9 - Chemical and Electrochemical Waves

Summary

Chemical waves arise when reaction and diffusion occur simultaneously. We shall see that chemical reaction is represented in the governing equations by nonlinear source terms. Molecular diffusion tends to smear out chemical concentrations, whilst suitable chemical reactions tend to sharpen them. We will concentrate here on chemical reactions occurring in solution, which are not strongly exothermic. We have already seen in subsection 7.2.5 that exothermic reactions can generate detonation waves.

We begin by examining the effects of molecular diffusion and chemical reaction in isolation. Even without molecular diffusion, many chemical reactions exhibit surprising nonlinear behaviour, including limit cycles and chaos. The most famous of these is the Belousov–Zhabotinskii reaction or BZ reaction. When the reactants are mixed in a beaker, the colour of the solution changes repeatedly from red to blue with a period of the order of minutes. We then consider the mathematics of diffusion, before combining these two processes to obtain systems of reaction–diffusion equations. Travelling wave solutions of this type of equation describe the propagation of many chemical waves, and form the basis for the analysis of more geometrically complex chemical waves. From these travelling wave solutions we can determine the speed and stability of the waves. A typical chemical wave is shown in figure 9.1. We also consider an example of the use of the method of matched asymptotic expansions to construct an approximate solution. Finally, we consider the propagation of waves in the nervous system.