axially symmetric vibrations of a liquidfilled poroelastic thin cylinder saturated with two immiscible liquids surrounded by a liquid

This paper studies axially symmetric vibrations of a liquidfilled poroelastic thin cylinder saturated with two immiscible liquids of infinite extent that is surrounded by an inviscid elastic liquid. by considering the stress free boundaries, the frequency equation is obtained. particular case, namely, liquidfilled poroelastic cylinder saturated with single liquid is discussed.  when the wavenumber is large, the frequency equation is reduced to that of rayleightype surface wave at the plane boundary of a poroelastic halfspace. in this case, the asymptotic expressions of bessel functions and modified bessel functions are used. in both general and particular cases, the case of the propagation of rayleigh waves in a poroelastic halfspace is obtained.  the parameter values of columbia fine sandy loam saturated with airwater mixture are used for the numerical evaluation.  in all the cases, phase velocity as a function of  wavenumber is computed and presented graphically. from the numerical results, some inferences are drawn.