With the advent of large storage and high speed capabilities of digital computers, computations previously considered intractable have become feasible. One area in which this fact is especially relevant is in the numerical modeling of wave propagation. In this paper, a sequel to a previous one [T. J. Cavicchi, ‘‘Transient high‐order ultrasonic scattering,’’ J. Acoust. Soc. Am.88, 1132–1141 (1990)] a time‐domain moment method expansion is, through computer simulations, demonstrated to be capable of solving a temporal forward scattering problem for small, fairly strong cylindrical scatterers. It is, however, very computationally intensive, so detailed studies of the matrix structure are presented. Such studies are useful in optimizing the current method in order to make feasible more practical medical imaging studies.