Shock waves and absorption of general nonlinear progressive waves

Abstract

Many physical phenomena are concerned with the propagation of weak nonlinear waves that can be modeled under the form of a generalized Burgers equation. Nonlinearities can be quadratic (nonlinear waves in fluids), cubic (nonlinear shear waves in soft solids), or non‐polynomial (Buckley‐Leverett equation for diphasic fluids, models for car traffic). Thermoviscous absorption in fluids is characterized by a quadratic dependence with frequency, but other types of absorption are encountered in media such as soft media (biological tissues), relaxing media (air at audible frequencies), and suspensions. In these last two cases, dispersion is also associated with absorption. For the inviscid case, a new weak shock formulation of the generalized Burgers equation using the potential is developed. The formulation is a generalization to non‐quadratic nonlinearities of the method originally proposed by Burgers himself in 1954 for his own equation, and later applied to sonic boom applications by Hayes et al. (1969). It ...