## 3 Plane Wave Expansion from Finite Element Data
[PERSON] and [PERSON] [27] proposed the following relation between the area of accurate reconstruction, the number of plane waves and the frequency of the field:  
\[L=\left(\left\lceil 2\pi\frac{R}{\lambda}\right\rceil+1\right)^{2}, \tag{14}\]  
in which \(L\) is the number of plane waves, \(\left\lceil\cdot\right\rceil\) indicates the round operator, \(\lambda\) is the wavelength and \(R\) is the radius of a sphere within which the reconstruction is accurate. Numerical simulations have been performed to evaluate the effects of discretizing the plane wave expansion. Figure 2 shows the real part of the target and the reconstructed acoustic pressure (Pa) in a cross-section of the domain using 36, 64 and 144 plane waves in the expansion. The frequency of the field corresponds to 250 HzThe black circle displayed in the figure represents the area of accurate reconstruction predicted by solving Equation (14) for \(R\). Figure 2 indicates that the region in which the reconstruction is accurate increases when a higher number of uniformly-distributed plane waves is considered in the expansion. Good agreement was found between the radius predicted by Equation (14) and the area where the reconstruction is correct. A preliminary analysis was performed to establish the size of the microphone array and the number of plane waves required to generate an inverse matrix whose condition number is smaller than \(10^{6}\). The condition number is defined as the ratio between the largest and the smallest singular value of the propagation matrix \(\mathbf{H}(\omega)\). It has been shown in [26] that the stability of the solution provided by the inverse method is determined by the condition number; therefore, high values of this parameter indicate that errors in the model, such as noise or non-linearity of the system, will affect the result for \(\mathbf{q}(\omega)\) significantly. The criterion of a condition number smaller than \(10^{6}\) was motivated due to the fact that the data come from numerical simulations, which are free from measurement noise. Although the model is still affected by numerical inaccuracies, the level of this type of noise is expected to be much lower than in the case of measured noise. Firstly, a simple incoming plane wave of 63 Hz (\(\theta=90^{\circ}\), \(\phi=45^{\circ}\)) in free field was selected as a target. The sound field was analytically calculated in a rectangular domain of dimensions (5 m, 10 m, 3 m) and captured by four different virtual cube arrays with linear dimensions of 1.2 m, 1.6 m, 2 m and 2.4 m, respectively. The spatial resolution between microphones corresponded to 0.2 m. A frequency of 63 Hz was selected as a reference since the condition number decreases with frequency, and 63 Hz therefore provides a reasonable lower threshold. Table 1 shows the condition number of matrix \(\mathbf{H}(\omega)\) for different sizes of the array and numbers of planes waves uniformly distributed over a unit sphere [25]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Length of the Array** & \(\boldsymbol{L=64}\) & \(\boldsymbol{L=144}\) & \(\boldsymbol{L=324}\) \\ \hline
1.2 m (343 mics) & \(2.39\times 10^{7}\) & \(2.86\times 10^{13}\) & \(2.13\times 10^{18}\) \\
1.6 m (729 mics) & \(3.09\times 10^{6}\) & \(8.88\times 10^{11}\) & \(1.32\times 10^{17}\) \\
2 m (1331 mics) & \(6.11\times 10^{5}\) & \(6.13\times 10^{10}\) & \(7.65\times 10^{16}\) \\
2.4 m (2197 mics) & \(1.56\times 10^{5}\) & \(6.89\times 10^{9}\) & \(3.01\times 10^{16}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Condition number of the matrix \(\mathbf{H}(\omega)\) as a function of the size of the microphone array and the number of plane waves. Figure 2: Reconstructed acoustic field using different numbers of plane waves in the expansion. (**a**) target field; (**b**) reconstructed field \(L=36\); (**c**) reconstructed field \(L=64\); (**d**) reconstructed field \(L=144\). As expected, the condition number decreases as the size of the array increases.