= Solution
Let $V=U-c$ and take $k>0$ without loss of generality. The <Boussinesq approximation> replaces density by $\rho_0$ in inertia and pressure acceleration, while retaining the density perturbation in <buoyancy>. Linearizing about the <hydrostatic equilibrium> and substituting the <normal mode> gives
$$
ikV\widehat u+U'\widehat w=-ik\widehat p/\rho_0,\qquad
ikV\widehat w=-\widehat p'/\rho_0-g\widehat\rho/\rho_0,
$$
$$
ik\widehat u+\widehat w'=0,\qquad
ikV\widehat\rho+\rho'\widehat w=0.
$$
The last two equations give $\widehat u=i\widehat w'/k$ and $\widehat\rho=i\rho'\widehat w/(kV)$. The horizontal momentum equation gives
$$
\frac{\widehat p}{\rho_0}=\frac{V\widehat w'-U'\widehat w}{ik}.
$$
Substituting in the vertical momentum equation yields the <Taylor–Goldstein equation>:
$$
\boxed{\widehat w''-k^2\widehat w-\frac{U''}{U-c}\widehat w+\frac{N^2}{(U-c)^2}\widehat w=0,\qquad
N^2=-\frac g{\rho_0}\rho'.}
$$
The <buoyancy frequency> is real for <stable density stratification>. The impermeable walls impose $\widehat w(\pm L)=0$.
Back to article page
