Nanoptera in nonlinear woodpile chains with zero precompression

TL;DR

This paper investigates travelling waves in nonlinear woodpile chains with zero precompression, revealing they are typically nanoptera with exponentially small oscillatory tails, and explores conditions under which these tails cancel or persist as precompression varies.

Contribution

It introduces a hybrid numerical-analytic method to approximate solutions and explicitly calculates tail oscillations, showing how precompression affects wave behavior and tail cancellation.

Findings

Travelling waves are nanoptera with exponentially small tails.

Tail oscillations can cancel, producing solitary waves.

Increasing precompression causes tail cancellation to cease.

Abstract

We use exponential asymptotics to study travelling waves in woodpile systems modelled as singularly perturbed granular chains with zero precompression and small mass ratio. These systems are strongly nonlinear, and there is no analytic expression for their leading-order solution. We instead obtain an approximated leading-order solution using a hybrid numerical-analytic method. We show that travelling waves in these nonlinear woodpile systems are typically "nanoptera", or travelling waves with exponentially small but non-decaying oscillatory tails which appear as a Stokes curve is crossed. We demonstrate that travelling wave solutions in the zero precompression regime contain two Stokes curves, and hence two sets of tailing oscillations in the solution. We calculate the behaviour of these oscillations explicitly, and show that there exist system configurations which cause the oscillations to cancel entirely, producing solitary wave behaviour. We then study the behaviour of travelling waves in woodpile chains as precompression is increased, and show that there exists a value of the precompression above which the two Stokes curves coalesce into a single curve, meaning that cancellation of the tailing oscillations no longer occurs. This is consistent with previous studies, which showed that cancellation does not occur in chains with strong precompression.