Propagation of vibration in a nonlinear dissipative medium

A simple problem, namely the problem of longitudinal vibration in a homogeneous rod, will be studied in this chapter. Most attention will be given to the analysis of vibration in a semi-infinite rod loaded at one of its ends. A traditional method of nonlinear mechanics, namely a series expansion in terms of normal modes, which was used in Chapters 4 and 6 is not applicable here. The approach used in this chapter is based on the method of propagating waves which overcomes some difficulties caused mainly by the wave heterogeneity due to the energy dissipation in the rod’s material. The problem of the vibration of a very long rod excited by a high frequency load is a related problem. A peculiarity of this problem lies in the fact that distortion of the normal modes due to small dissipative forces cannot be ignored.