Seismic surveys are often conducted using dynamite charges buried near the surface in unconsolidated material. In such material a large zone near the source should exist wherein nonlinear anelastic wave propagation, can be expected to take place, and have a significant impact on the way in which a seismic pulse forms and how its energy gets distributed into the surrounding medium. To obtain a solution for a propagating pulse in this zone, the equations of motion for nonlinear anelastic wave propagation, good to second order in the displacements, are solved numerically for the problem of a Gaussian pressure pulse acting on the interior cavity of a cylindrically symmetric hole in the medium. An implicit finite-difference algorithm is used for the solution to the equations of motion for this problem. The anelastic medium is characterized by multivalued stress-strain relations that exhibit hysteresis, and therefore a loss of energy per cycle, corresponding to a medium with a constant Q factor.Several numerical examples are calculated contrasting the nonlinear anelastic, linear anelastic, and linear elastic propagating pulses to one another. The nonlinear anelastic propagating pulse is found to have an amplitude that is several times larger than would be expected for a pulse in a linear medium and has a peak propagation velocity that is slightly less than that for a linear pulse. Dispersive effects are also evident for the nonlinear pulse.