## 1 Introduction
Due to the coherent property of ultrasonic radiation, the back propagation calculation of sound waves is possible, eliminating the diffraction-induced inaccuracies ([PERSON] [17]). A successful method to identify flaws is the re-radiation method ([PERSON] et al. [18], [PERSON] et al. [19; 20; 21], [PERSON] et al. [22; 23]). This is suitable for flaws down to the sub-wavelength range ([PERSON] et al. [19], [PERSON] et al. [22]) and can be carried out quickly in the experiment by using interpolation methods ([PERSON] et al. [22]). Besides the identification of flaws, it is also possible to identify objects in a sound field ([PERSON] and [PERSON] [24], [PERSON] et al. [25]). This method is based on the Rayleigh-Sommerfeld diffraction integral which has made its way from optics to acoustics. It offers a well interpretable mathematical expression by the representation in the spatial domain. Another method used in the field of optics to propagate the light waves is the angular spectrum method ([PERSON] and [PERSON] [26]). This was used first by [PERSON] and [PERSON] [27] and then considered by [PERSON] [28] for application in the diffraction of light rays in the ionosphere. [PERSON] et al. [29] showed that it is possible to apply this method in the field of acoustics to compute the propagation of sound waves similar to light waves. Identification of holes in the plate was carried out using a 5 MHz transmitter in a through transmission measurement and a steel plate with holes as a sound mirror. Since then, this method has been used in various areas of acoustics. [PERSON] and [PERSON] [30] and [PERSON] and [PERSON] [31] employed the method to characterize transmitters. A program called \"FOCUS\" has been developed to simulate transmitters ([PERSON] et al. [32], [PERSON] and [PERSON] [33], [PERSON] et al. [34]). It is used for the analysis of tissue harmonic imaging ([PERSON] and [PERSON] [35]), as well as the identification of objects ([PERSON] et al. [36]). As a finite element model, it has already been used by [PERSON] et al. [37]. [PERSON] and [PERSON] [38] have described the forward and backward propagation of sound waves with this method. Fast and reliable algorithms are indispensable for the idea of a real-time analysis of particleboards in production. With the established re-radiation method, this has been partially achieved. However, the hardware is not economically viable for the required number of computing operations. In this publication, we present the spec-radiation, which takes a further step towards real-time analysis by using the angular spectrum method for the detection of flaws in particleboards. We show that the spec-radiation method offers good detection possibilities even with a coarse measurement grid, in contrast to the re-radiation method, where this is only achievable with additional effort. We also show that this method can be applied with the extension for increasing the detectability of a coarse measuring grid as presented in [PERSON] et al. [22], as it is used in the re-radiation method. We show that the spec-radiation method delivers good reproductions of the sound field, includingthe area of the measuring plane. We show this through two experiments with flaw imitations in two dimensions. One has a larger diameter than the wavelength \(\lambda\) and the other has a smaller one. Large distances (approximately 500 mm) between the transmitter through the particleboard to the receiver have to be bridged with the sound attenuation in air depending on the frequency ([PERSON] and [PERSON] [39], [PERSON] [40]). For this, we use a low excitation frequency of 50 kHz. The presented spec-radiation method is up to 88% faster then the re-radiation method. We compare the re-radiation method with the spec-radiation method on the basis of measured data from a transmission test of a wooden particleboard with and without flaws. We show the influence of the measuring grid point distance on both methods. Using an academic piston transducer as an example, we investigate the calculation speed of both methods.