 Solution

The vertical parcel displacement has ${\partial }_t\eta =w$, so its complex amplitude is $\mathcal{E}=iW/\omega$. The horizontal momentum equation and incompressibility giveAt a material density interface the displacement is continuous, hence $[W]=0$. Continuity of pressure must be imposed on the displaced interface. Since the mean hydrostatic gradient is ${\bar{p}}_z=-\rho g$, continuity of Lagrangian pressure gives $[P]+\mathcal{E}[{\bar{p}}_z]=0$. A stable upward jump $[\bar{b}]=\Delta b$ gives $[{\bar{p}}_z]={\rho }_0\Delta b$, and thereforeThe same derivative jump follows by integrating the amplitude equation through the delta-function contribution $N^2={\bar{b}}_z$ at the interface. At a continuous-buoyancy stratification change, $\Delta b=0$ and both $W$ and $W^{\prime }$ are continuous. Assuming Eulerian pressure itself continuous across a density jump would miss the restoring interfacial force.