Abstract
A new analytical formulation is prescribed to solve the Helmholtz equation in 2D with arbitrary boundary. A suitable diffeomorphism is used to annul the asymmetries in the boundary by mapping it into an equivalent circle. This results in a modification of the metric in the interior of the region and manifests itself in the appearance of new source terms in the original homogeneous equation. The modified equation is then solved perturbatively. At each order the general solution is written in a closed form irrespective of boundary conditions. This method allows one to retain the simple form of the boundary condition at the cost of complicating the original equation. When compared with numerical results the formulation is seen to work reasonably well even for boundaries with large deviations from a circle. The Fourier representation of the boundary ensures the convergence of the perturbation series.
Highlights
The three dimensional Helmholtz equation is encountered frequently by physicist and engineers in different areas − like the eigenanalysis in acoustic and electromagnetic cavities, transmission of acoustic waves through ducts and in quantum mechanics
Canonical example of Neumann boundary condition (NBC) is the acoustic cavity where the sound velocity is set to zero on the boundary
The finite difference method (FDM), the finite element method (FEM) [17] and the boundary element method (BEM) [18] are popular ones but they consume huge amount of time to generate the mesh for a complicated geometry
Summary
The three dimensional Helmholtz equation is encountered frequently by physicist and engineers in different areas − like the eigenanalysis in acoustic and electromagnetic cavities, transmission of acoustic waves through ducts and in quantum mechanics. This analytic formulation is an extension of our earlier work for two dimensions [1] where a suitable diffeomorphism in terms of a Fourier series was chosen to map the general problem into an equivalent one where the boundary was circular but the equation gets complicated due to the deformation of the metric in the interior and was solved by perturbation technique of quantum mechanics In this method, an arbitrary domain in three dimensions is mapped to a regular closed region (in which the Helmholtz equation is exactly solvable) by a suitable co-ordinate transformation resulting the deformation of the interior metric.
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