## 3 Results
### Results with 2 mm Grid Point Distance  
In this section, the grid point distance is 2 mm in the \(x\) and \(y\) direction. First, the experimental result without a flaw imitation is evaluated. The size of the evaluation window corresponds to the measurement window and has 101 \(\times\) 101 grid points. A single frequency (50 kHz) is evaluated here to draw a straightforward comparison. In Figure 4, the computed pressure distribution on the surface of the particleboard utilizing the spec-radiation method is shown. In Figure 4a, the amplitude is depicted in dB related to the maximum value in the plane. The red area represents the position of the transmitter. In Figure 4b, the corresponding phase distribution is depicted. The position of the transmitter can be seen at the red circle area. In Figure 5, the same measurement is evaluated using the re-radiation method. The result is again given in amplitude (in Figure 5a) and angle (in Figure 5b). When comparing Figures 4 and 5, no difference can be seen at first glance. Upon closer inspection, however, a difference in both phase and amplitude is noticeable. In order to present this difference, an difference image \(DI\) of the difference function of the amplitudes was generated (see Figure 6), according to:  
\[DI=20 log_{10}\left(\frac{||p(\mathbf{x},\omega)_{spec}|-|p(\mathbf{x},\omega)_ {re}||}{max(|p(\mathbf{x},\omega)_{re}|)}\right), \tag{15}\]  
where \(|p(\mathbf{x},\omega)_{spec}|\) is the pressure amplitude determined by the spec-radiation method and \(|p(\mathbf{x},\omega)_{re}|\) is the pressure amplitude determined by the re-radiation method. The difference image is now scaled to \(-40\) dB to make the difference clearer. To find the reason for these differences, several horizontal planes with a distance-step of 1 mm were calculated utilizing both methods. Each method is used to calculate the forward and backward propagation, to and from the measuring plane away. The measuring plane is at \(z=0\) m. The result is given in Figure 7. The maximum of each calculated plane is depicted over the propagation direction \(z\) in the range \(-0.36\) m to \(+0.36\) m. The values are shown in dB related to the maximum of the re-radiation method of all calculated planes. Figure 4: 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 101 \(\times\) 101 points. Figure 5: Amplitude (**a**) and angle (**b**) at the top of the wooden particleboard without a flaw. Evaluated with the re-radiation method for the frequency 50 kHz, with \(\mathrm{d}x=\mathrm{d}y=2\) mm and a grid of the size 101 \(\times\) 101 points. Figure 6: Difference image \(DI\) of the amplitudes of the back propagated sound field of the spec- and re-radiation method. It can be seen in Figure 7a that the results of the spec-radiation and the re-radiation method are not identical. Due to a pole at the measurement plane in the re-radiation method, significant differences can be found, with distance to the measurement plane being similar but not equal. The maximum value of both methods changes in relation to the size of the evaluation window. To show this, the measurement window, with a size of 101 \(\times\) 101 grid points, was inserted into a larger evaluation window which is produced by zero padding. The grid point distance remains d\(x\) = d\(y\) = 2 mm. In Figure 7b, the respective results for 128 \(\times\) 128 and 151 \(\times\) 151 grids are given. With increasing grid size, the difference become smaller. The results depicted in Figure 7c show that for a further increase in the measurement grids, the differences nearly vanish, with only the difference around the measurement plane remaining. The possible reason for this pole in the re-radiation method is the calculation.