Vibrotransport Solutions of Wave Equations and Their Properties by Subsonic Velocities

Abstract

The vibrotransport sources of disturbances in various media are the most common. They are associated with moving objects of oscillation whose speed can be subsonic, sonic, supersonic, and in media with several sonic speeds (elastic, for example) and transonic. Here, fundamental and regular vibrotransport solutions of the wave equation are constructed, which describe the dynamics of the medium during the movement of a source, which is concentrated at a point, moves at a constant speed V , and vibrates at a constant frequency ω . The type of equations depends on the Mach number M = V / c , where c is the sound speed in the medium. The value of M essentially affects the type of equations to be solved: elliptical for M M = 1, and hyperbolic for M > 1. Green’s functions are constructed that describe the dynamics of the medium during the motion of a vibration source concentrated at a point in the subsonic range of speeds in 3D spaces. On this basis, general solutions of the vibrotransport equation are constructed under the action of both spatially distributed moving vibration sources and concentrated on moving surfaces and lines. A mathematical description of the Doppler effect with a graphical illustration is given. The constructed solutions allow one to construct solutions to many equations of continuum mechanics and field theory for studying wave processes excited by various types of moving sources of oscillations in media, and should find wide application in solving various engineering and technical problems.