5. High Frequency Approximation: WFS

The single-layer potential (3.1) satisfies the homogeneous Helmholtz equation both in the interior and exterior regions \(V\) and \(V^* {\mathrel{\!\mathop:}=}{\mathbb{R}}^n \setminus (V \cup \partial V)\,\). If \(D(\x_0,\w)\) is continuous, the pressure \(P(\x,\w)\) is continuous when approaching the surface \(\partial V\) from the inside and outside. Due to the presence of the secondary sources at the surface \(\partial V\), the gradient of \(P(\x,\w)\) is discontinuous when approaching the surface. The strength of the secondary sources is then given by the differences of the gradients approaching \(\partial V\) from both sides after [FN13] as

(5.1)\[D(\x_0,\w) = \partial_\n P(\x_0,\w) + \partial_{-\n} P(\x_0,\w),\]

where \(\partial_\n{\mathrel{\mathop:}=}\scalarprod{\nabla}{\n}\) is the directional gradient in direction \(\n\) – see Fig. 3.1. Due to the symmetry of the problem the solution for an infinite planar boundary \(\partial V\) is given as

(5.2)\[D(\x_0,\w) = -2 \partial_\n S(\x_0,\w),\]

where the pressure in the outside region is the mirrored interior pressure given by the source model \(S(\x,\w)\) for \(\x\in V\). The integral equation resulting from introducing (5.2) into (3.1) for a planar boundary \(\partial V\) is known as Rayleigh’s first integral equation. This solution is identical to the explicit solution for planar geometries (4.18) in \({\mathbb{R}}^3\) and for linear geometries (4.22) in \({\mathbb{R}}^2\).

A solution of (5.1) for arbitrary boundaries can be found by applying the Kirchhoff or physical optics approximation [CK83], p. 53–54. In acoustics this is also known as determining the visible elements for the high frequency boundary element method [HMWS03]. Here, it is assumed that a bent surface can be approximated by a set of small planar surfaces for which (5.2) holds locally. In general, this will be the case if the wave length is much smaller than the size of a planar surface patch and the position of the listener is far away from the secondary sources. [1] Additionally, only one part of the surface is active: the area that is illuminated from the incident field of the source model.

The outlined approximation can be formulated by introducing a window function \(w(\x_0)\) for the selection of the active secondary sources into (5.2) as

(5.3)\[P(\x,\w) \approx \oint_{\partial V} \!\! G(\x|\x_0,\w) \, \underbrace{-2 w(\x_0) \partial_\n S(\x_0,\w)}_{D(\x_0,\w)} \d A(\x_0).\]

In the SFS Toolbox we assume convex secondary source distributions, which allows to formulate the window function by a scalar product with the normal vector of the secondary source distribution. In general, also non-convex secondary source distributions can be used with WFS – compare the appendix in [LF47] [2].

One of the advantages of the applied approximation is that due to its local character the solution of the driving function (5.2) does not depend on the geometry of the secondary sources. This dependency applies to the direct solutions presented in Special Geometries: NFC-HOA and SDM.

[1]Compare the assumptions made before (15) in [SZ13], which lead to the derivation of the same window function in a more explicit way.
[2]The solution mentioned by [LF47] assumes that the listener is far away from the radiator and that the radiator is a physical source not a notional one as the secondary sources. In this case the selection criterion has to be chosen more carefully, incorporating the exact position of the listener and the virtual source. See also the related discussion.