Abstract:
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 in the case when the Lagrangian of a two-dimensional elastic system depends on generalized coordinates, their derivatives up to the second order on spatial variables, and mixed derivatives on spatial and time variables are given. Expressions through the density of the Lagrangian function for the density tensor of the wave momentum flux, the densities of the wave energy and wave momentum fluxes, the work of forces changing the system parameters, and the distributed recoil forces arising from wave propagation in an inhomogeneous system are found. The dispersion and energy characteristics of waves propagating in plates on an elastic base described by different models are compared. The conditions and frequency range of existence of so-called backward waves, in which phase and group velocities have opposite directions and essentially changing the character of energy flow behavior, are determined. The minimum phase velocities of waves in plates under consideration, when exceeded by a moving constant source in an elastic system, the Vavilov–Cherenkov radiation begins. Their dependence on the stiffness coefficients of the elastic base (often called bed coefficients) and physical and mechanical properties of the plate is established. For the mean values, relations linking the energy flux density and the wave momentum flux density tensor are given. It is found that for systems whose dynamic behavior is described by linear equations or nonlinear with respect to an unknown function, the ratio of the moduli of the mean values of the energy flux density to the wave momentum flux density is equal to the product of the moduli of the phase and group velocities of the waves.
Key words:distributed system, generalized elastic basis, wave, phase velocity, energy transfer rate, energy flux density, wave momentum flux density.
