A qualitative analysis of the mechanical response caused by smooth attenuating wave fronts and by continuous attenuating wave fronts degrading in rate‐dependent media is performed by an underdetermined system of equations, that is, by equations of conservation of mass and momentum only. Degradation is defined as the decrease of the number of partial waves in a smooth wave front or of independent waves in a continuous wave front. For the smooth wave front the inceptive wave front with which the degradation commences is a wave that is composed of four partial waves found previously and named a contraction wave, mesocontraction waves I and II, and a rear rarefaction wave. The analysis reveals one possible degradation process that is caused by another rarefaction wave which first follows the rear wave and then outruns the rear wave, mesocontraction waves II and I in sequence. The continuous wave degrades from the inceptive continuous wave front which is composed of four independent waves found previously and named a contraction wave, degenerate contraction waves I and II, and a rear rarefaction wave. The analysis indicates that degradation from the inceptive wave front to the contraction wave is possible to progress in six different patterns. In short, the degradation can occur through six different processes. Each process gives rise to four different sets of strain, particle velocity, and stress profiles, and accordingly four different types of stress‐strain paths and four different types of stress‐particle velocity paths. The stress‐strain paths induced during some of the processes are similar to the paths in sandstone computed by Seaman using the Lagrangian analysis. The stress‐particle velocity paths induced during some of the processes are also similar to the paths in LASL graphite computed by Seaman.