Laboratory experiments on wave propagation through saturated and partially saturated porous media have often been conducted on porous cylinders that were initially fully saturated and then allowed to dry while continuing to acquire data on the wave behavior. Since it is known that drying typically progresses from the outside to the inside, a sensible physical model of this process is concentric cylinders having different saturation levels--the simplest example being a fully dry outer cylindrical shell together with a fully wet inner cylinder. We use this model to formulate the equations for wave dispersion in porous cylinders for patchy saturation (i.e., drainage) conditions. In addition to multiple modes of propagation obtained numerically from these dispersion relations, we find two distinct analytical expressions for torsional wave modes.