In this paper we consider a uniform gas bubble-liquid mixture rising under buoyancy. When the gas volume flux is decreased, while keeping bubble size constant, a smooth transition is formed between the region of lower concentration by volume and the region of initial concentration. This transition travels through the mixture as a permanent wave. We start by discussing the mechanisms which make possible such a permanent wave. The first is its tendency to steepen at the low concentration side. At the root of this is the decrease of the uniform rise velocity, under buoyancy, with increasing concentration. Associated with the motion of the bubbles is the liquid impulse. It is shown that this increases with increasing concentration, producing a reactive force on the bubbles which counteracts buoyancy and reduces the force available to overcome friction. In the transition a balance between these two effects occurs. The internal structure following from this balance is analysed in detail and it is shown that under certain conditions all its properties can be derived from knowledge of the average rise velocities of bubbles in uniform mixtures as a function of concentration. Measurements on these are reported subsequently, followed by a discussion of our experiments on transitions of the kind mentioned in which velocity, thickness etc. of the waves have been measured. The data are compared with the results of the analysis. Order-of-magnitude agreement is found but there are differences as well, requiring further research.