On a Complex Wave Equation Arising in Isotropic Incompressible Elastodynamics

Abstract

In isotropic and incompressible elastodynamics, Carroll’s solutions for plane, circularly polarized shear waves represent an important class of controllable, non-universal solutions. It has already been shown that these solutions can be obtained in a compact way by rewriting the equations governing transverse waves as a single complex differential wave equation. In this work, we show that a complex wave equation formally identical to Carroll’s one can be used to construct new classes of solutions, still in the isotropic and incompressible setting, but in a more general case involving plane wave superimposed to homogeneous deformations. We analyze these new solutions and observe that they also exist in an interesting asymptotic regime frequently encountered in non-linear acoustics. Moreover, we demonstrate that these solutions are closely linked to symmetry properties of the complex wave equation and satisfy a nonlinear universal relation—a noteworthy and rare result.