Revisiting the Love hypothesis for introducing dispersion of longitudinal waves in elastic rods

Abstract

We re-examine the Love equation, which forms the first historical attempt at improving on the classical wave equation to encompass for dispersion of longitudinal waves in rods. Dispersion is introduced by accounting for lateral inertia through the Love hypothesis. Our aim is to provide a rigorous justification of the Love hypothesis, which may be generalized to other contexts. We show that the procedure by which the Love equation is traditionally derived is misleading: indeed, proper variational dealing of the Love hypothesis in a two-modal kinematics (the Mindlin–Herrmann system) leads to the Bishop–Love equation instead. The latter is not asymptotically equivalent to the Love equation, which is in fact a long wave low frequency approximation of the Pochhammer–Chree solution. However, the Love hypothesis may still be retrieved from the Mindlin–Herrmann system, by a slow-time perturbation process. In so doing, the linear KdV equation is retrieved. Besides, consistent approximation demands that a correction term be added to the classical Love hypothesis. Surprisingly, in the case of isotropic linear elasticity, this correction term produces no effect in the correction term of the Lagrangian, so that, to first order, the same Bishop–Love equation is the Euler–Lagrange equation corresponding to a family of Love-like hypotheses, all being different by the correction term. Besides, ill-posedness coming from non-standard (namely non static) natural boundary conditions is now amended.