Asymptotically approximate model equations for weakly nonlinear long waves in compressible elastic rods and their comparisons with other simplified model equations

Weakly nonlinear long waves in a cylindrical elastic rod composed of a compressible Murnaghan material are studied in this paper. Four types of model equations describing nonlinear dispersive waves are derived, two of which can satisfy the lateral boundary conditions asymptotically. The dispersion relations for these types of model equations are obtained and compared with the exact dispersion relation from the three-dimensional elasticity. Solitary wave solutions of the far-field equations of these models are shown. By comparison of these four types of model, some conclusions about their advantages and disadvantages are drawn. © 2004 Sage Publications