When considering nonlinear wave propagation through a medium, it is important to consider which linear attenuation mechanisms significantly affect wave profiles. Specifically, it must be asked whether a particular physical attenuation is sufficient to prevent the overturning of finite‐amplitude waves. Thermoviscous dissipation, however small, will always prevent such overturning, but relaxation effects alone do not necessarily ensure single‐valued solutions. In this paper, the attenuation effects of a relaxing medium are investigated. The model equation considered contains two parameters [D. G. Crighton, Ann. Rev. Fluid Mech.11, 11–33 (1979)] related to a characteristic relaxation time and to the difference between equilibrium and frozen sound speeds. A method using intrinsic coordinates is described as a means of investigating the phenomenon of overturning. This is then applied to the case of a relaxing medium. In this way, the set of parameter values is divided into those for which single‐valued solutions are obtained, and those for which overturning occurs. The latter case corresponds to the physical situation of normally insignificant thermoviscous dissipation becoming locally as important as the relaxation effects.