On Some Kinematic and Energy Relations for Waves Propagating in Elastic Systems

Regularities inherent in waves propagating in structural elements modeled as one-dimensional and two-dimensional elastic systems are revealed. Local laws of energy and wave momentum transfer are given in the case when the Lagrangian of a two-dimensional elastic system depends on generalized coordinates, their derivatives up to the second order with respect to spatial variables, and mixed derivatives with respect to spatial and time variables. Expressions are found in terms of the Lagrangian density for the wave momentum flux density tensor, wave energy and wave momentum flux densities, work of forces changing system parameters, and distributed recoil forces arising during wave propagation in an inhomogeneous system. The dispersion and energy characteristics of waves propagating in plates on an elastic foundation described by different models are compared. The conditions and frequency range of the existence of the so-called backward waves, in which the phase and group velocities have opposite directions and significantly change the behavior of the energy flux, are determined. The minimum phase velocities of waves in the plates under consideration are found, the excess of which by a moving constant source gives rise to Vavilov–Cherenkov radiation in the elastic system. Their dependence on the stiffness modulus of the elastic foundation (often called the coefficient of subgrade reaction) and the physical and mechanical properties of the plate is established. For average values, relations between the energy flux density and the wave momentum flux density tensor are given. It has been established that, for systems whose dynamic behavior is described by linear or nonlinear equations with respect to an unknown function, the ratio between the average values of the energy flux density to the wave momentum density is equal to the product of the magnitudes of the phase and group wave velocities.