Contributions to the theory of longitudinal vibrations and wave propagation in rods and tubes: I. A mathematical model for linear elastic and hereditary elastic materials

After a brief historical survey of some work done on the linear theory of longitudinal vibrations and wave propagation in rods and tubes of uniform cross-section, a simple mathematical model for rods and tubes of linear elastic materials is proposed. Three suitably selected propagation modes (one extensional and two shear modes) with dispersion relations corresponding to mixed boundary conditions are coupled in order to approximately comply with zero-stress boundary conditions. The coupling gives a set of partial differential equations in the mode amplitudes, with time and a single space coordinate (along the axis of symmetry of the rod or tube) as independent variables. Then, the model is generalized to a set of partial integral-differential equations in order to be able to describe vibrations and wave propagation in rods and tubes made of linear hereditary-elastic solids. In this first part of the work, the focus is in either very low frequency or very high frequency phenomena using a simple model with only two coupled modes. The model allows a fairly elegant and comparatively powerful analytical approach to longitudinal vibrations and to longitudinal pulse propagation in solid waveguides. Analytical formulae for group velocities are derived, as well as asymptotic expressions for the propagation of mode amplitudes. The limitations and pitfalls of the model are assessed, and new experiments and digital simulations are suggested to test some of its predictions, wave propagation; elastic and hereditary-elastic materials; propagation modes in rods and tubes