The absorptive full fluid elastic scalar wave equation is developed by deriving the full fluid elastic scalar wave equation from the first principles and adding an absorptive term to the result. The time domain solution to the initial value forward scattering problem is developed for this equation. The solution algorithm is an exact numerical solution and is useful in pressure wave modeling for media with spatially varying velocity, density, and attenuation profiles. The number of operations per time step is of the order ofN log2 N, whereNis the number of spatial points into which the model has been discretized.