Quadratic and Cubic Reaction–Diffusion Fronts*

Robert Luther’s student, Herr cand. Meinecke, moved a wire loop along the test tube to mark the position of the propagating front, with the image projected onto a screen so the entire audience could see. It was May 21, 1906, at the Main Meeting of the Deutsche Bunsengesellschaft für Angewandte Physikalische Chemie held in Dresden, Germany. Luther, who was Director of the Physical Chemistry Laboratory in Leipzig, read his paper ‘‘Propagation of Chemical Reactions in Space’’ on the discovery and analysis of chemical waves [1,2]. His lecture demonstration showed a propagating front in the permanganate oxidation of oxalic acid, with the purple solution becoming colorless at the moving boundary. He also described fronts in other autocatalytic chemical reactions, such as the bromate oxidation of arsenous acid. Today, many chemical systems are known to exhibit such waves, in which autocatalytic reaction couples with molecular diffusion to give constant waveform, constant velocity fronts [3]. This is the simplest type of chemical wave: Reactants are converted into products as the front propagates through the reaction mixture. Virtually all of the chemical reaction occurs in the thin boundary between the reactant and product zones. The basic mechanism of front propagation is also found in more complex waves, such as those of the spectacular Belousov–