Spontaneous pattern formation in an acoustical resonator

A dynamical system of equations describing parametric sound generation (PSG) in a dispersive large aspect ratio resonator is derived. The model generalizes previously proposed descriptions of PSG by including diffraction effects, and is analogous to the model used in theoretical studies of optical parametric oscillation. A linear stability analysis of the solution below the threshold of subharmonic generation reveals the existence of a pattern forming instability, which is confirmed by numerical integration. The conditions of emergence of periodic patterns in transverse space are discussed in the acoustical context.