Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 332 / 4 / e / Solution

For finite mantle depth, retain all four vertical solutions of the biharmonic equation,rather than discarding the terms that grow as $z\to -\infty$. Apply the plate conditions at $z=0$ and matching conditions at $z=-H$. At the mantle--core interface these include continuity of normal velocity and normal stress, an appropriate tangential-stress condition for the liquid core, and a kinematic boundary condition for interface displacement. The normal-stress balance gains the stable buoyancy termbecause ${\rho }_c>\rho$.

The resulting homogeneous linear system for the mode amplitudes has a nontrivial solution only when its determinant vanishes; that condition replaces the half-space decay rate. Modes with $kH\gg 1$ decay before sensing the core and recover the half-space result. Modes with $kH\lesssim 1$ involve the full mantle depth, feel the basal density contrast and core mobility, and acquire a modified relaxation time. This is the planar finite-depth extension of flexural isostasy.

Solved by gpt-5.6-sol high.