# The Spec-Radiation Method for Layered Fluid Media
## 2 Material and Methods
### Practical Implementation  
To describe the calculation from one interface to the next, the flowchart in Figure 2 is used. In the first step, the constant known parameters such as the speed of sound, density, the \(z\)-position of the interfaces, and the measured sound pressure must be set.  
Then, the sound pressure is transformed in the frequency domain. The sound field generated by the incident wave at the first interface layer is determined by Equation (9) (spec-radiation method). In the block \"Interface j\" (with the interface number \(j=1\), \(2\), \(3\),...), the reflected and transmitted part is calculated. A flowchart of the block \"Interface j\" is depicted in Figure 3.  
In the block \"Interface j\", the first decision is among which frequencies the calculation has to be performed. For the representation of a harmonic signal, the calculation of a single frequency can be sufficient, while for the calculation of a single sinusoidal wave, several frequencies are needed. Here, all frequencies from 1.2 kHz up to 120 kHz were used. This means that 164 frequency step points were taken into account with the conjugate complex parts as well. All other frequencies were set to 0. This procedure saves much computing time. Then, the different wavelengths have to be calculated. Next, the sound pressure was transformed with a 2D FFT over the spatial coordinates. Any value that was not on the Ewald sphere was set to 0. Then, the two different spatial frequencies for the two media at the interface in the propagation direction had to be calculated. In the next step, the specific impedances and, with the specific impedances, the reflection and transmission coefficient were determined in the Fourier domain. The propagation function \(H\) was then calculated with the spatial frequency for the medium from the incident wave. In the following step, the sound pressure of the reflected and transmitted part at the interface can be calculated. If one is only interested in the sound pressure values, then the block \"Interface j\" could be left out. Here also, the deflection potenti  
Figure 2: Flowchart for the practical realization of the spec-radiation method for layered fluid media.  
needed, for the reflected and transmitted part. At last, in the block, the question is: Is one interested in the particle deflection? If so, then the particle deflection could be calculated from the deflection potentials in the Fourier domain as in Equation (22). Now, the block can be left alone. It has to be decided whether another interface should be determined or not. If not, then the generated results can be displayed. If so, then the block \"Interface j\" must be executed again. The transmitted wave up to the second interface results from the transmitted part at the first interface at \(z=h_{1}\). The calculation has to be considered, now that the transmitted and reflected part starts again at a local \(z=0\). The incident wave at Interface 2 is the transmitted part of Interface 1. As at Interface 1, the incident wave is the start for the transmitted and reflected part, and they start again at a local \(z=0\). After the calculation, it must be decided again whether there is another interface layer or not. It must also be decided whether the reflected sound field should be calculated back to the first interface. If this is the case, then the procedure from Figure 3 is carried out again, whereby the incident sound wave is now the reflected part of Interface 2. It was again assumed that the calculation starts at a local \(z=0\). For the calculation, the indices at the impedances were now simply exchanged. Similarly, for the transmission factor \(H\), the spatial frequency of the second medium was now used, due to the fact that the incident sound wave comes from the second medium. Depending on how many reflections and transmissions through different boundary layers are to be performed, this procedure must be repeated.