 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2026

Paper 314

Set $\delta\mathbf u=-i\omega\boldsymbol\xi$. The continuity and adiabatic equations give the Lagrangian perturbations
$$
\frac{\Delta_L\rho}{\rho}=-\Delta,
\qquad
\frac{\Delta_Lp}{p}=-\gamma\Delta,
\qquad
\Delta=ik_x\xi_x+\frac{d\xi_z}{dz}.
$$
Because the equilibrium is a <neutrally stratified polytropic atmosphere>, $p\propto\rho^\gamma$, so the displacement terms cancel and
$$
\boxed{\frac{\delta p}{p}=\gamma\frac{\delta\rho}{\rho}}.
$$
Using $dp/dz=-\rho g$ then gives
$$
\boxed{\Psi=\frac{\delta p}{\rho}
=g\xi_z-v_s^2\Delta},
\qquad
v_s^2=\frac{\gamma p}{\rho}.
$$

Eliminating $\xi_y$ from the two horizontal momentum equations and combining gravity with the vertical pressure force yields
$$
\boxed{-(\omega^2-4\Omega^2)\xi_x=-ik_x\Psi},
\qquad
\boxed{-\omega^2\xi_z=-\frac{d\Psi}{dz}}.
$$
Thus
$$
\xi_x=\frac{ik_x\Psi}{\omega^2-4\Omega^2},
$$
and the requested coupled first-order system is
$$
\boxed{
\frac{d\Psi}{dz}=\omega^2\xi_z,
\qquad
\frac{d\xi_z}{dz}
=\frac{g}{v_s^2}\xi_z
+\left(\frac{k_x^2}{\omega^2-4\Omega^2}
-\frac1{v_s^2}\right)\Psi}.
$$

Solution

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

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

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

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

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)