Analysis of Acoustic Scattering Problems in Flexible Duct Involving Step Discontinuity

Abstract

This dissertation addresses a class of boundary value problems arising m the modelling of scattering of fluid-structure coupled waves in ducts or channels with abrupt geometric changes and discontinuous material properties. The main aim is to invoke the semi-analytic methods such as mode-matching and Galerkin techniques to render the solutions to these archetypal boundary value problems. The envisaged scattering problems are governed by Laplace or Helmholtz equations together with high-order boundary conditions describing flexible duct or channel boundaries. It is substantiated that the separation of variables reduces these boundary value problems to non-Strum-Liouville eigen-systems wherein the eigen-functions are neither orthogonal nor linearly independent. In this effect, generalized orthogonal relations are established which, together with a mode-matching technique, render a solution by matching modes across the interfaces of the geometric and material discontinuities. The emphasis in the dissertation is on membrane and elastic plate bounded ducts involving step discontinuities at interfaces with varying properties. Precisely, soft, rigid and flexible step discontinuities are taken into account. The viability of the modematching technique from the perspectives of the generalized orthogonal relations to address aforementioned acoustic scattering problems in ducts or channels with soft, rigid or flexible vertical strips is mathematically analysed and is supported with apposite numerical experiments. Due to the slow convergence rate of mode-matching solutions for scattering problems involving flexible strips, Galerkin method is employed as an attractive and potential alternative. The expressions for energy fluxes in different regimes are obtained and the analysis of power balance is performed.