Past exam of the physics of the Earth as a Planet course of the University of Cambridge / 2014 / Paper 1 / 2 / a / Solution

Subtract the background hydrostatic pressure. A relief amplitude $h$ produces a restoring pressure perturbation $p^{\prime }\sim \rho gh$ and horizontal gradient $\rho gh/\lambda$. Incompressibility makes the disturbance penetrate a depth of order its horizontal scale. For sinusoidal viscous rebound, with relaxation speed $v\sim h/\tau$, the viscous force density is $\eta v/{\lambda }^2$. Balancing givesThis is a small-amplitude viscous half-space estimate; numerical factors from the sinusoidal geometry are absent from a scaling argument.