Nonlinear Propagation of Wave-Packets on Fluid Interfaces

Abstract

The method of multiple scales is used to derive two partial differential equations which describe the evolution of two-dimensional wave-packets on the interface of two semi-infinite, incompressible, inviscid fluids of arbitrary densities, taking into account the effect of the surface tension. These differential equations can be combined to yield two alternate nonlinear Schrödinger equations; one of them contains only first derivatives in time while the second contains first and second derivatives in time. The first equation is used to show that the stability of uniform wavetrains depends on the wave length, the surface tension, and the density ratio. The results show that gravity waves are unstable for all density ratios except unity, while capillary waves are stable unless the density ratio is below approximately 0.1716. Moreover, the presence of surface tension results in the stabilization of some waves which are otherwise unstable. Although the first equation is valid for a wide range of wave numbers, it is invalid near the cutoff wave number separating stable from unstable motions. It is shown that the second Schrödinger equation is valid near the cutoff wave number and thus it can be used to determine the dependence of the cutoff wave number on the amplitude, thereby avoiding the usual process of determining a new expansion that is only valid near the cutoff conditions.