 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 givesThe last two equations give $\hat{u}=i{\hat{w}}^{\prime }/k$ and $\hat{\rho }=i{\rho }^{\prime }\hat{w}/(kV)$. The horizontal momentum equation givesSubstituting in the vertical momentum equation yields the Taylor–Goldstein equation:The buoyancy frequency is real for stable density stratification. The impermeable walls impose $\hat{w}(\pm L)=0$.