## 3 Plane Wave Expansion from Finite Element Data  
Although the plane wave expansion is an integral representation, for the implementation of the method, Equation (1) is discretized into a finite number of \(L\) plane waves, namely:  
\[p(\mathbf{x},\omega)=\sum_{l=1}^{L}e^{i\mathbf{x}\cdot\mathbf{\hat{y}}_{l}}q_ {l}(\omega)\Delta\Omega_{l}, \tag{8}\]  
where \(\Delta\Omega_{l}\) corresponds to the portion of the unit sphere that is associated with the plane wave \(l\). The discretization of Equation (1) is performed by using a predefined uniform distribution of \(L\) plane waves over a unit sphere [25]. Based on a finite plane wave expansion, an inverse method can be implemented to estimate a discrete set of plane waves whose complex amplitudes synthesize the target acoustic field. To that end, the acoustic pressure calculated with the FEM at a specific location of the domain can be understood as corresponding to the output of an omnidirectional microphone. The combination of acoustic pressures at discrete points generates a virtual microphone array that is used to extract spatial information of the sound field. Based on that information, the amplitude \(q_{l}\) of each plane wave is determined by the inversion of the transfer function matrix between the microphones and the plane waves [26]. This principle is explained as follows: the complex acoustic pressure predicted with the FE model at \(M\) virtual microphone positions is denoted in vector notation as:  
\[\mathbf{p}(\omega)=\left[p_{1}(\omega),p_{m}(\omega),\dots,p_{M}(\omega) \right]^{T}, \tag{9}\]  
where \(p_{m}\) is the acoustic pressure at the \(m\)-th virtual microphone. Likewise, the complex amplitudes of \(L\) plane waves used to reconstruct the sound field are represented by the vector:  
\[\mathbf{q}(\omega)=\left[q_{1}(\omega),q_{l}(\omega),\dots,q_{L}(\omega) \right]^{T}. \tag{10}\]  
Finally, the transfer function that describes the sound propagation from each plane wave to each virtual microphone is arranged in matrix notation as:  
\[\mathbf{H}(\omega)=\left| \begin{array}{ccc}h_{11}(\omega)&\dots&h_{1L}( \omega)\\ \vdots&h_{ml}(\omega)&\vdots\\ h_{M1}(\omega)&\dots&h_{ML}(\omega)\end{array} \right|\]  
in which \(h_{ml}=e^{i\mathbf{k}\mathbf{x}_{m}\cdot\mathbf{\tilde{y}}_{l}}\). Consequently, the relationship between the plane wave amplitudes and the virtual microphone signals is:  
\[\mathbf{p}(\omega)=\mathbf{H}(\omega)\mathbf{q}(\omega). \tag{11}\]  
The amplitude of the plane waves is calculated by solving Equation (11) for \(\mathbf{q}(\omega)\). This is carried out in terms of a least squares solution, which minimizes the sum of the squared errors between the reconstructed and the target sound field [26]. In the case of an overdetermined problem (more virtual microphones than plane waves), the error vector can be expressed as:  
\[\mathbf{e}(\omega)=\mathbf{\tilde{p}}(\omega)-\mathbf{p}(\omega), \tag{12}\]  
where \(\mathbf{\tilde{p}}(\omega)\) is the pressure reconstructed by the plane wave expansion and \(\mathbf{p}(\omega)\) is the target pressure from the FE model. The least squares solution is achieved by the minimization of a cost function \(J(\omega)=\mathbf{e}^{H}(\omega)\mathbf{e}(\omega)\) in which \((\cdot)^{H}\) indicates the Hermitian transpose. The minimization of the cost function \(J(\omega)\) is given by [26]:  
\[\mathbf{q}(\omega)=\mathbf{H}^{\dagger}(\omega)\mathbf{p}(\omega), \tag{13}\]  
in which \(\mathbf{H}^{\dagger}(\omega)\) is the Moore-Penrose pseudo-inverse of the propagation matrix \(\mathbf{H}(\omega)\)[26]. The use of a finite number of plane waves leads to artefacts in the sound field reconstruction.