The ringing of resonances in acoustic scattering was experimentally observed by Faran in selected cases. In our resonance scattering theory [G. C. Gaunaurd and H. Überall, J. Appl. Phys. 50, 4642 (1979)], this is obtained simply as the Fourier transform of a pole term in the scattering amplitudes, with poles located in the complex-frequency plane corresponding to the eigenfrequencies of the target system, as described in the radar case by the “Singularity Expansion Method” (SEM). These SEM poles are found to occur in distinct “layers,” and each such layer corresponds to the manifestations of a given circumferential wave, as obtained by us previously from a Watson transformation. We show for impenetrable spheres and cylinders that summing the pole terms over a corresponding layer leads to creeping pulses with arrival times governed by the group velocity, including multiple circumnavigations which constitute the ringing. Phase and group velocity dispersion curves are obtained directly from the SEM poles. [H. Überall is also at Catholic University, Washington, DC, supported by the Office of Naval Research.]