The characteristics of tube waves in a borehole, with and without a solid cylindrical mandrel, which may be either elastic or poro-elastic, are compared. With an elastic mandrel, the tube waves are slower, more dispersive, and more sensitive to the formation shear modulus than without. Similarly extensional mode characteristics are compared in the presence of a formation. In the presence of an elastic formation the extensional mode is faster, more dispersive than without, and it is only slightly sensitive to the formation shear properties. These calculated characteristics are in excellent agreement with our measured data. Additionally, the characteristics of tube waves and extensional modes are studied in a borehole in the presence of a concentric liquid-saturated porous mandrel whose acoustic properties are calculated using the Biot theory. The coupling of the tube wave propagating in the annulus with the slow wave propagating in the porous mandrel introduces attenuation and additional dispersion to the tube waves; the effect on the extensional mode is slight. Theoretically calculated and experimentally measured dispersion and attenuation are in excellent agreement with each other, with no adjustable parameters. 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.