Certain aspects of the acoustics of a strongly nonlinear discrete lattice

Abstract

The aim of this paper is to analytically investigate certain aspects of the nonlinear acoustics, especially traveling breathers, of a one-dimensional, non-dissipative, essentially nonlinear lattice. The considered semi-infinite lattice is formed by coupling linearly grounded single-degree-of-freedom oscillators by means of essentially nonlinear stiffnesses with purely cubic force–displacement characteristics and is subject to an impulsive excitation at its free boundary. In the limit of small energy, the complexification-averaging method is employed to analytically prove the existence of a nonlinear, energy-dependent propagation zone (PZ) and two complementary attenuation zones (AZs) in frequency–energy domain of the corresponding infinite lattice (i.e., the one without a free boundary). Excitations within the PZ yield waves that propagate in the far field of the lattice, whereas excitations within either one of the AZs generate exponentially decaying, spatially localized, standing wave close to the excitation site. Therefore, unlike in the linear case nonlinear lattices possess energy-tunable filtering properties. After determining the region in the frequency–energy domain where wave (and energy) propagation within the lattice is allowed, the excitation of traveling breathers in the semi-infinite lattice is studied. It is analytically and numerically shown that such breathers can exist only close to, but outside of the upper boundary of the PZ of the lattice in frequency–energy domain. Moreover, due to the localized nature of the traveling breather, each oscillator experiences a continuous and smooth change in its total energy as the breather passes. It is analytically shown and explained that this smooth energy transition results in a similar form of transition in the instantaneous frequency of each oscillator.