Resonances and beats in nonlinear structural acoustic waveguides

Abstract

In this thesis that is entirely analytical in nature, nonlinear wave propagation in various structural-acoustic waveguide geometries is studied. The waveguides considered include a circular cylindrical waveguide with a rigid wall in one case and a flexible thin wall in the other, and a 2-D rectangular waveguide with one wall being a flexible plate in one case and a flexible membrane in the other. The objective of this thesis is to address the resonance and the beating phenomena generated due to the nonlinearity. The fluid and the structure (whenever flexible) are considered to be weakly nonlinear. The fluid is inviscid with zero mean flow. All the calculations are done till second order.

In each of these waveguides, a piston oscillating mostly at a single frequency is considered. The magnitude of oscillations is such that the nonlinear phenomenon becomes manifest. Due to the piston excitation, several modes start propagating. In the presence of nonlinearity, these propagating waves interact with each other and generate higher order frequencies and wavenumbers (higher harmonics) of the input waves.

Using the regular perturbation method, the nonlinear system is decomposed into two sets of equations: one of them represents the linear system and the other represents the nonlinear order. The linear (primary) order wavenumbers are derived analytically using the asymptotic method. These waves interact at the nonlinear order and create higher order waves. These higher order waves could again interact with the primary order waves and create resonances or beats. Self-mode and cross-mode interactions at single and multiple frequencies (in a single case) have also been considered.

In all cases, closed form solutions for the nonlinear equations are obtained. Mostly, the higher harmonic nonlinear solutions are found to be of the nature of beating. The parameters affecting the beating length and the beating amplitude are presented. Occasionally, under certain conditions, the nonlinear solution becomes resonant. The solutions and the conditions for the resonance are derived in the thesis. Also, the relation between the group/phase speed of the primary interacting waves and the generated higher harmonic wave for the resonance condition is obtained.

Lastly, the effect of the fluid loading parameter µ (or the structural flexibility) is presented. As µ increases, i.e., the boundary of the waveguide becomes softer, the occurrence of resonances reduces. Also, as µ increases, the amplitude of the beating/non-resonant solutions reduces. This holds good for all the waveguides considered in this work.