Abstract
The phenomenon of irregular frequencies or spurious modes when solving the Kirchhoff–Helmholtz integral equation has been extensively studied over the last six or seven decades. A class of common methods to overcome this phenomenon uses the linear combination of the Kirchhoff–Helmholtz integral equation and its normal derivative. When solving the Neumann problem, this method is usually referred to as the Burton and Miller method. This method uses a coupling parameter which, theoretically, should be complex with nonvanishing imaginary part. In practice, it is usually chosen proportional or even equal to [Formula: see text]. A literature review of papers about the Burton and Miller method and its implementations revealed that, in some cases, it is better to use [Formula: see text] as coupling parameter. The better choice depends on the specific formulation, in particular, on the harmonic time dependence and on the fundamental solution or Green’s function, respectively. Surprisingly, an unexpectedly large number of studies is based on the wrong choice of the sign in the coupling parameter. Herein, it is described which sign of the coupling parameter should be used for different configurations. Furthermore, it will be shown that the wrong sign does not just make the solution process inefficient but can lead to completely wrong results in some cases.
Highlights
The Burton and Miller formulation[1] for exterior acoustic problems is well-known since it is free of fictitious resonances, see Ref. 2
It will even be shown that many authors are using a coupling parameter which is not optimal and which can lead to wrong results
The Burton and Miller method is based on a linear combination of Eqs. (13) and (20) for which the coupling parameter η is introduced
Summary
The Burton and Miller formulation[1] for exterior acoustic problems is well-known since it is free of fictitious resonances, see Ref. 2. When reading the literature applications of the method, it seems clear to choose this parameter to be i/k,3–29 or at least positively This is an Open Access article published by World Scientific Publishing Company. Kress[14] and Amini[3] were both using this kind of harmonic time dependence and clearly found that i/k is a very good and even close to optimal choice to minimize the condition number for a sphere at high frequencies. It is not completely clear, what the optimal coupling parameter for the Burton and Miller formulation is. It will even be shown that many authors are using a coupling parameter which is not optimal and which can lead to wrong results
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