# The Spec-Radiation Method for Layered Fluid Media
## 2 Material and Methods
### Spec-Radiation Method  
The spec-radiation method is based on the angular spectrum method, and the derivation described here was mainly based on that of [PERSON] [48] and [PERSON] [46]. Due to the nature of the spec-radiation method, only harmonic waves were used for the calculation. Starting from a source wave field described by \(p(x,y,z=0,\omega)\) in a plane, the FFT of it can be written as follows:  
\[ \begin{split} P(f_{x},f_{y},z=0,\omega)&=\tilde{ \mathfrak{F}}_{x,y}(p(x,y,z=0,\omega))\\ &=\iint_{-\infty}^{\infty}p(x,y,z=0,\omega)\mathrm{e}^{-2\pi(f_{ x}x+f_{y}y)}\mathrm{d}x\mathrm{d}y.\end{split}  \tag{1}\]Here, \(x\), \(y\), and \(z\) are spatial coordinates. \(\mathfrak{F}_{x,y}\) is the Fourier transform operator for the coordinates \(x\) and \(y\). \(p(x,y,z=0,\omega)\) is the sound pressure wave field at \(z=0\). \(\omega\) is the time-dependent angular frequency; \(\mathrm{i}\) is the imaginary unit, and \(f_{x}\), \(f_{y}\), and \(f_{z}\) are the spatial frequencies with:  
\[\left(f_{x},f_{y},f_{z}\right)=\left(\frac{k_{x}}{2\pi},\frac{k_{y}}{2\pi}, \frac{k_{z}}{2\pi}\right). \tag{2}\]  
\(k_{x}\), \(k_{y}\), and \(k_{z}\) are the wave numbers in the different spatial dimensions, and therefore, they are the components of the wave vector. The frequency \(f_{z}\) is not independent of the other spatial frequencies and the wavelength. For the determination of \(f_{z}\), we have to take a deeper view into the wave vector. The components of the wave vector are dependent on the following condition:  
\[\left|\mathbf{k}\right|=\frac{2\pi}{\lambda} \tag{3}\]  
with \(\mathbf{k}\) as the wave vector. This means that the spatial frequencies always satisfy:  
\[f_{x}^{2}+f_{y}^{2}+f_{z}^{2}=\left(\frac{1}{\lambda}\right)^{2}. \tag{4}\]  
By changing Equation (4), it becomes obvious that \(f_{z}\) is dependent on the other two spatial frequencies:  
\[f_{z}(f_{x},f_{y},\lambda)=\sqrt{\left(\frac{1}{\lambda}\right)^{2}-f_{x}^{2} -f_{y}^{2}}. \tag{5}\]  
This equation represents the so-called [PERSON] sphere, and each plane wave is represented by a point on the surface of the sphere. Another important connection of spatial frequencies in the description of the propagation of plane waves is the relationship with the directional cosine of the wave propagation:  
\[ \begin{split} f_{x}&=\frac{\cos(\alpha)}{\lambda} \\ f_{y}&=\frac{\cos(\beta)}{\lambda}\\ f_{z}&=\frac{\cos(\gamma)}{\lambda}\end{split}  \tag{6}\]  
which means that a point on the Ewald sphere can also be described by three directional cosines of the propagation angles of the plane wave. To calculate the sound wave field of another plane, Equation (1) can be multiplied by the propagation function \(H(f_{z},z)\). \(P(f_{x},f_{y},z,\omega)\) becomes:  
\[ \begin{split} P(f_{x},f_{y},z,\omega)&=H(f_{z},z) \;\mathfrak{F}_{x,y}(p(x,y,z=0,\omega)\\ &=\mathrm{e}^{-\mathrm{i}2\pi f_{z}}\iint_{-\infty}^{\infty}p(x, y,z=0,\omega)\mathrm{e}^{-\mathrm{i}2\pi(f_{x}x+f_{y}y)}\mathrm{d}x\mathrm{d}y. \end{split}  \tag{7}\]  
While computing the equation, it has to be considered that the square root of Equation (5) can also become complex. Thus, two cases have to be discussed:  
\[ \begin{split} f_{x}^{2}+f_{y}^{2}&\leq\left(\frac{1 }{\lambda}\right)^{2}\text{, leads to complex solutions}\\ f_{x}^{2}+f_{y}^{2}&>\left(\frac{1}{\lambda}\right) ^{2}\text{, leads to real solutions}.\end{split}  \tag{8}\]  
In the first case, when the square root of Equation (5) becomes complex, the information is outside the Ewald sphere and represents exponential decaying evanescent waves. For the through-transmission technique used in this publication to identify flaws, their contribution is rather small, and it was neglected in the further steps. The second case describes the information that is located directly on the surface of the Ewald sphere. With the propagation function \(H(f_{z},z)\), it is possible to calculate the sound propagation forward \((t\geq 0)\) and backward \((t\leq 0)\) in time.