The travelling waves that can develop in a chemical system in which there is an ionic autocatalytic reaction, with a quadratic rate law, are considered when a constant (dimensionless) current I is applied. The equations for the travelling waves are discussed in detail from which it is seen that the structure and the propagation speed of these waves depend on both the magnitude and direction of the current as well as on the parameter δB, the ratio of diffusion coefficients of autocatalyst and substrate. An initial‐value problem, whereby the electric current is switched on after travelling waves in the absence of the electric field have become fully developed, is set up and solved numerically, with particular attention being paid to the long‐time structures which are formed. These are seen to depend crucially on whether δBgreater than 1 or δB less than 1. For δB greater than 1 there is a transition from essentially kinetic front waves to Kohlrausch electrophoretic fronts as I is varied. For δB less than 1, both kinetic waves and Kohlrausch fronts are seen as well as an additional type of wave where a much enhanced reaction takes place.