# 4. Special Geometries: NFC-HOA and SDM¶

The integral equation (3.1) states a Fredholm equation of first kind with a Green’s function as kernel. This type of equation can be solved in a straightforward manner for geometries that have a complete set of orthogonal basis functions. Then the involved functions are expanded into the basis functions \(\psi_n\) after [MF81], p. (940) as

where \(\tilde{G}_n, \tilde{D}_n, \tilde{S}_n\) denote the series expansion coefficients, \(n \in \mathbb{Z}\), and \(\langle\psi_n, \psi_{n'}\rangle = 0\,\) for \(n \ne n'\). If the underlying space is not compact the equations will involve an integration instead of a summation

where \(\d\mu\) is the measure in the underlying space. Introducing these equations into (3.1) one gets

This means that the Fredholm equation (3.1) states a convolution. For geometries where the required orthogonal basis functions exist, (4.7) follows directly via the convolution theorem [AW05], eq. (1013). Due to the division of the desired sound field by the spectrum of the Green’s function this kind of approach has been named SDM [AS10]. For circular and spherical geometries the term NFC-HOA is more common due to the corresponding basis functions. “Near-field compensated” highlights the usage of point sources as secondary sources in contrast to Ambisonics and HOA that assume plane waves as secondary sources.

The challenge is to find a set of basis functions for a given geometry. In the following paragraphs three simple geometries and their widely known sets of basis functions will be discussed.

## 4.1. Spherical Geometries¶

The spherical harmonic functions constitute a basis for a spherical secondary source distribution in \({\mathbb{R}}^3\) and can be defined after [GD04], eq. (12.153) [1] as

where \(P_n^{|m|}\) are the associated Legendre functions. Note that this function may also be defined in a slightly different way, omitting the \((-1)^m\) factor, see for example [Wil99], eq. (6.20).

The complex conjugate of \(Y_n^m\) is given by negating the degree \(m\) as

For a spherical secondary source distribution with a radius of \(R_0\) the sound field can be calculated by a convolution along the surface. The driving function is then given by a simple division after [Ahr12], eq. (3.21) [2] as

where \(\breve{S}_n^m\) denote the spherical expansion coefficients of the source model, \(\theta_\text{s}\), \(\phi_\text{s}\), and \(r_\text{s}\) its directional dependency, and \(\breve{G}_n^0\) the spherical expansion coefficients of a secondary monopole source located at the north pole of the sphere \(\x_0 = (\frac{\pi}{2},0,R_0)\). For a point source this is given after [SS14], eq. (25) as

where \(\hankel{2}{n}{}\) describes the spherical Hankel function of \(n\)-th order and second kind.

## 4.2. Circular Geometries¶

The following functions build a basis in \(\mathbb{R}^2\) for a circular secondary source distribution, compare [Wil99]

The complex conjugate of \(\Phi_m\) is given by negating the degree \(m\) as

For a circular secondary source distribution with a radius of \(R_0\) the driving function can be calculated by a convolution along the surface of the circle as explicitly shown by [AS09a] and is then given as

where \(\breve{S}_m\) denotes the circular expansion coefficients for the source model, \(\phi_\text{s}\), and \(r_\text{s}\) its directional dependency, and \(\breve{G}_m\) the circular expansion coefficients for a secondary monopole source. For a line source located at \(\x_0 = (0,R_0)\) this is given as

where \(\Hankel{2}{m}{}\) describes the Hankel function of \(m\)-th order and second kind.

## 4.3. Planar Geometries¶

The basis functions for a planar secondary source distribution located on the \(xz\)-plane in \(\mathbb{R}^3\) are given as

where \(k_x\), \(k_z\) are entries in the wave vector \(\k\) with \(k^2 = (\wc )^2\). The complex conjugate is given by negating \(k_x\) and \(k_z\) as

For an infinitely long secondary source distribution located on the \(xz\)-plane the driving function can be calculated by a two-dimensional convolution along the plane after [Ahr12], eq. (3.65) as

where \(\breve{S}\) denotes the planar expansion coefficients for the source model, \(y_\text{s}\) its positional dependency, and \(\breve{G}\) the planar expansion coefficients of a secondary point source after [SS14], eq. (49) with

for \((\wc )^2 > (k_x^2+k_z^2)\).

For the planar and the following linear geometries the Fredholm equation is solved for a non compact space \(V\), which leads to an infinite and non-denumerable number of basis functions as opposed to the denumerable case for compact spaces [SS14].

## 4.4. Linear Geometries¶

The basis functions for a linear secondary source distribution located on the \(x\)-axis are given as

The complex conjugate is given by negating \(k_x\) as

For an infinitely long secondary source distribution located on the \(x\)-axis the driving function for \({\mathbb{R}}^2\) can be calculated by a convolution along this axis after [Ahr12], eq. (3.73) as

where \(\breve{S}\) denotes the linear expansion coefficients for the source model, \(y_\text{s}\), \(z_\text{s}\) its positional dependency, and \(\breve{G}\) the linear expansion coefficients of a secondary line source with

for \(0<|k_x|<|\wc |\,\).

[1] | Note that \(\sin\theta\) is used here instead of \(\cos\theta\) due to the use of another coordinate system, compare Figure 2.1 from [GD04] and Fig. 2.1. |

[2] | Note the \(\frac{1}{2\pi}\) term is wrong in [Ahr12], eq. (3.21) and eq. (5.7) and omitted here, compare the errata and [SS14], eq. (24). |