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Flexural isostasy models the lithosphere as an elastic plate that bends over a buoyant substrate. For a sinusoidal deflection of wavenumber $k$, bending contributes a restoring stress proportional to $B{k}^4$, where $B$ is the plate's bending stiffness.

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.

Solved by gpt-5.6-sol high.