Analytical Investigations on Linear And Nonlinear Wave Propagation in Structural-acoustic Waveguides
Vijay Prakash, S
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This thesis has two parts: In the first part, we study the dispersion characteristics of structural-acoustic waveguides by obtaining closed-form solutions for the coupled wave numbers. Two representative systems are considered for the above study: an infinite two-dimensional rectangular waveguide and an infinite fluid- filled orthotropic circular cylindrical shell. In the second part, these asymptotic expressions are used to study the nonlinear wave propagation in the same two systems. The first part involves obtaining asymptotic expansions for the fluid-structure coupled wave numbers in both the systems. Certain expansions are already available in the literature. Hence, the gaps in the literature are filled. Thus, for cylindrical shells even in vacuo wavenumbers are obtained as part of the objective. Here, singular and regular perturbation methods are used by taking the thickness parameter as the asymptotic parameter. Valid wavenumber expressions are obtained at all the frequencies. A transition in the behavior of the flexural wavenumbers occurs in the neighborhood of the ring frequency. This frequency of transition is identified for the orthotropic shells also. The closed-form expressions for the orthotropic shells are obtained in the limit of slight orthotropy for the circumferential orders n > 0 at all the frequency ranges. Following this, we derive the coupled wavenumber expressions for the two systems for an arbitrary fluid loading. Here, the two-dimensional rectangular waveguide is considered first. This rectangular waveguide has a one-dimensional plate and a rigid surface as its lateral boundaries. The effects due to the structural boundary are studied by analyzing the phase change due to the structure on an incident plane wave. The complications due to the cross-sectional modes are eliminated by ignoring the presence of the other rigid boundary. Dispersion characteristics are predicted at various regions of the dispersion diagram based on the phase change. Moreover, the also identified. Next, the rigid boundary is considered and the coupled dispersion relation for the waveguide is solved for the wavenumber expressions. The coupled wavenumbers are obtained as the coupled rigid-duct, the coupled structural and the coupled pressure-release wavenumbers. Next, based on the above asymptotic analysis on a two-dimensional rectangular waveguide, the asymptotic expansions are obtained for the coupled wavenumbers in isotropic and orthotropic fluid- filled cylindrical shells. The asymptotic expansions of the wavenumbers are obtained without any restriction on the fluid loading. They are compared with the numerical solutions and a good match is obtained. In the second part or the nonlinear section of the thesis, the coupled wavenumber expressions are used to study the propagation of small but a finite amplitude acoustic potential in the above structural-acoustic waveguides. It must be mentioned here that for the rst time in the literature, for a structural-acoustic system having a contained fluid, both the structure and the acoustic fluid are nonlinear. Standard nonlinear equations are used. The focus is restricted to non-planar modes. The study of the cylindrical shell parallels that of the 2-D rectangular waveguide, except in that the former is more practical and complicated due to the curvature. Thus, with regard to both systems, a narrow-band wavepacket of the acoustic potential centered around a frequency is considered. The approximate solution of the acoustic velocity potential is found using the method of multiple scales (MMS) involving both space and time. The calculations are presented up to the third order of the small parameter. It is found that the amplitude modulation is governed by the Nonlinear Schr•odinger equation (NLSE). The nonlinear term in the NLSE is analyzed, since the sign of the nonlinear term in the NLSE plays a role in determining the stability of the amplitude modulation. This sign change is predicted using the coupled wavenumber expressions. Secondly, at specific frequencies, the primary pulse interacts with its higher harmonics, as do two or more primary pulses with their resultant higher harmonic. This happens when the phase speeds of the waves match. The frequencies of such interactions are identified, again using the coupled wavenumber expressions. The novelty of this work lies firstly in considering nonlinear acoustic wave prop-agation in nonlinear structural waveguides. Secondly, in deriving the asymptotic expansions for the coupled wavenumbers for both the two-dimensional rectangular waveguide and the fluid- filled circular cylindrical shell. Then in using the same to study the behavior of the nonlinear term in NLSE. And lastly in identifying the frequencies of nonlinear interactions in the respective waveguides.
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