The present paper concerns the propagation of distributional travelling waves in models ruled by the equation u_t+[φ(u)]_x=0$, where φ stands for an entire function. Using the concept of α-solution defined in the setting of a distributional product, it is possible to establish a deeper insight about the propagation of such waves. The set of α-solutions contains all weak solutions for this equation and allows us to understand that, in the nonlinear case, the propagation is not possible for a large class of important profiles (all nonconstant continuous profiles are included). However some profiles that are not continuous functions, others that are not functions, and also others that are not measures may propagate. As a particular case, the characterization of all profiles that, locally, are bounded variation functions, is easily established. A brief survey of ideas and formulas used to compute φ(u) when u is a distribution is included.

The author is greatly indebted to the referee whose questions helped to improve the text and the mathematical results presented. The present research was supported by FCT, UIDB/04561/2020.