Stability of gravity-capillary waves generated by a moving pressure disturbance in water of finite depth

In previous work, we investigated two-dimensional steady gravity-capillary waves generated by a localized pressure distribution moving with constant speed U in water of finite depth h . Localized solitary waves can only exist in subcritical flows where the Froude number F = U/(gh) < 1 , and were found using a combination of numerical simulations of the fully nonlinear inviscid, irrotational equations, and analytically from a weakly nonlinear long-wave model, the steady forced Korteweg-de Vries equation. The solution branches depended on three parameters, the Froude number, F < 1 , the Bond number, τ > 1/3 , and the magnitude and sign of the pressure distribution, ǫ . In this paper, we examine the two-dimensional stability of these waves using numerical simulations of the fully nonlinear unsteady equations. The results are favourably compared with analogous numerical solutions of the unsteady forced Korteweg-de Vries equation. We find that, for ǫ > 0 , the small-amplitude steady depression wave is stable whereas the large-amplitude steady depression wave is unstable. The depression wave with a dimple at its crest, which occurs only when ǫ < 0 is unstable, but the small-amplitude elevation wave with ǫ < 0 is stable.