This talk is on a unique numerical solution method for the Kirchhoff–Helmholtz integral equation. It is based on a highly efficient representation of the sound field using the reproducing kernel in the solution space of the homogeneous Helmholtz equation. When compared to the conventional boundary element method, the proposed method offers several advantages: no need to evaluate singular integrals, independent control over the boundary discretization and unknown variables, compatibility with higher-order impedance boundary conditions, and more. This talk presents a brief overview of the analysis theory underlying the proposed method. Then, we demonstrate its validity by calculating bounded sound fields. The accuracy of the calculation depends on several settings, including the arrangement and number of reproducing kernel nodes and collocation points, as well as the numerical treatment of boundary integrals. Another significant issue is that the coefficient matrix of the proposed system suffers from rank deficiency when the number of reproducing kernel nodes exceeds a certain criterion. This is partly understood in terms of the relationship between truncation in spherical harmonic expansion and the size of the domain being represented.
Inoue et al. (2025) studied this question.