Acoustic waves are dispersed by two distinct mechanisms as they propagate through a fiber‐reinforced viscoelastic material. The first mechanism, viscoelastic dispersion is characterized by an increase in phase velocity with an increase in frequency and an exponential attenuation of wave amplitude with propagation distance. The second, geometric dispersion may be characterized as a wave‐filtering phenomenon in which periodic waves are selectively transmitted and reflected. Regions of transmission are commonly called “passbands” and regions of reflections have been designated “forbidden bands.” When both mechanisms are present in one material, one or the other may be dominate. In this study, the interplay between these two mechanisms is examined experimentally by determining the dispersion characteristics of two fiber‐reinforced viscoelastic materials (viscoelastic dispersion dominating in one and geometric dispersion dominating in the other).