Linear stability of transversely modulated finite-amplitude capillary waves on deep water

We investigate the three-dimensional linear stability of the periodic motion of pure capillary waves progressing in permanent form on water of infinite depth for the whole range of wave amplitudes. After introducing a coordinate transformation based on a conformal map for two-dimensional steady capillary waves, we perform linear stability analysis of finite-amplitude capillary waves in the transformed space. To solve the linearized equations for small amplitude disturbances, it is assumed that the wavelengths of the disturbances in the transverse direction are much longer than those in the propagation direction and, therefore, the disturbances are weakly three-dimensional. This assumption along with the periodicity of solutions allows us to write the linearized equations as an eigenvalue problem in matrix form. Following a perturbation theory for matrices, we expand the solutions of this eigenvalue problem in terms of a small parameter measuring the weak three-dimensionality, and numerically obtain approximate eigenvalues. For weakly three-dimensional superharmonic disturbances, the numerical results demonstrate that the pure capillary waves are two-dimensionally stable, but three-dimensionally unstable for almost all wave amplitudes. On the other hand, for subharmonic disturbances that are known to be two-dimensionally unstable, it is found that the long-wavelength disturbances in the transverse direction reduce the two-dimensional growth rate near the critical amplitude beyond which the pure capillary waves are unstable.