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

The kinematic boundary condition is ${\zeta }_t=w$ at $z=0$. Part c therefore givesand henceIn terms of wavelength $\lambda =2\pi /k$,Without elasticity, $\tau \sim 2\mu k/(\rho g)$: shorter wavelengths penetrate less deeply but relax more slowly because their viscous gradients are larger. At short wavelength, elastic bending dominates andso the strong $B{k}^4$ restoring stress makes very short waves relax rapidly. The graph therefore tends to zero at both wavelength extremes and has one maximum. Differentiating with respect to $k$ places that maximum atThis exponential relaxation is the simplest half-space model of postglacial rebound and glacial isostatic adjustment.

Solved by gpt-5.6-sol high.