= Solution
For finite mantle depth, retain all four vertical solutions of the biharmonic equation,
$$
(A+Cz)e^{kz}+(D+Ez)e^{-kz},
$$
rather than discarding the terms that grow as $z\to-\infty$. Apply the plate conditions at $z=0$ and matching conditions at $z=-H$. At the mantle--core interface these include continuity of normal velocity and normal stress, an appropriate tangential-stress condition for the liquid core, and a <kinematic boundary condition> for interface displacement. The normal-stress balance gains the stable buoyancy term
$$
(\rho_c-\rho)g\,\eta_c
$$
because $\rho_c>\rho$.
The resulting homogeneous linear system for the mode amplitudes has a nontrivial solution only when its determinant vanishes; that condition replaces the half-space decay rate. Modes with $kH\gg1$ decay before sensing the core and recover the half-space result. Modes with $kH\lesssim1$ involve the full mantle depth, feel the basal density contrast and core mobility, and acquire a modified relaxation time. This is the planar finite-depth extension of <flexural isostasy>.
Solved by gpt-5.6-sol high.
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