Appropriate boundary conditions for use with Biot's equations have been considered by Deresiewicz and Skalak (1963), Berryman and Thigpen (1985), and Pride and Haartsen (1996) and we make use of these results here.

At the external surface *r*=*R _{2}* where the outer porous material
contacts the surrounding air, it is appropriate to use the free
surface conditions

(32) |

The internal interface at *r*=*R _{1}* needs more precise definition.
We assume that all the meniscii that are separating the inner fluid from
the outer fluid are contained within a thin layer (shell) of thickness
(a few grain sizes in width) straddling the surface

(33) |

(34) |

(35) |

The rate at which energy fluxes radially through the porous material
is given by with implicit
summation over the index *i*. The difference in the rate at which
energy is entering and leaving the interface layer is due to
work performed in stretching the meniscii.
Each meniscus
has an initial mean curvature *H*_{o} that is determined by
the initial fluid pressures (those that hold before the wave arrives)
as where is the surface tension.
As the wave passes, the ratio between the actual mean curvature
*H* and *H*_{o} is a small quantity on the order of the capillary number
[see Pride and Flekkoy (1999)]
where is some estimate of
the induced Darcy flow and that goes as wave strain times wave velocity ( m/s).
Since Pam for air-water interfaces,
we have , which can be considered
negligible. By integrating the energy flux rate over a Gaussian shell that
straddles *r*=*R _{1}*, it is straightforward to obtain

(36) |

To apply the boundary conditions (45) and (49), we need in addition to (48) the result

(37) |

(38) |

To apply the boundary conditions (63), we need the explicit expressions for the displacement which follow from (23). The results are of the form

(39) |

(40) |

(41) |

Both (ur) and (uz) are needed for extensional waves, while the remaining component,

(43) |

It follows from (46)-(49), (65), and (utheta) that (for the inner cylinder) and the corresponding coefficients for the cylindrical shell are all completely independent of the other mode coefficients and, therefore, relevant to the study of torsional waves, but not for extensional waves. Pertinent equations for the torsional wave dispersion relation are continuity of the angular displacement, , and stress, , at the internal interface, and vanishing of the stress, , at the external surface.

The final set of equations for the extensional wave dispersion
relation involves nine equations with nine unknowns. The nine unknowns
are: , , (coefficients of *J _{0}* in the
central cylinder), plus three 's
(coefficients of

11/11/2002