## 4 Discussion  
In this paper, a new innovative method called spec-radiation for the detection of flaws in wooden particleboards was presented and evaluated. With this method, flaws near the theoretical resolution limit of \(\lambda/2\) can be identified. We showed in the analytical derivation that the re-radiation method can be easily interpreted by its representation with spatial coordinates, but requires a lot of computational effort. While the spec-radiation method is more complicated to interpret by using the spatial angular frequency coordinates, it is much more efficient to compute. A comparison of the two methods showed that the spec-radiation method can be 88% faster in computation than the re-radiation method. We showed this using an example of an idealized circular piston transmitter. We also showed that the re-radiation method is less accurate than the spec-radiation method, especially in the peripheral areas (see Figure 20). Another important aspect when it comes to time efficiency is the number of measuring points needed. We showed that with the re-radiation method, it is possible to identify the flaw up to a certain grid point distance, but that there are artifacts that make the interpretation less clear (see Figures 13 and 14). In order to get good interpretable results with the re-radiation method despite a large grid point distance, one has to rely on techniques, as in [PERSON] et al. [22]. For the spec-radiation method, it was shown that a large grid point distance does not produce the same artifacts as with the re-radiation method and that a \(\alpha\)20 mm flaw can still be clearly identified with a 16 mm grid point distance. For even larger grid point distances, we showed that the technique from [PERSON] et al. [22] can also be used to increase detectability. We also showed that the re-radiation method has a pole near the measuring plane, which makes the calculation results inaccurate close to the measurement plane. This pole position does not exist in the spec-radiation method and at the position of the measuring plane exactly the measuring plane is returned. With the help of the attached videos, we showed that we can determine and interpret the amplitude sound fields in space as well as the transient sound fields with the spec-radiation method. We are confident that this is a big step towards real-time analysis in the non-destructive air-coupled ultrasonic testing of wood-based materials.  
The following are available online at [[http://www.mdpi.com/2076-3417/10/19/6663/s1](http://www.mdpi.com/2076-3417/10/19/6663/s1)]([http://www.mdpi.com/2076-3417/10/19/6663/s1](http://www.mdpi.com/2076-3417/10/19/6663/s1)), Video S1: Soundfield 3D 20 mm Flaw particleboard. Video S2: 3D Soundfield of the piston transducer.  
Conceptualization, A.S.S. and J.T.; methodology, A.S.S.; software, A.S.S.; validation, A.S.S. and J.T.; formal analysis, A.S.S.; investigation, A.S.S.; resources, A.S.S.; data curation, A.S.S.; writing--original draft preparation, A.S.S.; writing--review and editing, A.S.S. and J.T.; visualization, A.S.S.; supervision, J.T.; project administration, J.T.; funding acquisition, J.T. All authors have read and agreed to the published version of the manuscript.  
The publication of this article was funded by the Open Access Fund of Leibniz Universitat Hannover.  
This research was supported by [PERSON].  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
GFRP Glass fiber reinforced polymer  
CFRP Carbon fiber reinforced polymer  
NDT Non destructive testing  
ACU Air-coupled Ultrasound  
MDF Medium-density fibreboard  
FFT Fast Fourier Transformation  
IFFT inverse Fast Fourier Transformation  
NI National Instruments