ARTICLE OVERVIEW
This article revealed why three famous integral formulations suffer numerical difficulties in predicting acoustic radiation and scattering from a finite object. These integral formulations are based on (1) potential theory or simple-source formulation, (2) the Helmholtz integral theory or surface Helmholtz integral formulation, and (3) the interior Helmholtz integral formulation. Most significantly, it presents the combined Helmholtz integral equation formulation (CHIEF) algorithm to overcome these numerical difficulties and proves that the CHIEF solution is unique.
1. Simple-source formulation
Therefore, by Smithies's theorem, the simple-source formulation has no solution when the frequency is near certain characteristic frequencies.
2. Surface Helmholtz integral formulation
Equation (6) has a unique solution except when k = , where represents a characteristic wavenumber. Unlike the source-density function , the surface acoustic pressure in Eq. (5) has a physical significance and, therefore, the compatibility condition will hold true. In other words, Eq. (6) has a solution for all wavenumber k. However, Eq. (6) has no unique solution to the surface acoustic pressure (see Fig. 1).
Maximum relative error in surface pressure as a function of ka for a uniformly vibrating sphere. (—) Surface Helmholtz integral formulation; ---, combined Helmholtz integral equation formulation with one interior point at the center. Reprinted with permission from H. A. Schenck, J. Acoust. Soc. Am. 44, 41–58 (1968). Copyright 1968 Acoustical Society of America (Ref. 5).
Maximum relative error in surface pressure as a function of ka for a uniformly vibrating sphere. (—) Surface Helmholtz integral formulation; ---, combined Helmholtz integral equation formulation with one interior point at the center. Reprinted with permission from H. A. Schenck, J. Acoust. Soc. Am. 44, 41–58 (1968). Copyright 1968 Acoustical Society of America (Ref. 5).
3. Interior Helmholtz integral formulation
4. CHIEF
It is emphasized that the difficulties encountered in the simple-source formulation, the surface Helmholtz integral formulation, and the interior Helmholtz integral formulation occur in numerical computations. These theories themselves are well defined.
IMPACT OF THE ARTICLE
Harry A. Schenck does not have many published research papers, yet his name is a household brand, and the CHIEF algorithm is a synonym in numerical computation methods for predicting acoustic radiation from finite objects in free space. This is because his seminal paper revealed the root causes of numerical difficulties inherent in three famous computational methodologies and, most importantly, has mathematically proven that the CHIEF algorithm always produces unique solutions.
Since publication, the article by Schenck has garnered 1032 citations with 33 highly influential citations.5