The classic Weyl-van der Pol (WVDP) formula is a well-known asymptotic solution for accurately predicting sound fields above a locally reacting ground surface. However, the form of the WVDP formula is inadequate for predicting sound fields in the vicinity of non-locally reacting surfaces; a correction term is often required in the formula to provide accurate numerical solutions. Even with this correction, there is a singularity in the diffraction wave term when the source is located directly above or below the receiver. This paper explores a heuristic method to remove this singularity and suggests an analytical form comparable to the WVDP formula. This improved formula offers a physically interpretable solution and allows for accurate predictions of the total sound field above locally and non-locally reacting surfaces for all geometrical configurations.
In the absence of atmospheric effects, the sound fields above a locally reacting ground can be accurately predicted by the Weyl-van der Pol formula. The solution is based on an asymptotic analysis to yield two terms: a direct and a ground reflected wave terms. The reflected wave term can be written as a product of a spherical wave reflection coefficient and the sound reflected from a rigid ground. In the contrary, it is more challenging to derive a similar formula for the sound fields above non-locally reacting grounds. In the past, an approximation in the same form as the Weyl-van der Pol formula has been used which becomes inadequate for layered grounds. In this presentation, a brief review of the asymptotic analysis will be discussed. An overview of the analytical and numerical approaches will be presented for obtaining accurate prediction of sound fields above the non-locally reacting ground. It will be further demonstrated that the reflection coefficient can be split exactly into two terms - a plane wave reflection coefficient and a ground wave term involving the boundary loss factor. The correlation between the numerical distance and the location of the surface wave pole will be examined.
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