A new approach to the complex Helmholtz equation with applications to diffusion wave fields, impedance spectroscopy and unsteady Stokes flow

Abstract

A new transform pair representing solutions to the complex Helmholtz equation in a convex 2D polygon is derived using the theory of Bessel's functions and Green's second identity. The derivation is a direct extension of that given by Crowdy (2015, A transform method for Laplace's equation in multiply connected circular domains. IMA J. Appl. Math., 80, 1902-1931) for 'Fourier-Mellin transform' pairs associated with Laplace's equation in various domain geometries. It is shown how the new transform pair fits into the collection of ideas known as the Fokas transform where the key step in solving any given boundary value problem is the analysis of a global relation. Here we contextualize those global relations from the point of view of 'reciprocal theorems', which are familiar tools in the study of the effective properties of physical systems. A survey of the many uses of this new transform approach to the complex Helmholtz equation in applications is given. This includes calculation of effective impedance in electrochemical impedance spectroscopy and in other spectroscopy methods in diffusion wave field theory, application to the 3$\omega $ method for measuring thermal conductivity and to unsteady Stokes flow. A theoretical connection between this analysis of the global relations and Lorentz reciprocity in mathematical physics is also pointed out.