An Eulerian model for nonlinear waves in elastic rods, solved numerically by the CESE method

In this paper, we use a comprehensive perturbation theory to develop an Eulerian model for nonlinear longitudinal waves in tapered elastic rods. The equations of motion, when cast in their Eulerian form, are ideal for fixed-grid numerical techniques such as finite-difference methods, finite-volume methods, and modern high-resolution shock-capturing schemes. The leading-order equations in our perturbation formalism are verified using a control-volume analysis, then linearized to recover a classical model for longitudinal waves in ultrasonic horns. We assess the mathematical structure of our leading-order model, deduce the eigenvalues and eigenvectors of the Jacobian matrix, and verify its hyperbolicity. Solutions are obtained using the space-time conservation element and solution element (CESE) method, a novel numerical technique for first-order hyperbolic systems. In our simulations, the CESE method is shown to effectively capture the evolution of traveling waves such as shocks and rarefactions, as well as salient features of nonlinearity.