Subsonic Vibrotransport Solutions of D’Alembert Equation in Spaces of Dimensions N = 3, 2, and 1

Abstract

Among active disturbance sources in various environments, the most common are transport and vibrotransport phenomena, which are associated with moving objects, whose speeds can be subsonic, sonic, supersonic, and in environments with several sonic speeds (elastic, for example) also transonic. Here, fundamental and regular vibrotransport solutions of the wave equation are constructed at subsonic speeds of the disturbance source in spaces of physical dimensions (N = 3, 2, 1). Green’s functions are constructed to describe the dynamics of the medium during the movement of a source concentrated at a point, moving at constant speed and vibrating at constant frequency. Based on these results, general solutions of the vibration transport 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 of many equations of continuum mechanics for this type of moving sources of disturbances in media and can be applied extensively in solving various engineering and technical problems.