Wave reflection at the interface between a nonlinear discrete lattice and a matching linear system

This contribution considers the wave reflection at the interface between a discrete lattice that describes the nonlinear near-field response to a dynamic load and a matching system that describes the linear far-field response. The near-field system is modelled as a one-dimensional discrete system of particles that is capable of describing nonlinear behaviour by including dry friction. The purpose of coupling this nonlinear near-field system to a linear far-field system is to optimize for computational efficiency as, in the time domain, the linear far-field system can be replaced by a single integral force-displacement relation. In the frequency domain, this relation is commonly known as the dynamic stiffness, or inversely, as the dynamic compliance. For an ideal coupling between the two systems an incident wave should propagate through the interface undisturbed and without reflections. For long incident waves, it would suffice to describe the far-field system by a classical continuum, however for short incident waves, the reflections of the classical continuum are significant. Therefore, the linear far-field system is described as a semi-infinite discrete particle system, also known as a semi-infinite cascade, that matches the discrete nature of the near-field lattice. To assess the quality of the applied coupling, we consider the reflection of an incident wave at the lattice-cascade interface, using both time and frequency domain approaches, and compare it to the reflection at the interface of a system where the far-field system is described by a semi-infinite continuum.