In this paper, an explicit acoustical wave propagator (AWP) is introduced to described the time-domain evolution of acoustical waves. To implement its operation on an initial state of wave motion, the acoustical wave propagator is approximated as a Chebyshev polynomial expansion, which converges to machine accuracy. The spatial gradient in each polynomial term is evaluated by a Fourier transformation scheme. Analysis and numerical examples demonstrated that this Chebyshev-Fourier scheme is highly accurate and computational effective in predicting time-domain acoustical wave propagation and scattering. (C) 2000 Acoustical Society of America.