Resonant interfacial capillary–gravity waves in the presence of damping effects

Abstract

An investigation is made into the waves of small and moderate amplitude which may occur at the interface of two inviscid fluids of different densities. The external forces are those of gravity and surface tension and the waves are due to the resonant interaction between the Mth and Nth harmonics of the fundamental mode. In contrast to previous studies, damping effects are taken into account. Important parameters in the problem are the velocity and density ratios. A pair of coupled nonlinear Schrodinger-type partial differential equations for the wave amplitudes is derived which model the evolution of the waves, correct up to third order. A wide variety of sinusoidal solutions to the equations is shown to exist, irrespective of the values assigned to the parameters. The stability of these solutions to small modulational perturbations is considered. It is found that when the damping is due to dissipation then the waves are stabilized.

On Two Types of Stability of Solutions to a Pair of Damped Coupled Nonlinear Evolution Equations

“…rm ul + . Remark 3.2 Similar results to Thm 3.1 can be found in [3] [20]. Equation (3.6) is of Sturm-Louiville type, for a general treatment of such equations see [30].…”

“…Our results prove that any amount of dissipation (however small) has the effect of stabilising the waves in the presence of small perturbations known as sidebands. In the absence of dissipation, it was shown [3] that the waves may or may not be stable depending on the precise modes involved in the interactions, ie the values of M and N, and also whether the direction of the perturbations is normal, transverse or oblique to the main wavetrain. This is in contrast to the nonresonant case when the classical Benjamin-Feir theory predicts instability in the absence of dissipation [23] while the later results of Segur et al [20] showed that such waves are rendered stable by dissipative effects.…”

“…Later Jones used the method of multiple scales to derive the system (I), first to model resonant waves on the surface of deep water [1] and then those which manifest themselves on the interface of two fluids [2]. More recently [3] he conducted a detailed study of the stability of the solutions (1.4) of (I) when 0 δ = ie the undamped case. His findings showed that the stability is dependent on the values of the parameters in the problem, but physically one important parameter is V which is the ratio of the horizontal speed of the two fluids when the system is in its quiescent state.…”

“…We shall be concerned with the following pair of coupled nonlinear partial differential equations: Such systems have been used to model the slow evolution of small amplitude capillary-gravity waves caused by the interaction between the Mth and Nth harmonics of the fundamental mode and in the presence of damping. They can be applied both to waves on a free surface of an infinitely deep fluid or to those arising at the interface of two fluids each of infinite vertical extent, for more details and a derivation consult Jones [1] [2] [3]. Other authors who have considered such systems as applied to water waves include Carter [4], Dhar & Das [5] Osborne & Serio [9].…”

“…It is hoped to cover the case of nonlinear stability in subsequent work. A detailed analysis of the stability of (1.2) in the absence of dissipation, ie when 0 δ = was carried out by Jones [3]. His results showed that the solutions may be stable or unstable to plane wave perturbations depending on the signs of the coefficients , , , , , r u v s p q and also on the nature of the perturbations themselves ie whether they are longitudinal, transverse or oblique to the carrier wave.…”