Theoretical and experimental investigations are described of the linearized wave equation of supersonic flow with a single relaxation process. Firing-range experiments using a known model gas mixture with a single relaxation process are discussed. Qualitatively, it is shown that weak conical and spherical waves decay as predicted by theory. In addition, this decay can be determined quantitatively for conical disturbances; and, satisfactory agreement with theory is found. For application to unknown relaxation processes, it is suggested that observation of the damping of weak waves provides an additional tool for the study of fast rate processes.