LONGITUDINAL WAVE PROPAGATION IN A VISCOELASTIC ROD ACCORDING TO THE MAXWELL MODEL. PART 1. ANALYSIS OF DISPERSION CHARACTERISTICS AND FREQUENCY DEPENDENT DAMPING WHEN SOLVING BOUNDARY-VALUE PROBLEMS

The dynamics of a rod, the material of which obeys Maxwell's law of deformation of a medium, is considered. The propagation of a longitudinal wave in such a rod is described by a one-dimensional wave equation supplemented by a term characterizing the viscosity of the material. The solution to the equation is found in the form of a traveling harmonic wave. From the original partial differential equation, a transition is made to an algebraic complex dispersion equation that connects frequency and wave number, allowing one to calculate the phase and group velocities of the wave and determine the patterns of its propagation and attenuation. When analyzing the dispersion equation, two problems are highlighted: 1) frequency is considered a real quantity, and the wave number is considered a complex quantity (as is customary when solving boundary value problems); 2) the wave number is considered a real quantity, and the frequency is considered a complex quantity (as is customary when solving the Cauchy problem). The first task is considered in detail. It has been revealed that a longitudinal wave that satisfies its conditions has the following propagation features: as the real part of the wave number increases, its frequency increases (at low viscosity it increases slowly, at high viscosity it increases quickly), the phase velocity first increases and then reaches a horizontal asymptote, the group velocity increases, reaches a maximum, then decreases, reaching a different horizontal asymptote than the phase velocity. Over the entire range of changes in the real part of the wave number, anomalous dispersion of the longitudinal wave is observed (i.e., the group velocity is greater than the phase velocity); The attenuation of the wave (determined by the imaginary part of the wave number) first increases with increasing frequency, then reaches a horizontal asymptote and becomes frequency-independent.