Wave-particle duality coming from a bead oscillator in an elastic medium, theoretical study and quantum similarities

We introduce a dual wave-particle macroscopic system, where a bead oscillator oscillates in an elastic medium which obeys the Klein-Gordon equation. This theoretical system is mostly inspired by bouncing droplets experiments and bead sliding on a vibrating string experiments. This system is studied using a common and simple mathematical formalism. We compute the motion equation of the bead as well as the wave equation of the system. We introduce the effective velocity of the bead with respect to the elastic medium and the wave ψ, created by the bead, which modulates the natural wave of the medium. Provided some conditions, ψ obeys an equation analogous to the free Schrödinger equation. In the case of linear and spherical cavities, the particle-like characteristics of the bead, expressed with its effective velocity, are proportional to the corresponding wave-like characteristics of the system. This paper is a translation of Dualité onde-corpuscule formée par une masselotte oscillante dans un milieu élastique : étude théorique et similitudes quantiques.