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

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.