Self and cross-mode interactions in a weakly nonlinear two-dimensional structural-acoustic waveguide

In this study, that is entirely semi-analytical, weakly nonlinear wave propagation in an infinite two-dimensional structural-acoustic waveguide is studied. The lower boundary of the waveguide is a rigid wall, while the top boundary is a flexible plate. Both in-plane and transverse displacements are included in the flexible plate model. The acoustic fluid and the plate are modeled using standard nonlinear equations. The waveguide is driven by an oscillating piston at the origin. The regular perturbation method is used to separate the equations at linear and nonlinear order. The self- and cross-mode interactions of the non-orthogonal propagating modes are solved for. It is found that in the self-mode interactions, resonances occur both in pressure and in the in-plane plate displacement at their own different frequencies. Whereas in the cross-mode interactions, there is no in-plane plate resonance and mostly beats (in space) occur except at some frequencies where resonance in pressure is seen. Also, boundary flexibility reduces the number of resonances drastically in comparison to the rigid waveguide. The relevant conditions and closed form solutions are derived for the resonance cases. It is also observed among the nonresonant solutions (or beats) that below the coincidence frequency, the bending mode dominates, and above the coincidence frequency, the acoustic mode dominates.