The present communication is devoted to the study of characteristic solutions in the neighbourhood of the leading frozen characteristics in steady hypersonic flows of fluids. The effects of relaxation on the global behaviour of the wave amplitude have been studied. It is concluded that all compressive waves with initial amplitude greater than the critical one will grow and terminate into shock waves due to nonlinear steepenning, while all expansion waves will decay out. A critical stage is also discussed when the compressive wave will either grow or decay.