The change in the thickness of an interface between two immiscible fluids due to the propagation of an internal capillary-gravity wave along the interface is considered using a Bhatnagar, Gross and Krook (BGK) lattice Boltzmann model of a binary of fluid. The vertical thickness of the interface is recorded from the simulations since this is the most easily measured quantities in any simulation or experiment. The vertical thickness is then related to the actual thickness (perpendicular to the interface) which is seen to vary with the phase of the wave. The positions of the maxima and minimum thicknesses are seen to be approximately constant relative to the phase of the propagating wave and the range of variation of the thickness decreases at approximately the same rate as the wave amplitude is damped. A simplified model for the interface is considered which predicts a similar variation due to the interface being stretched as the internal wave propagates.