# DIFFRACTION OF ACOUSTIC WAVES BY A HALF-PLANE SUBJECT TO VARYING ROBIN BOUNDARY CONDITIONS

• Marianthi Moschou

Student thesis: Phd

### Abstract

This thesis discusses the canonical problem of acoustic diffraction by a half-plane subject to what we call `varying' Robin boundary conditions, terminology which is used throughout this work. The main motivation is to study the quarter-plane problem and the development of a full solution in the case of the `varying' Robin boundary conditions. Customarily, the standard, impedance (Robin) boundary condition is an adequate approximation for the scattering by a rough surface or a physical boundary from a uniformly porous material, whereas physical boundaries with non-uniform porosity which smoothly varies along the boundary can be approximated with a`varying' Robin boundary condition, which similarly to the standard, Robin boundary condition is a linear combination of the Dirichlet and Neumann ones. In this study, we bring together an assortment of advanced mathematical techniques and endeavour to provide mathematical solutions derived using analytic methods. This can be incredibly effective, since computational costs can be altogether reduced and cases in which numerical models struggle, such as generation of high-frequency noise, can be addressed. We first review the relatively new mathematical technique of the embedding formula. The embedding formula is a powerful tool which enables us to evaluate the diffraction coefficient, which contains all the necessary information for the description of the acoustic far-field (high-frequency approximation of the wave-field). In the second part of this study, we address the various challenges imposed by these alternative boundary conditions, focusing on the half-plane geometry since its examination might provide valuable insight for the solution of the quarter-plane problem. This involves studying various techniques, such as the well-known method of images and its variation, the complex image method, as well as the Mellin and Kontorovich-Lebedev transformations. To the best of our knowledge, there is no solution available to the problem of diffraction by a half-plane subject to these non-standard, varying Robin boundary conditions, making this work novel and providing a good understanding of the nature of the problem and the mathematics it involves.
Date of Award 31 Dec 2021 English The University of Manchester William Parnell (Supervisor) & Raphael Assier (Supervisor)

### Keywords

• diffraction
• Robin boundary conditions
• impedance boundary conditions
• Mellin transform
• Kontorovich-Lebedev transform
• canonical problems

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