We show from analytical analysis of the basic stimulated Brillouin scattering (SBS) equations in the time domain that the SBS amplification process does not amplify an external Stokes pulse and therefore cannot induce group delay of the Stokes pulse as claimed in the literature. Rather, the delayed output Stokes pulse is the pump radiation reflected by the induced acoustic wave, the amplitude of which determines the rate of the amplification process and time delay of the pulse. The latter is predominantly a consequence of the SBS buildup process determined by the inertia of the acoustic wave excitation. Analytical solutions of the SBS equations in the frequency domain show that spectral broadening of the pump radiation leads to only negligible broadening of the SBS spectral bandwidth and so does not provide an effective means to achieve broadband pulse delay.