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

To first order, vanishing tangential velocity on the elastic plate is simply $u(0)=0$. In the mode from part a this givesConsequentlyso $u(0)=0$, $w(0)=ikA{e}^{ikx}$, and $w_z(0)=0$. The dynamic pressure at the surface is

Evaluating the hydrostatic pressure at $z=\zeta$ contributes the restoring normal stress $\rho g\zeta$. Since ${\nabla }^4\zeta =k^4\zeta$ for this mode, the linearized normal-stress balance isIt follows thatThe two restoring effects are buoyancy and the plate's bending stiffness.