Title page for ETD etd-07082010-083729


Type of Document Master's Thesis
Author Lanz, Colleen B
URN etd-07082010-083729
Title The Use of Schwarz-Christoffel Transformations in Determining Acoustic Resonances
Degree Master of Science
Department Mathematics
Advisory Committee
Advisor Name Title
Rogers, Robert C. Committee Chair
Borggaard, Jeffrey T. Committee Member
Lin, Tao Committee Member
Keywords
  • Laplacian
  • Eigenvalues
  • Schwarz-Christoff el Transformations
  • Polygons
Date of Defense 2010-07-06
Availability unrestricted
Abstract
In this thesis, we set out to provide an enhanced set of techniques for determining the eigenvalues of the Laplacian in polygonal domains. Currently, finite-element methods provide a numerical means by which we can approximate these eigenvalues with ease. However, we would like a more analytic method which may allow us to avoid a basic parameter sweep in finite-element software such as COMSOL to determine what could possibly be an "optimal" distribution of eigenvalues.

The hope is that this would allow us to draw conclusions about the acoustic quality of a pentagonally-shaped room. First, we find the eigenvalues using a common finite-element method through COMSOL Multiphysics. We then examine another method which makes use of conformal maps and Schwarz-Christoffel transformations with the prospect that it might provide a more analytic understanding of the calculation of these eigenvalues and possibly allow for variation of certain parameters. This method, as far as we could find, had not yet been developed on the pentagon. We end up carrying this method through nearly all of the steps necessary in finding these eigenvalues. We find that the finite-element method is not only easier to use, but is also more efficient in terms of computing power.

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