The results of the linear theory for the flow of a supersonic relaxing gas past a slender body of revolution are analysed in regions where its predictions of wavelet position begin to break down. In this way new variable systems can be found which make it possible to discuss the correct nonlinear wave behaviour far from the body. The situation depends upon three especially important parameters, namely the thickness ratio ε of the body, the ratio δ of relaxing-mode energy to thermal energy and the ratio λ of a relaxation length to a typical body length. After establishing general results from the linear theory, the conical body is treated in some detail. This makes it possible to demote λ as an important parameter, although its restoration does prove useful at one point in the analysis, and results are derived for shock-wave behaviour when ord 1 [ges ] δ > ord ε4, δ = ord ε4 and δ < ord ε4. In the first range of δ fully dispersed waves are essential, although they are fully established only at great distances from the cone; in the second range of δ partly dispersed waves seem to be the most likely to appear, and in the third range relaxation effects are second-order modifications of a basically frozen-flow field. Practical situations may well fall into the first of these categories.