Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 333 / 3 / iii / Solution

When $\mu \gg 1$, the boundary coupling $B/H=\mathrm{csch}\mu /\mu$ is exponentially small. The two eigenvalues are therefore the diagonal boundary-wave speeds through order $1/\mu$:The lower wave is damped at the imposed rate, while the upper wave is exponentially isolated from the lower boundary and remains neutral to algebraic order. Damping destroys the phase-locked counterpropagating-wave interaction at short horizontal wavelength rather than damping both waves equally.