The classic work of Pochhammer (1876) and Chree (1886) gave exact solutions for wave propagation in elastic rods. When the rod is instead a porous cylinder with fluid-filled pores, the equations of linear elasticity do not describe all possible motions of the fluid/porous-solid mixture. Biot's theory of fluid-saturated porous media provides a continuum theory, permitting the fluid and solid components to move independently and accounts approximately for the attenuation of waves due to viscous friction. Gardner (1962) used Biot's theory (Biot, 1956a,b) to study long-wavelength extensional waves in circular cylinders. Gardner considered only the low-frequency regime where the second bulk compressional mode predicted by Biot's theory is diffusive in character. Gardner also limited consideration to the case of open-pore surface boundary conditions.
The present work is based in part on another paper by Berryman (1983), in which both open-pore and closed-pore surface boundary conditions for the fluid-saturated porous cylinder were studied. Here we consider only the open-pore surface, but we allow non-uniform or patchy saturation (Berryman, 1988; Knight and Nolen-Hoeksema, 1990; Knight et al., 1998; Johnson, 2001) inside the cylinder. In particular, it is quite common to study partial saturation in the laboratory under drainage or drying conditions wherein an initially fully saturated porous cylinder is allowed to dry while continuing to acquire data on the cylinder's modes of oscillation. We want to model this behavior explicitly. The simplest such model is concentric cylinders with a fully dry outer cylindrical shell enclosing a fully liquid-saturated inner cylinder. A more realistic model would involve many layers with various degrees of partial saturation between the dry outer shell and the saturated inner cylinder, but such complications will not be treated here. We find that studies of the two-layer case have all the important physical complications expected in this problem, while still having enough simplicity that some of the analysis can be done semi-analytically -- thereby providing soughtafter insight into the problem.
We present the equations of poroelasticity, and then show the forms of the equations needed for cylindrical geometry. Appropriate boundary conditions for our problem are discussed. Equations are subsequently formulated to determine both the extensional and torsional modes of concentric poroelastic cylinders under conditions of partial saturation. Solutions of these equations are computed and discussed here for torsional waves, while the harder problem of extensional waves will be treated fully in a later publication.