Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 314 / 2 / d / Solution

For an incompressible perturbation, $\Delta =0$, so $\Psi =g{\xi }_z$. The vertical equation givesIncompressibility together with the horizontal equation also givesEquating the two expressions yieldsThe positive-frequency branch is the surface gravito-inertial waveIts vertical displacement decays as $e^{{\omega }^{2}z/g}$ into the atmosphere $z<0$.