Asymptotic solutions of the problem on standing capillary-gravity waves in a fluid at near-critical modes

We construct asymptotic solutions of the nonlinear 2D problem on standing capillary-gravity waves at the interface between two fluids of different densities in the form of expansions in powers of the wave amplitude, which is assumed to be small. The solutions of the problem and nonlinear corrections to the wave frequency are obtained in Lagrange variables up to fourth-order infinitesimals. The solutions are studied in the problem parameter range where the second-order frequency corrections are zero (the "critical mode") or nearly zero ("near-critical modes"). We show that the amplitude-frequency curves in near-critical modes have two branches. A detailed analysis of the problem on waves on the free surface of a homogeneous fluid of finite depth as well as of the problem on internal waves on the interface between two semi-infinite fluid layers of different densities is carried out. The amplitude-frequency curves and wave profiles are presented graphically. The results are compared with laboratory experiments for gravity waves

Hydroelastic stationary problem on tsunami waves generation

A two-layered system is considered: a compressible fluid layer of constant depth overlies an homogeneous elastic half-space containing a source of disturbances, which are assumed to be longitudinal steady waves. The vertical displacements of the fluid's free surface and of the sea-bottom interface are calculated by asymptotic methods in the far field approximation away from the epicenter and by numerical methods based on Maple in the epicentral zone. The model is applied to the tsunami wave generation problem. At moderate depths of the hypocenter, it is found that noticeable tsunami waves can be generated if the spectrum of the source includes long-period components. The results are compared with those obtained previously in the long-wave theory approximations. Copyright © 2004 by MAIK "Nauka/Interperiodica"

Hydroelastic stationary problem on tsunami waves generation

A two-layered system is considered: a compressible fluid layer of constant depth overlies an homogeneous elastic half-space containing a source of disturbances, which are assumed to be longitudinal steady waves. The vertical displacements of the fluid's free surface and of the sea-bottom interface are calculated by asymptotic methods in the far field approximation away from the epicenter and by numerical methods based on Maple in the epicentral zone. The model is applied to the tsunami wave generation problem. At moderate depths of the hypocenter, it is found that noticeable tsunami waves can be generated if the spectrum of the source includes long-period components. The results are compared with those obtained previously in the long-wave theory approximations. Copyright � 2004 by MAIK "Nauka/Interperiodica"

Asymptotic solutions of the problem of steep capillary-gravitational Faraday waves on the interface between fluids

We construct asymptotic solutions of the 2D problem on weakly nonlinear capillary-gravitational Faraday waves and the interface between two fluids (of distinct density) subjected to vertical oscillations. The problem is formulated in the Lagrange variables, and the solutions are sought in the form of expansions in powers of the small parameter characterizing the steepness of the waves, and it is assumed that the ratio of the amplitude of the vertical acceleration of the fluids to the acceleration of gravity is of the order of some power of this small parameter. We construct solutions in the domains of parameters of the problem that correspond to both ordinary regimes (in which one can restrict oneself to two terms of the expansion) and critical and near-critical regimes (in which one must consider fourth-order terms). The resonance curves and wave profiles of the waves of maximal steepness are represented graphically. The results for ordinary regimes are compared with the known asymptotic solutions for free standing waves and with data of laboratory experiments concerning Faraday waves