Transverse Waves in Nonlinearly Elastic Solids and the Milne-Pinney (or Ermakov) Equation

We establish a connection between the general equations of nonlinear elastodynamics and the nonlinear ordinary differential equation of Pinney [Proc. Amer. Math. Soc. 1 (1950) 681]. As a starting point, we use the exact travelling wave solutions of nonlinear elasticity discovered by Carroll [Acta Mechanica 3 (1967) 167]. The connection provides a method for finding new exact and approximate dynamic solutions for neo-Hookean and Mooney-Rivlin solids, and for the general third- and fourth-order elasticity models of incompressible solids.