Shock Waves

The aim of this book is to present the fundamentals of shock wave theory.  In most respects this is a subject within the scope of partial differential equations, but this text goes well beyond the standard theory. Stokes and Riemann were pioneers in this area. Their efforts, as with much of the succeeding work, were driven by analysis of physical systems. The earliest work was on gas dynamics, and Stokes was the first to show that singularities can arise even with smooth initial data. Since that time shock waves have been found in many different kinds of physical environments. 

 

The overall focus of the book is to analyze shock waves that arise from partial differential equations that represent conservation laws. For a simple example, consider the partial differential equation \( u_{t}=\phi_{x}+\lambda(u) u_{x} \), the transport equation represents a physical conservation law (like conservation of mass or energy) and makes the assumption that the solution \( u \) has continuous first derivatives. The method of characteristics can be used to construct such a solution, but only up to the point where characteristic lines converge (when \( \lambda_{x} < 0\) ) and then intersect. If that happens, a singularity develops. Such a solution can be extended beyond the singularity by permitting \( u(x, t) \) to be a piecewise smooth function. To do this, one must return to the original integral form of the conservation law at points \( (x, t) \) where \( u(x, t) \) is discontinuous. The discontinuous solution of the conservation law that results is called a shock wave.

 

Shock waves arise in many circumstances. For example, diffusion of a pollutant in water, the flow of automobile traffic on a single-lane road, and compression of acoustic waves in gas flow can lead to shock wave solutions.  

 

The book has three parts. The first establishes the basic ideas of shock wave theory for conservation laws in one spatial dimension. The second part develops the general theory for systems of hyperbolic conservation laws.  Then the author returns to the original motivation of shock wave theory by focusing on specific physical models. Here he considers questions like the effects of dissipation in gas dynamics, and applies his analysis first to Burgers equation. In this section he also explores viscous conservation laws for nonlinear waves in dissipative systems, as well as relaxation models and non-linear resonance phenomena.

 

This is an advanced text. Although the author suggests that it is accessible to upper level undergraduates, the reader would need a very strong background in partial differential equations. It is a more appropriate book for advanced graduate students or specialists. Plenty of exercises are provided, and there is an extensive bibliography.

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Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.