## 3 Results
### Results with 2 mm Grid Point Distance
The re-radiation method  
Figure 7: Maximum value of several calculated planes using the re-radiation method for calculation (blue) and with the spec-radiation method for calculation (red). For the grid of the size 101 \(\times\) 101 points (**a**), for the grid of the size 128 \(\times\) 128 points and for the grid of the size 151 \(\times\) 151 points (**b**) and for the grid of the size 256 \(\times\) 256 points and for the grid of the size 300 \(\times\) 300 points (**c**). starts from Equation (14), where the term for the free space Green's function for the center of the coordinate system (\(x=y=0\)) at a distance for \(z=0\) with \(R(0,0,0)=0\) takes the following value:  
\[\frac{1}{2\pi}\frac{\partial}{\partial z}\left[\frac{\mathrm{e}^{ViR}}{R} \right]\Bigg{|}_{x=0,y=0,z\to 0}= \begin{cases}-\infty&z<0\\ +\infty&z>0\end{cases} . \tag{16}\]  
On the other hand, Equation (12) is for \(H(k_{x},k_{y},z=0)=1\), which returns the measurement results again with the spec-radiation method. Since a pole location at the position of the measuring plane does not make sense, due to physical reasons, the spec-radiation method reflects the reality better than the re-radiation method in the region of the measurement plane. The calculated sound fields at the surface of the flawless particleboard with a calculation window size of 256 \(\times\) 256 grid points are depicted for the spec-radiation method in Figure 8 and for the re-radiation method in Figure 9, respectively. These are optically almost identical, with only phase differences visible at the corners. However, these differences are irrelevant, because this area around the measurement window was only filled with zeros and, therefore, depending on the numerical deviations, large differences in a phase can be expected there. The difference image \(DI\) of the two amplitudes is shown in Figure 10. The scaling had to be extended to \(-60\) dB in order to make the minute differences recognizable. In [PERSON] et al. [48], the influence of zero-padding for the fictitious increase of the frequency resolution is described in detail. The results are now considered identical for a calculation window size of 256 \(\times\) 256 at a sufficient distance from the measuring plane. In Figure 11, the amplitude distribution of the measurement with a \(\omega\)4 mm flaw on the top (the side closer to the microphone) of the particleboard is depicted. In the middle of the red area, i.e., in the middle of the transmitter influence area, a yellow spot can be seen in a size of approximately \(\omega\)4 mm. This identifies the \(\omega\)4 mm flaw which is smaller than the wavelength \(\lambda=6.9\) mm in air. In Figure 12, the \(\omega\)20 mm flaw is well identified by the blue area in the influence area of the transmitter. This shows that with the spec-radiation method, it is possible to identify flaws both larger and smaller than the wavelength by evaluating a single frequency reference. The resolution limit for both the re-radiation method and the spec-radiation method, without further processing, is theoretically \(\lambda/2\), according to [PERSON] [49] and [PERSON] [50]. To visualize the result from the spec-radiation method, we provide a Video Supplementary Materials S1 within the supplementary materials, where the amplitude distribution of each plane above the particleboard can be seen in 3D space. The influence of the flaw and the diffraction behind can be seen. Figure 8: Amplitude (**a**) and angle (**b**) at the top of the wooden particleboard without a flaw. Evaluated with the spec-radiation method for the frequency 50 kHz, with \(\mathrm{d}x=\mathrm{d}y=2\) mm and a grid of the size 256 \(\times\) 256 points. Figure 11: Amplitude at the top of the wooden particleboard with a e4 mm piece of paper. Evaluated with the spec-radiation method for the frequency 50 kHz, with \(\mathrm{d}x=\mathrm{d}y=2\) mm and a grid of the size 256 \(\times\) 256 points.