Analysis of Acoustic Problems with Different Boundary Conditions and Step Discontinuities
The present thesis discusses a class of physical problems that contains the propagation and attenuation of fluid structure coupled waves through discontinuous flexible waveguides which support structure-borne as well as fluid-borne vibrations. The modelled configurations of flexible waveguides comprise thin elastic elements such as elastic membranes and/or plates joined to the structural discontinuities with or without flanges. The associated boundary value problems are governed by Helmholtz’s equation and have Dirichlet, Neumann, Robin and/or higher order boundary conditions. The Mode-Matching (MM) scheme is used to solve the governing boundary value problems. This technique relies on the eigenfunction ansatz, which are based on the eigenvalue problem corresponding to the given boundary value problems. The eigenvalue problems having rigid, soft or impedance types of boundary conditions reveal orthogonal eigenfunctions, and the resulted eigen-sub-systems undergo Sturm-Liouville (SL) category. However, if the eigenvalue problems involve higher order boundary conditions the eigenfunctions are non-orthogonal in nature and, the resulted eigen-sub-systems underlie nonSturm-Liouville category. In such systems, the development and use of generalized orthogonal characteristics is indispensable to ensure the point-wise convergence of the solution. The orthogonal characteristics are incorporated in the process of conversion of differential systems to the linear algebraic systems. The application of generalized orthogonality relation (OR) governs additional constants that are found through application of appropriate edge conditions. The systems are truncated and inverted to explain physical characteristics of the modeled structures. The numerical computations are performed by using the truncated solutions to see how the choice of appropriate edge conditions, structural variations and bounding wall conditions affect acoustic attenuation for structure-borne as well as fluidborne vibrations. Furthermore, the Low-Frequency Approximation (LFA) solution which is valid only in low frequency regime is developed. The performance of LFA is compared with the benchmark MM method and is found in a good agreement with relative merits.