Viscous cross-waves: An analytical treatment

Viscous effects on the excitation of cross-waves in a semi-infinite box of finite depth and width are considered. A formalism using matched asymptotic expansions and an improved method of computing the solvability condition is used to derive the relative contributions of the free-surface, sidewall, bottom, and wavemaker viscous boundary layers. This analysis yields an expression for the damping coefficient previously incorporated on heuristic grounds. In addition, three new contributions are found: a viscous detuning of the resonant frequency, a slow spatial variation in the coupling to the progressive wave, and a viscous correction to the wavemaker boundary condition. The wavemaker boundary condition breaks the symmetry of the linear neutral stability curve at leading order for many geometries of experimental interest.