Release:
2021. Vol. 7. № 1 (25)About the authors:
Konstantin Yu. Basinsky, Cand. Sci. (Phys.-Math.), Associate Professor, Department of Fundamental Mathematics and Mechanics, University of Tyumen; k.y.basinskij@utmn.ruAbstract:
This article deals with a problem that describes the propagation of surface waves in a layer of an inhomogeneous fluid. The authors present a mathematical model that describes wave motions on the surface of an ideal exponentially stratified fluid. In the equations and boundary conditions, the transition to dimensionless variables and quantities has been completed. Next, a linear version of the problem follows, the solution of which is in the form of progressive waves of a steady-state form with unknown amplitude coefficients. This type of solution is substituted into the equations and boundary conditions of the linear problem, which makes it possible to reduce the determination of unknown quantities to the problem of solving a system of ordinary differential equations. Solving the system has allowed identifying two areas of physical parameters with different nature of wave motion. Expressions are obtained for the unknown components of the fluid velocity, pressure, free surface shape, and wave frequency.
This article contains the analysis of the influence of various parameters of the problem on the wave motion: graphs of the dependence of the phase velocity of the wave on the stratification parameter are constructed for different layer depths and wavelengths. For a better understanding of the nature of wave motion, the expressions for the trajectories of liquid particles are determined. This has required writing the equations of motion of particles using the obtained expressions for the components of the velocity vector; these equations are solved with the method of asymptotic approximations. A graphical analysis of the effect of the stratification parameter value on the particle trajectory shape is carried out. The results have revealed that an increase in stratification leads to a compression of the trajectory in the vertical direction.
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