Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 333 / 3 / iv / Solution

Suppose $ϵ=N^{2}H/g\ll 1$. The smallest root satisfiesso the leading external gravity wave eigenfunction is depth-independent:For $n\ge 1$, the roots lie just above $n\pi$:and henceIn particular, up to an arbitrary normalization and sign,The hierarchy $c_0>c_1>c_2>\cdots$ separates the fast, nearly barotropic free-surface mode from the internal baroclinic modes.