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An immiscible, binary fluid lattice Boltzmann model is described and its associated equations of motion are given. It is seen that the lattice Boltzmann scheme is totally isotropic and that it does not suffer from the problems of noisy results and a lack of Galilean invariance which plagued its predecessor: the lattice gas model. The incorporation of a body force into the lattice Boltzmann technique is considered. A method which introduces the body force directly into the governing equation is proposed and is seen to have the desired effect without destroying the Galilean invariance of the original model and without introducing any dependency on the grid orientation. The immiscible, binary fluid models, with the body force incorporated, are applied to simulate interfacial waves between the two fluids. The model parameters allow the interface thickness, the fluid viscosity, the gravitational strength and the relative density of the two fluids to be varied. The wavelength of the wave can also be set during the wave initialisation. Standing waves are simulated for a wide range of the variable parameters and progressive waves for a subset of the parameters. The results are seen to compare well with linear wave theory. When compared with available experimental results the behaviour is seen to be similar.