On the determination of the Umov – Poynting vector in the propagation of acoustic waves in a hypoelastic medium

Abstract

For hypoelastic media with initial stresses, the propagation of acoustic waves is considered
from the point of view of superposition of small perturbations on finite strains. The initial
state of the medium is characterized by homogeneous fields of finite strains and stresses, wave
propagation is described by small perturbations of the displacement field. In the article, the
formulation of the theorem on the change in the kinetic energy of the medium, linearized in
the vicinity of the initial state, and, as a consequence, the formulation of the acoustic Poynting
theorem for a hypoelastic medium are obtained. An expression for the Umov – Poynting vector
for a hypoelastic medium is written in terms of a generalized true stress tensor.
For plane monochromatic waves, the change in the stress tensor associated with the
passage of a wave in a medium with initial stresses is determined, and an expression for the
Umov–Poynting vector is obtained through the second Christoffel tensor and the initial stresses
acting in the medium. An expression for the radial velocity vector that takes into account the
initial stresses acting in the medium is obtained. It is shown that under the action of initial
stresses, the Umov – Poynting vector deviates from the radial velocity vector. This result does
not allow to use the radial velocity vector to determine the direction of energy flows during the
propagation of acoustic waves in hypoelastic media with initial stresses.