Using the finite-difference time-domain method, we theoretically study elastic waves propagating in a two-dimensional composite consisting of aluminum cylinders embedded in mercury. The importance of the transverse components of the aluminum scatterers in the creation of the gaps is shown by switching off the transverse waves inside the aluminum. In that case, there are very narrow gaps or no gaps at all. In contrast, when the transverse components are on, as they should be, wide full band gaps appear for longitudinal incident waves. Since mercury supports only longitudinal waves, these are full elastic band gaps. © 2000 American Institute of Physics.