## 2 Materials and Methods
### Practical Implementation  
Figures 2 and 3 show the program flow and clarifies the practical realization of the spec-radiation method and the difference to the re-radiation method. The extension for the enhancement of the detectability of a coarse measuring grid made by [PERSON] et al. [22] is already included in the flow chart.  
Figure 1: Experimental setup with a flaw imitation at the top of the wooden particleboard. NI = National Instruments, [PERSON] et al. [22].  
Figure 2: Flow chart for the practical realization of the spec-radiation and re-radiation technique.  
The program starts in Figure 2 by loading the measured transient sound pressure field \(p(x^{\prime},y^{\prime},0,\omega)\) in the measuring plane at \(z^{\prime}=0\) into the working memory. Afterwards, important parameters are set, such as the sound velocity of air, which can be calculated according to [PERSON] [47]. In these carried out experiments, the speed of sound was approximately 343 m/s. In the next section, a decision has to be made whether an enhancement of the detectability is desired on the basis of whether a coarse measuring grid is to be used or not. If an enhancement of the detectability is desired, the data is interpolated by a spline interpolation to a finer virtual grid and so \(p(x^{\prime},y^{\prime},0,t)\) is interpolated to \(p(x^{\prime}_{Virt},y^{\prime}_{Virt},0,)\). Afterwards, the transient sound pressure field is brought into the time frequency domain by a one-dimensional FFT, whereby the complex amplitudes in the spatial domain are obtained. These amplitudes then go into the block \"Radiation Technique\", depicted in Figure 3. Here, the re-raditaion and the spec-radiation techniques use different codes. For the re-radiation technique, a grid with the spatial coordinates is created first. With this grid, the term can then be calculated from Equation (13). Next, the 2D FFT of this term is calculated to determine the propagation function \(H(k_{x},k_{y},z)\). A 2D FFT of the complex amplitudes in the spatial domain is performed as well to transfer the complex amplitudes into the Fourier domain. Then, the multiplication of \(H(k_{x},k_{y},z)\) and \(p(k_{x},k_{y},0,\omega)\) is performed, which corresponds to the convolution, and the complex amplitudes in the Fourier domain at the distance \(z\) are obtained. The complex amplitudes are transferred from the Fourier domain back into the spatial domain utilizing a 2D inverse Fourier Transform (iFFT). For the calculation with the spec-radiation method, a grid of the spatial angular frequencies \(k_{x}\) and \(k_{y}\) is created first. Then, a two-dimensional FFT of the complex amplitudes in the spatial domain is performed to obtain the complex amplitudes in the Fourier domain. In contrast to the re-radiation method, the propagation function \(H(k_{x},k_{y},z)\) can be calculated directly through the grid of the spatial frequencies according to Equations (5) and (12), making a 2D FFT not necessary. The wavenumber step is defined with the size of the measurement field to \(\Delta k_{x}=\frac{2\pi}{y_{max}}\) and \(\Delta k_{y}=\frac{2\pi}{y_{max}}\). By multiplying the complex amplitudes in the Fourier domain with the propagation function, i.e., the convolution,  
Figure 3: Flow chart of the flow chart block of the radiation technique (see Figure 2) for the practical realization of the spec-radiation and re-radiation technique.  
the complex amplitudes in the Fourier domain at the distance \(z\) are obtained. The complex amplitudes are transferred from the Fourier domain back into the spatial domain utilizing a 2D iFFT. The results can now be displayed as the amplitude or phase in the spatial domain to identify flaws. To compute the time data, the results must be brought back into the time domain with a corresponding iFFT. It is easy to see that the spec-radiation method is faster in the numerical calculation than the re-radiation method because only one 2D FFT is necessary. The advantages of the spec-radiation method will be discussed in the following section.