Negative flow of energy in a mechanical wave

TL;DR

This paper demonstrates a classical mechanical system where energy flows backward against the overall direction due to dispersion and interference, analogous to quantum probability backflow, with measurable effects confirmed through theoretical analysis.

Contribution

It introduces a classical mass-spring system exhibiting energy backflow, linking dispersion, interference, and local modes to quantum-like probability backflow phenomena.

Findings

Backflow of energy occurs in a simple mass-spring chain.

Energy can flow back to the external force during excitation.

Backflow magnitude exceeds quantum probability backflow by an order of magnitude.

Abstract

A classical system, which is analogous to the quantum one with a backflow of probability, is proposed. The system consists of a chain of masses interconnected by springs, as well attached by other springs to fixed supports. Thanks to the last springs the cutoff frequency and dispersion appears in the spectrum of waves propagating along the chain. It is shown that this dispersion contributes to the appearance of a backflow of energy. In the case of the interference of the two waves, the magnitude of this backflow is an order of magnitude higher than the value of the probability backflow in the mentioned quantum problem. The equation of Green's function is considered, and it is shown that the backflow of energy is also possible when the system is excited by two consecutive short pulses. This classical backflow phenomenon is explained by the branching of energy flow to local modes, what is confirmed by the results for the forced damped oscillator. It is shown that even in such a simple system the backflow of energy takes place (both an instantaneous and on average) and the energy comes back to external force.