# The Spec-Radiation Method for Layered Fluid Media
If, in this way, the information of a plane is measured and known, methods of acoustic holography can be applied to calculate from that plane to another plane directly above the object under test. The acoustic holography originates from the holography in optics invented by [PERSON] [25]. Acoustic holography enables the calculation of sound fields starting from a known plane in the sound field. Due to the coherent properties of ultrasonic waves, diffraction-induced inaccuracies can be eliminated in this way ([PERSON] [26]). An important method of acoustic holography for the detection of flaws is the re-radiation method ([PERSON] et al. [27], [PERSON] et al. [28], [PERSON] et al. [29], [PERSON] et al. [30], [PERSON] et al. [31], [PERSON] et al. [32], [PERSON] et al. [33]). With the re-radiation method, flaws in the sub-wavelength range can be identified ([PERSON] et al. [28]). Under the assumption that the solid test object is similar to a fluid and only longitudinal waves can propagate, a well-interpretable result has already been obtained on the side facing away from the receiver ([PERSON] et al. [28]), and objects in a sound field have also been determined ([PERSON] et al. [31], [PERSON] and [PERSON] [34]). The re-radiation method is based on the Rayleigh-Sommerfeld diffraction integral and thus offers a well-interpretable description in the spatial domain. Due to the description in the spatial domain, the computation effort is not low, since several two-dimensional Fourier transformations (FFTs) and inverse Fourier transformations (iFFTs) have to be performed for the evaluation. The spec-radiation method ([PERSON] and [PERSON] [35]) offers the possibility to perform faster computations, since the data are processed in the spatial frequency domain, so fewer FFTs and iFFTs have to be performed. The smallest detectable flaws that can be detected theoretically are, according to [PERSON] [36], in the size of \(\lambda/2\) with the re-radiation, as well as the spec-radiation method. The spec-radiation method is based on the angular spectrum method, which was first used by [PERSON] and [PERSON] [37] to describe light propagation. [PERSON] [38], then it was possible to describe the diffraction of light in the ionosphere with this method. [PERSON] et al. [39] then showed that it is possible to apply this method to the propagation of sound waves, and he was able to identify holes in steel plates with a 5 MHz transmitter. To characterize the sound field of ultrasonic transmitters, [PERSON] and [PERSON] [40] and [PERSON] and [PERSON] [41] proposed the use of the angular spectrum method. While [PERSON] et al. [42] used it for object identification, [PERSON] and [PERSON] [43] used it for the analysis of tissue harmonic imaging. A finite-element model based on the angular spectrum method was built by [PERSON] et al. [44], and [PERSON] and [PERSON] [45] described the forward and backward propagation of sound waves using the method. [PERSON] et al. [2] showed the influence of zero-padding. [PERSON] [46] showed the application of the angular spectrum method in the field of computer holography. [PERSON] and [PERSON] [35] utilized it for the spec-radiation method to identify flaws in panel materials. In [PERSON] and [PERSON] [47], the spec-radiation method was extended to the calculation of tilted planes in sound fields, to identify flaws in tilted panels. However, all these applications assume that the sound waves propagate in one medium, so that there are no layered media considered. In this publication, we present an extension to the spec-radiation method to calculate the sound field in layered fluid media. We show this in the first step including the determination of the particle deflection. The objective of the publication was to obtain an increased accuracy in the detection of flaws in thicker materials and on the side of the particleboard facing away from the receiver.