The essence of Biot waves in an oscillator with two degrees of freedom

Abstract

In poroelastic media, i.e., porous structures whose pores contain fluid, a kind of waves can be observed that does not occur in elastic media, the so-called slow P-waves or Biot waves, which may be perceived as opaque when first encountered. In this paper, we pursue two goals: firstly, we want to provide a simple explanatory model of these waves and, secondly, we want to prepare the reader for Biot’s seminal paper. We discretize a finite poroelastic waveguide by Galerkin’s method to arrive at a mechanical system with 2 degrees of freedom and solve the eigenvalue problem of free oscillations. This oscillator representation (ODE) is simpler than the wave representation (PDE) while maintaining salient features of poroelastodynamics and offering a different perspective. In fact, an oscillation is a standing wave with wave velocity and wave length being related to frequency and domain length. In this reduced model, slow P-waves, when they exist, correspond to an oscillation with large phase shift and fast P-waves to an oscillation with small phase shift.

The intended audience are engineering or physics graduate students with basic knowledge of linear oscillations, linear differential equations and some understanding of biphasic media.