The complex Helmholtz operator: a new transform approach
File(s)
Author(s)
Hauge, Jordan
Type
Thesis
Abstract
In this thesis we introduce a new transform approach to solving the complex Helmholtz equation in polygonal domains.
We show how to retrieve analytical solutions for effective parameters by using Green's identity and strategic choices of particular solutions to the governing equation. We extend this idea to more complicated polygonal domains by deriving a new transform method for arbitrary convex polygons. We derive the transform method by adapting a geometric construction of the unified transform method for Laplace's equation, by using Green's identity and an integral representation for the associated Green's function valid for half-planes.
We demonstrate how to use the new transform method on various bounded and unbounded polygonal domains: general rectangle, right isosceles triangle, upper-half plane, channel and quarter plane. We proceed by using the new transform method to obtain new analytical solutions for electrochemical impedance spectroscopy and the 3ω method. Finally, we demonstrate how we can generalise the new transform method to higher order operators, where we validate the approach by solving problems in unsteady Stokes flow.
We show how to retrieve analytical solutions for effective parameters by using Green's identity and strategic choices of particular solutions to the governing equation. We extend this idea to more complicated polygonal domains by deriving a new transform method for arbitrary convex polygons. We derive the transform method by adapting a geometric construction of the unified transform method for Laplace's equation, by using Green's identity and an integral representation for the associated Green's function valid for half-planes.
We demonstrate how to use the new transform method on various bounded and unbounded polygonal domains: general rectangle, right isosceles triangle, upper-half plane, channel and quarter plane. We proceed by using the new transform method to obtain new analytical solutions for electrochemical impedance spectroscopy and the 3ω method. Finally, we demonstrate how we can generalise the new transform method to higher order operators, where we validate the approach by solving problems in unsteady Stokes flow.
Version
Open Access
Date Issued
2019-03
Date Awarded
2019-06
Copyright Statement
Creative Commons Attribution NonCommercial Licence
Advisor
Crowdy, Darren Christian
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)