Waves in the theory of elasticity for triple porosity materials

The elastic model with triple porosity is considered in connection with the propagation of harmonic in time waves with assigned wavelength. It is shown that there are two transverse waves that are not damped over time and are not affected by the dispersive nature of the triple porosity model. As regard the longitudinal harmonic in time wave solutions, a dispersion relation is explicitly established like a fifth degree equation with real coefficients and hence it always allows some real roots. Consequently, the existence of four longitudinal wave solutions subject to the effects of dispersion and damping over time is shown: a longitudinal quasi-elastic wave and three longitudinal quasi-pore modes that decrease exponentially over time. The influence of taking into account the pores on the propagation of the longitudinal elastic waves is materialized in: an increase of the propagation speed corroborated with the appearance of the effect of their damping in time. The propagation of the Rayleigh surface waves is addressed and the corresponding secular equation is explicitly established.