An equivalent physical model of three acoustic waves interaction in nonlinear medium

Abstract

Among the parametric effects, three waves interaction has been widely investigated in many fields of physics, such as nonlinear optics and plasma. The interaction of three acoustic waves in the medium with quadratic nonlinearity, which must satisfy conditions on their frequencies and wavenumbers, is essentially confined to energy exchange between three components. Based on Burgers equation, the propagation rules of these three acoustic waves which are called a resonant triad, are governed by a drastically simplified system of three coupled nonlinear ordinary differential equations. A mathematical equivalence between the equations for an acoustic triad and a simple parametric vibration system, the undamped elastic pendulum, is discussed in this paper by a multiple time-scale analysis. We study the dynamics of this system, drawing analogies between its behavior and that of the acoustic triad. Finally, it is certified that three acoustic waves interaction can be described by Mathieu type equations in case one acoustic wave is much stronger than the others in three waves. This means that experiments with an elastic pendulum can give us new insights into dynamics of mechanism in three acoustic waves interaction.