Abstract Based on the general theory of the non-equilibrium statistical mechanics, we have discussed the influences of waves on the relaxation processes in macroscopic systems, especially. about the non-exponential asymptotic behaviour of the correlation functions.Because, how to select the representation is important for discussing the dissipative processes in non-equilibrium statistical mechanics, it is desirable to start from the elementary excitation representation. Using the Basic idea of C* algebra theory, we have developed a method for operator expansion, and employed it to express the flow operators in second quantized elementary excitation representation. Taking the residual interaction between the elementary excitations as perturbation in the calculation of the resolvent of the Liouville operator, and using the method of projection operator, a general expression of the non-exponential decaying part of the correlation functions have been obtained.By analysing the procedure of treating the elementary excitation spectrum, we devided the residual interaction into two parts. For pure dissipative mode, taking into account the linear interaction, the main terms of the non-exponential decaying part(t(-d/2) term) were obtained. Detailed analysis shows that, the non-linear interaction (mode-mode coupling) is very small, and leads to a relatively faster decaying process (t(-d) or so), hence it can be neglected.This article emphasizes that, for the wave processes, the local fluctuation correlation, the particle self-correlation and the transport processes are different in nature. Their relative magnitudes (relative to the corresponding quasi-particle parts) may also be quite different.With these general results, we have analysed briefly the relaxation processes in liquid, and made some discussion about the so called mode-mode coupling theory.