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.

The simple-source formulation is based on the potential theory that states that the acoustic pressure at any field point px can be expressed as in integral of some unknown source-density function σξ,
(1)
where k is the acoustic wavenumber and d indicates the distance between a surface point ξ and a field point x. The source-density function σξ in Eq. (1) can be obtained by solving the following integral equation:
(2)
Because the source-density function σξ and normal surface velocity vξ are arbitrary, the following integral cannot be true:
(3)
where σ0ξ is a nontrivial solution that satisfies the following adjoint homogeneous integral equation:1 
(4)

Therefore, by Smithies's theorem, the simple-source formulation has no solution when the frequency is near certain characteristic frequencies.

Copley2 was the first to come up with the name of surface Helmholtz integral formulation, which states that the acoustic pressure at any field point can be represented by the Helmholtz integral formulation,
(5)
As the normal surface velocity vξ is specified, we need to first determine the surface acoustic pressure pξ in Eq. (5), which can be performed by solving an integral equation by letting the field point approach the surface, xξ,
(6)

Equation (6) has a unique solution except when k = k, where k represents a characteristic wavenumber. Unlike the source-density function σξ, the surface acoustic pressure pξ 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 pξ (see Fig. 1).

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).

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).

Close modal
The interior Helmholtz integral formulation is given by
(7)
where X implies an interior point. Because the normal surface velocity vξ is given, the surface acoustic pressure pξ can be determined by solving Eq. (7) such that
(8)
Once this is performed, the field acoustic pressure px is completely determined by Eq. (5). Kupradze3 has revealed this approach, and Copley4 has applied it to practical acoustic radiation problems. However, similar to Eq. (6), Eq. (8) does not have a unique solution at certain characteristic wavenumbers.1 Nevertheless, there is only one solution that can satisfy Eqs. (6) and (8) simultaneously. This leads to the CHIEF algorithm as described below.

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.

To implement the CHIEF algorithm, we discretize Eqs. (6) and (8), respectively, as follows:
(9)
(10)

Equations (9) and (10) are the key results of this paper.

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 

1.
F.
Smithies
,
Integral Equations
(
Cambridge University Press
,
Cambridge, UK
,
1958
), p.
51
52
.
2.
L. G.
Copley
, “
Fundamental results concerning integral representations in acoustic radiation
,”
J. Acoust. Soc. Am.
44
,
22
32
(
1968
).
3.
V. D.
Kupradze
, “
Fundamental problems in the mathematical theory of diffraction
,” (
NBS Report No. 2008
, October
1952
).
4.
L. G.
Copley
, “
Integral equation method for radiation from vibrating bodies
,”
J. Acoust. Soc. Am.
41
,
807
816
(
1967
).
5.
H. A.
Schenck
, “
Improved integral formulation for acoustic radiation problems
,”
J. Acoust. Soc. Am.
44
,
41
58
(
1968
).