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.

Weakly nonlinear acoustic wave propagation in a nonlinear orthotropic circular cylindrical waveguide.

Nonlinear acoustic wave propagation is considered in an infinite orthotropic thin circular cylindrical waveguide. The modes are non-planar having small but finite amplitude. The fluid is assumed to be ideal and inviscid with no mean flow. The cylindrical waveguide is modeled using the Donnell's nonlinear theory for thin cylindrical shells. The approximate solutions for the acoustic velocity potential are found using the method of multiple scales (MMS) in space and time. The calculations are presented up to the third order of the small parameter. It is found that at some frequencies the amplitude modulation is governed by the Nonlinear Schrödinger Equation (NLSE). The first objective is to study the nonlinear term in the NLSE, as the sign of the nonlinear term determines the stability of the amplitude modulation. On the other hand, at other specific frequencies, interactions occur between the primary wave and its higher harmonics. Here, the objective is to identify the frequencies of the higher harmonic interactions. Lastly, the linear terms in the NLSE obtained using the MMS calculations are validated. All three objectives are met using an asymptotic analysis of the dispersion equation.

Asymptotic expansions for wavenumbers in orthotropic fluid-filled circular cylindrical shells for intermediate fluid loading.

Coupled wavenumbers in infinite fluid-filled isotropic and orthotropic cylindrical shells are considered. Using the Donnell-Mushtari (DM) theory for thin shells, compact and elegant asymptotic expansions for the wavenumbers are found at an intermediate fluid loading for both the coupled rigid-duct modes ("fluid-originated") and the coupled structural wavenumbers ("structure-originated modes") over the entire frequency range where DM theory is valid. The coupled rigid-duct expansions are found to be valid for O(1) orthotropy and for all circumferential orders, whereas the coupled structural wavenumber expansions are valid for small orthotropy and for low circumferential orders. These two above results are then used to derive the expansions for a set of multiple complex roots that display a locking behavior at this intermediate fluid-loading. The expansions are matched with the numerical solutions of the coupled dispersion relation and the match is found to be good over most of the frequency range.

Asymptotic expansions for the coupled wavenumbers in an infinite orthotropic flexible fluid-filled cylindrical shell.

Analytical expressions are found for the coupled wavenumbers in flexible, fluid-filled, circular cylindrical orthotropic shells using the asymptotic methods. These expressions are valid for arbitrary circumferential orders. The Donnell-Mushtari shell theory is used to model the shell and the effect of the fluid is introduced through the fluid-loading parameter µ. The orthotropic problem is posed as a perturbation on the corresponding isotropic problem by defining a suitable orthotropy parameter ε, which is a measure of the degree of orthotropy. For the first study, an isotropic shell is considered (by setting ε=0) and expansions are found for the coupled wavenumbers using a regular perturbation approach. In the second study, asymptotic expansions are found for the coupled wavenumbers in the limit of small orthotropy (ε≪1). For each study, isotropy and orthotropy, expansions are found for small and large values of the fluid-loading parameter µ. All the asymptotic solutions are compared with numerical solutions to the coupled dispersion relation and the match is seen to be good. The differences between the isotropic and orthotropic solutions are discussed. The main contribution of this work lies in extending the existing literature beyond in vacuo studies to the case of fluid-filled shells (isotropic and orthotropic).