Seeded, transient, stimulated Brillouin scattering (SBS) is studied using a new noniterative, numerical algorithm to solve this two‐point boundary value problem. The coupled pump, Stokes, and phonon equations are solved numerically, including the effects of pump depletion and a finite phonon lifetime. The noniterative approach leads to efficient computation (<30 s of cpu time, VAX 8000 series computer for 500 by 5000 points in space and time, respectively). The numerical study predicts transient oscillations in the pump and the Stokes intensities. The frequency of these oscillations depends on the phonon lifetime and the depleted, exponential SBS gain. This work is the first known prediction of these oscillations.