Nonlinear Interaction of Capillary-Gravity Waves with a Subsonic Gas in the Presence of Magnetic Field

Abstract

The method of multiple scales is used to analyse the nonlinear propagation of waves on the interface between a liquid and a subsonic gas in the presence of magnetic field taking into account surface tension. The evolution of the amplitude is governed by a nonlinear Schr6dinger equation which gives the criterion for modulational instability. Numerical results are given in the graphical form. considerable attention during the last decade. We consider in this paper, the weakly nonlinear physical system, the interaction of capillary gravity waves with a subsonic flow moving uni- formly parallel to the undisturbed liquid surface in the presence of magnetic field. The same problem without magnetic field when the liquid is of finite depth has been investigated earlier by Nayfeh and Hassan (1). The inclusion of nonlinear terms results in amplitude modulation. In various problems of interest, it has been shown that the long-time slow modulation of wave amplitude is governed by the nonlinear Schr0dinger equation. In recent years, evolution of wave packets on the surface of an electrically conducting fluid has been investigated by a number of workers. El Shehawey (2) discussed the nonlinear condi- tions of stability and instability of electro-hydrodynamic Kelvin-Helmholtz mechanisms in the presence of a normal field in the absence of surface charges on the interface. Pusri and Malik (3) investigated the propagation of wave packets on the surface of an electrically conducting fluid of uniform depth in the presence of a tangential magnetic field in three dimensions by extending the analyses of Djordjevic and Redekopp (4) and Ablowitz and Segur (5) to incorporate magne- tohydrodynamic effects. The method of multiple scales was very successfully used by Hasimoto and Ono (6) to derive a single equation describing the long-time evolution of the envelope of a packet of plane finite amplitude gravity waves. In this presentation, by the multiple scale method, we plan to develop the nonlinear Schrodinger equation describing the evolution of the finite amplitude wave packet on the liquid surface in the presence of normal magnetic field with