On Mode-Matching Investigation of Two Dimensional Acoustic Waveguide Structures Scattering characteristics of wave phenomenon are important topics while dealing with the study of water waves, electromagnetic waves and acoustic waves. This thesis aims to investigate acoustic wave scattering problems of two-dimensional waveguide structures. Multiple canonical structures are considered so as to determine sound characteristics due to change in material properties and abrupt changes in height. These structures are extended by involving more complexities in terms of cavity and then the effect of metamaterials within a waveguide structure. The underlying structures have very much relevance to the study of heating, ventilation and air conditioning systems, modified silencers, exhausting systems and rainbow trapping. The mathematical formulation of such problems is governed by the Helmholtz equation subject to planar and non-planar boundary conditions. This together categorizes the underlying problems as Sturm-Liouville eigenvalue problems and non-Sturm-Liouville eigenvalue problems, respectively. The solutions to the class of boundary value problems addresses in this thesis involve the use of well-known mode-matching technique and Fourier transforms, just in case. In the process, the scattered amplitudes are determined resulting expressions for power distribution in different duct regions. The power distribution is then illustrated graphically subject to different parameters of interest. We observe that the results are well supporting the boundary value problems both mathematically and physically. In case of waveguide problem with a tapered metamaterial cavity, the results demonstrate that huge amount of acoustic wave energy can be damped by a tapered array over a broad range of frequencies. This signifies that the metamaterial cavity is an extremely effective broad banded absorber.