all of the characteristic speeds associated with this system The slow modulation of the interfacial capillary-gravity waves are real. From this criterion he derived the same results as of two superposed dielectric fluids with uniform depths and solid Benjamin (4). The effect of surface tension on Benjaminhorizontal boundaries, under the influence of a normal electric Feir instability has been investigated by Barakat (2). He field and in the absence of surface charges at their interface, is found that capillary-gravity waves are unstable to sideband investigated by using the multiple-time scales method. It is found modulation in a deep liquid, and he gives bounds on the that the complex amplitude of quasi-monochromatic traveling sideband frequencies for the growth of instability. Easwaran waves can be described by a nonlinear Schro ¨dinger equation in a (13) and Easwaran and Majumdar ( 14) extended the analyframe of reference moving with the group velocity. The stability sis to a liquid layer of arbitrary uniform depth. They found characteristics of a uniform wave train are examined analytically and numerically on the basis of the nonlinear Schro ¨dinger equa-that in the presence of surface tension there is always instation, and some limiting cases are recovered. Three cases appear, bility for some wavenumber and liquid depth, and they dedepending on whether the depth of the lower fluid is equal to, rived bounds on the sideband frequencies for unbounded greater than, or less than the depth of the upper fluid. The effect amplification.
of the normal electric field is determined for the three stability
The problem of nonlinear modulation of water waves has regions of the pure hydrodynamic case. It is found that the normal been investigated by several authors and some instructive electric field has a destabilizing influence in the first stability region conclusions and results obtained. An early and typical work and a stabilizing effect in the second and third stability regions.
using the method of multiple scales was done by Hasimoto Moreover, one new unstable region or two new stable and unstable and Ono (20), who studied two-dimensional water waves.
regions appear, all of which increase when the electric field in-They showed that first-order perturbation is governed by the creases. On the other hand, the complex amplitude of quasi-monononlinear Schro ¨dinger equation. A Stokes wave train soluchromatic standing waves near the cutoff wavenumber is governed by a similar type of nonlinear Schro ¨dinger equation in which the tion was found and thus its stability could be discussed easroles of time and space are interchanged. This equation makes it ily. Afterward, Davey and Stewartson (10) extended the possible to estimate the nonlinear effect on the cutoff wavenumber. results to three-dimensional cases, whereas Djordjevic and ᭧ 1998 Academic Press Redekopp (12) examined three-dimensional packets of capillary-gravity waves. They showed that the evolution is described by two partial differential equations: a nonlinear