An Improved Approach for Computing the Green Functions of the Helmholtz Equation in the 2D Impedance Half Space

The boundary element method (BEM) is an important numerical tool in outdoor acoustics. It can be used, for example, to compute the acoustic fields near noise barriers. In BEM simulations of such problems, the ground surface can be seen as an infinite plane with a known impedance [1], and the corresponding boundary value problems are said to be defied in the impedance half space (IHS). Therefore the Green functions of the Helmholtz equation in the IHSs, which satisfy the impedance boundary condition on the ground surface and the Sommerfeld condition, must be computed efficiently and exactly for various source-receiver positions and various possible values of the ground surface impedance. Methods for computing the Green functions of the Helmholtz equation in the IHS has been intensively investigated in the past decades. Among the numerous publications on this topic [2] is interesting because it presented the method of the steepest decent path for two dimensional problems in detail and the error was estimated theoretically and numerically. The method described in [3] is also very interesting because the Green functions were given in a simple form, i.e. as a sum of the Green function in the hard reflecting half space and a line integral in which the Hankel function is the essential part of the integrand. However, in its original form, the method can only be applied to the so called mass-like IHSs. It is well known that most real ground surfaces have spring-like impedances. Therefore, in the present paper, the method is reformulated so that it can also be applied to spring-like IHSs. By means of numerical examples, we demonstrate that our approach yields reliable results for IHSs with spring-like impedances as well as with mass-like impedances.