Reconsideration on Helmholtz-Kirchhoff Integral Solutions for Boundary Points in Radiation Problems

The Helmholtz-Kirchhoff integral (HKI) formula is very useful when designing transducers because it can be used to predict the acoustic pressure of a radiator at any position given only the acoustic pressure and velocity of the source. Many studies have been carried out to determine how to predict the acoustic pressure distributions generated by radiator sources using the HKI formula and boundary conditions. However, if the surface integration process includes radiator edges or vertices, then it is difficult to predict a consistent acoustic pressure distribution accurately, and the precise HKI formula to solve this problem and rigorous derivation are not known. In this article, to overcome these limitations, a formulation of the HKI for the boundary is proposed. This formulation is based on intuitive considerations and proven mathematically. Using the proposed expression of the HKI formula for the boundary, the acoustic pressures radiated by irregular surfaces were calculated and compared with the distributions obtained by the finite element method and theoretically exact solutions. The results obtained with the proposed formulation of the HKI were confirmed to be more accurate than those of the conventional HKI formula.