ВИПАДКОВІ ГРАВІТАЦІЙНІ ХВИЛІ У ДВОШАРОВІЙ ГІДРОДИНАМІЧНІЙ СИСТЕМІ

Abstract

The article is devoted to the study of the propagation of random gravitational waves in a three-dimensional hydrodynamic system half-space– half-space. An overview of studies on the analysis of the propagation of random waves in different systems is given. Mathematical statement of the problem contains second-order differential equations with respect to velocity potentials, kinematic and dynamic conditions on the contact surface. To study the problem, the field of deviations and the potentials of the wave velocities are presented in the form of expansions in Fourier-Stiltjes integrals. Stochastic amplitudes of the corresponding fields are expressed through the amplitude of the deviation field in the form of recurrent relations. Using the expansion in series in a small parameter for the stochastic field amplitude variations, the dynamic equation in integral form has been received. It should be noted that the use of a small parameter makes it possible to control the contribution of the nonlinearity of the corresponding terms. Subintegral functions of two- and three-wave interaction are obtained in symmetrized form. Based on the obtained equation, a linear dispersion relationship is derived. In the two-dimensional case, it degenerates into the dispersion relationship obtained by A. Naifehfor deterministic wave motions in a two-layer system. Using the equations for the amplitude of the deviation field and the ensemble averaging procedure, the equation for the spectrum of the first harmonics is obtained. The reliability of the obtained results is confirmed by a comparison with previous studies of the problem of propagation of random surface gravitational waves performed in the works of Masuda and others. The obtained results can be used in the study of the propagation of random internal waves in the oceans.