Polar Coordinate Solutions of the Helmholtz Equation in General Dimensions and an Orthonormal Basis

TL;DR

This paper develops a general-dimensional approach to solving the Helmholtz equation in polar coordinates, introducing an orthonormal basis within a reproducing kernel Hilbert space to unify and simplify solutions across dimensions.

Contribution

It provides a novel derivation of polar coordinate solutions in arbitrary dimensions and constructs an orthonormal basis linked to the addition theorem, enhancing theoretical understanding.

Findings

Derived polar coordinate solutions in general dimensions.

Established an orthonormal basis for the solution space.

Connected the addition theorem to reproducing kernel properties.

Abstract

In acoustical engineering, analytical methodologies are often restricted to two or three dimensions; however, a general-dimensional approach can enhance learning and implementation efficiency while providing a unified understanding of foundational principles. In this manuscript, we present a straightforward derivation of the polar coordinate solutions of the homogeneous Helmholtz equation in general dimensions. We define the radial function as a special function, with coefficients selected to maintain orthonormality within a reproducing kernel Hilbert space, which simplifies its kernel representation. Additionally, we derive an orthonormal basis for this space, thereby demonstrating that the addition theorem arises naturally from a property of the reproducing kernel.