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

Eliminate ${\xi }_z$ from the coupled system to obtainWriting $\Psi =\psi e^{kz}$ givesFor the neutral polytrope found in part (a),If $\psi$ is a polynomial of degree $n$ with leading term $z^n$, the coefficient of the highest power $z^n$ in the ODE isA polynomial solution therefore requiresThe factor $e^{kz}$ traps every mode below the free surface. The $n=0$ solution is the incompressible surface gravito-inertial or $f$ mode of part (d). The $n\ge 1$ solutions are vertically structured acoustic p modes; neutral stratification leaves no buoyancy-driven $g$-mode family.