Mathematical Models for the Propagation of Stress Waves in Elastic Rods: Exact Solutions and Numerical Simulation (Advances in Applied Mathematics and Mechanics, 2018)
In this work, the Bishop and Love models for longitudinal vibrations areadopted to study the dynamics of isotropic rods with conical and exponential cross-sections.Exact solutions of both models are derived, using appropriate transformations.The analytical solutions of these two models are obtained in terms of generalisedhypergeometric functions and Legendre spherical functions respectively. Theexact solution of Love model for a rod with exponential cross-section is expressed as asum of Gauss hypergeometric functions. The models are solved numerically by usingthe method of lines to reduce the original PDE to a system of ODEs. The accuracy ofthe numerical approximations is studied in the case of special solutions.