Methods are provided for estimating a quality of cement in the annuli of a multi-string wellbore. Wideband acoustic energy signals are generated and detected in the wellbore and are processed to obtain indications of wideband casing-formation phase slowness dispersions in the wellbore. The indications are compared to reference wideband model casing-formation phase slowness dispersions in order to estimate status of cement or lack of cement in the annuli at that location based on the results of the comparison.