Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod

In this paper, we study nonlinear axisymmetric waves in a circular cylindrical rod composed of a compressible Mooney-Rivlin material. The aim is to derive simplified model equations in the far field which include both nonlinearity and dispersion. We consider disturbances in an initially pre-stressed rod. For long finite-amplitude waves, the Korteweg-de Vries (KdV) equation arises as the model equation. However, in a critical case, the coefficient of the dispersive term in the KdV equation vanishes. As a result, the dispersion cannot balance the nonlinearity. On the other hand, the latter has the tendency to make the wave profile steeper and steeper. The attention is then focused on finite-length and finite-amplitude waves. A new nonlinear dispersive equation which includes extra nonlinear terms involving second-order and third-order derivatives is derived as the model equation. In the case that the rod is composed of a compressible neo-Hookean material, that equation is further reduced to the Benjamin-Bona-Mahony (BBM) equation, which is known as an alternative to the KdV equation for modelling long finite-amplitude waves. To the author's knowledge, it is the first time that the BBM equation is found to arise as a model equation for finite-length and finite-amplitude waves.