The authors investigate the stability of interfacial waves in conducting fluids under the influence of a vertical current density. A wave equation for the interface in aluminium reduction cells is derived from the shallow water theory, in which the Lorentz force is expressed explicitly in terms of the fluid motion. It is shown that travelling waves may become unstable in the presence of a uniform vertical magnetic field, and standing waves may also become unstable. The authors derive a simple energy criterion and give a sufficient condition for stability of standing waves in a finite domain.