The <kinematic boundary condition> is $\zeta_t=w$ at $z=0$. Part c therefore gives
$$
\dot\epsilon
=-\frac{\rho g+Bk^4}{2\mu k}\epsilon,
$$
and hence
$$
\boxed{
\epsilon(t)=\epsilon(0)e^{-t/\tau(k)}},
\qquad
\boxed{
\tau(k)=\frac{2\mu k}{\rho g+Bk^4}}.
$$
In terms of wavelength $\lambda=2\pi/k$,
$$
\boxed{
\tau(\lambda)=
\frac{4\pi\mu/\lambda}
{\rho g+B(2\pi/\lambda)^4}}.
$$
Without elasticity, $\tau\sim2\mu k/(\rho g)$: shorter wavelengths penetrate less deeply but relax more slowly because their viscous gradients are larger. At short wavelength, elastic bending dominates and
$$
\tau\sim\frac{2\mu}{Bk^3},
$$
so the strong $Bk^4$ 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 at
$$
\boxed{k=\left(\frac{\rho g}{3B}\right)^{1/4}}.
$$
This exponential relaxation is the simplest half-space model of <postglacial rebound> and <glacial isostatic adjustment>.

Solved by gpt-5.6-sol high.

Solution

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

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Postglacial rebound is uplift following removal of an ice load. It reflects viscous flow in the mantle and elastic or viscoelastic deformation of the lithosphere.


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 Incoming links: Postglacial rebound