We consider finite-amplitude acoustic disturbances propagating through media in which relaxation mechanisms, such as those associated with the vibration of polyatomic molecules, are significant. While the effect of these relaxation modes is to inhibit the wave steepening associated with nonlinearity, whether a particular mode is sufficient to prevent the occurrence of multi-valued solutions will depend on the form of the disturbance and on the characteristic parameters of the relaxation. Analysis of this condition is necessary in order to reveal which physical mechanisms actually determine the evolution of the wave profile. This then dictates the scaling of any embedded shock regions. Sufficient conditions for the occurrence of multi-valued solutions are obtained analytically for periodic waves, hence proving that in certain circumstances relaxation is in fact insufficient in fully describing the wave propagation. A much more precise criterion is then obtained numerically. This uses the techniques described in Part 1 for analysing the phenomenon of wave overturning using intrinsic coordinates. Illustrations are provided of the development of a harmonic signal for different classes of material parameters.