## 6 Conclusions  
A framework for the generation of an interactive auralization of an enclosure based on a plane wave expansion has been presented. This acoustic representation not only allows for interactive features, such as the translation and rotation of sound fields, but is also compatible with several sound reproduction techniques, such as binaural rendering, Ambisonics, WFS and VBAP. The directional impulse responses corresponding to this plane wave representation were predicted by means of the finite element method.  
An analysis of the reconstruction of the sound field in terms of monaural and B-format signals indicates that the interactive auralization system based on a plane wave representation is able to synthesize the acoustic field at low frequencies correctly, making it suitable for the auralization of enclosures, whose sound field is characterized by a modal behaviour.  
Figure 31: Comparison of full (a) and 1/3 octave band (b) frequency responses. Translated PWE (\(\gamma\)) and reference (\(\gamma\)) at translated Position 5.  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Receiver** & \(W\) **(dB)** & \(X\) **(dB)** & \(Y\) **(dB)** & \(Z\) **(dB)** & **Average (dB)** \\ \hline
2 (0.5 m) & 0.6 & 1.0 & 1.5 & 1.3 & 1.1 \\ \hline
5 (1.5 m) & 1.2 & 1.9 & 2.0 & 1.5 & 1.7 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Mean errors in the B-format signals for the interactive auralization at different receiver locations.  
Figure 32: Comparison of full (a) and 1/3 octave band (b) frequency responses. Translated PWE (\(Z\)) and reference (\(Z\)) at translated Position 5.  
The suitability of inverse methods to estimate the amplitude of a set of plane waves that reconstruct a target field has been proven. This technique is useful to extract directional information from data obtained from FE simulations. However, the discretization of the integral representation into a finite number of plane waves limits the spatial accuracy of the sound field reconstruction. The extent of the region in which the synthesis is accurate depends on the number of plane waves and the frequency of the field.  
The use of Tikhonov regularization has three main effects on the sound field representation. The first is that the energy of the plane wave density used for the synthesis of the sound fields is considerably lower than for the non-regularized solution. The second consequence is that the energy distribution of the plane wave density is much more directionally concentrated. The last effect is a reduction of the area where the sound field reconstruction is accurate compared to the non-regularized solution. Nevertheless, the implementation of regularization is convenient for an interactive auralization system because the translation operator can generate zones with high acoustic pressure if no regularization is applied.  
Future work will include the combination of finite element and geometrical acoustic results to extend the method outlined here to middle and high frequencies.  
The code developed for the interactive system can be download from \"[[https://drive.google.com/file/d/OBwuuNpQpY5](https://drive.google.com/file/d/OBwuuNpQpY5) UKZ2 htSW9 JbXNSOVU/view?usp=sharing]([https://drive.google.com/file/d/OBwuuNpQpY5](https://drive.google.com/file/d/OBwuuNpQpY5) UKZ2 htSW9 JbXNSOVU/view?usp=sharing)\".  
The current paper is the result of the research conducted by [PERSON] for the degree of doctor of philosophy at the Institute of Sound and Vibration Research, University of Southampton. [PERSON] and [PERSON] were the supervisors; their contributions correspond to the advice, support and review at all stages of the research.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
GA Geometrical Acoustics  
FEM Finite Element Method  
BEM Boundary Element Method  
FDTD Finite Difference Time Domain  
GPU Graphics Processor Unit  
PWE Plane Wave Expansion  
WFS Wave Field Synthesis  
VBAP Vector-Based Amplitude Panning