Pochhammer's method of analysis for waves in circular cylinders is extended to Biot's theory for an elastic porous solid saturated with a compressible viscous fluid. Attention is given mainly to slender rods vibrating at low frequencies. The boundary condition of zero stresses on the curved cylindrical surface is used to derive the frequency equation from which phase velocity and attenuation may be obtained. In general, two types of extensional waves exist, just as there are two types of dilatational waves. The attenuation per unit length of the first kind of wave is proportional to the square of the frequency at low frequencies except when the wavelength of dilatational waves of the second kind is about half the perimeter of the cylinder. In the neighborhood of this frequency the attenuation may show a maximum. Waves of the second kind behave like a diffusion phenomenon, except in the same neighborhood.