A major area of study, both experimental, theoretical and computational, within fluid dynamics is wave motion. This includes the study of surface gravity waves, which occur at the free surface between a liquid phase and a gaseous phase, and internal gravity waves which can either be interfacial, occurring at the interface between two fluids of the same phase, or can occur within a stratified fluid where a density gradient is produced by an external influence. Internal waves occur in both liquid and gaseous phases. Numerical studies of wave motion have followed the traditional route, common to most numerical problems, of solving the differential equation describing the motion subject to a set of boundary and initial conditions. A major problem when applying this to gravity waves is that a boundary condition needs to be applied at the interface at which the waves are propagating. The interface can be either the free-surface between a liquid and a gas or an internal interface between two fluids with the same phase. Either way the boundary conditions need to be applied at a boundary which is changing as the simulation develops. This can become even more complicated if wave breaking is being considered when the surface becomes a multi-valued function of the horizontal co-ordinates. This problem does not occur is a lattice gas or a lattice Boltzmann model exhibiting some form of fluid separation. Wave simulations have been performed [7] using a lattice gas model, however they were subject to the problems inherent in all lattice gas model. The results did however suggest that a lattice Boltzmann model, which overcomes the lattice gas problems, could be a useful tool in the study of wave motion.