The characteristics of tube waves in water-filled boreholes with and without an elastic cylindrical mandrel in the borehole are compared. With a mandrel, the tube waves are slower, more dispersive, and more sensitive to the formation shear modulus than without. Additionally, the characteristics of tube waves in a borehole in the presence of a liquid-saturated porous mandrel were also studied. The tube waves propagating in the annulus couple with the slow waves propagating in the porous mandrel. This coupling and the viscous dissipation of the slow waves introduce attenuation and additional dispersion to the tube waves. Whether the mandrel is porous or not, there are two modes which are weakly dispersive in the quasistatic limit. Very roughly, the two modes consist of an extensional mode in the rod and a fluid-based mode in the annulus which are coupled together.