frequency> $N=N_0e^{Mz}$, a stationary nonzero-horizontal-wavenumber <Taylor–Goldstein equation> reduces to the displayed constant-coefficient equation. Its structures are oscillatory, affine or exponential according to the sign of $N_0^2/U_0^2-M^2-k^2$. The disturbance <Laplacian> is a constant times its <streamfunction>, but its <buoyancy> anomaly is $N_0^2e^{Mz}\phi/U_0$. Therefore vorticity self-advection vanishes while buoyancy self-advection produces a nonzero second harmonic for $M\ne0$. These nontrivial real linear modes are not exact finite-amplitude solutions without additional fields. For $M=0$, both polarizations are constant and these nonlinear self-interactions vanish.
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