# The Spec-Radiation Method for Layered Fluid Media
## 2 Material and Methods
### Extended Spec-Radiation Method for Layered Fluid Media  
The extension is derived for a rather simple case first and is generalized later. There are two different fluid media layers, as depicted in Figure 1. According to [PERSON] and [PERSON] [49], the layer interface interaction is described by the part of the wave that has an influence at the interface. Hence, a corresponding impedance has to be defined; we call it here the specific impedance. The specific impedance for one single plane wave is calculated by:  
\[Z=\frac{p(x,y,z)}{v_{z}(x,y,z)}, \tag{10}\]  
which leads to the well-known formulation:  
\[Z=\frac{c\rho}{\cos{(\gamma)}}, \tag{11}\]  
with \(\gamma\) as the propagation direction angle, \(\rho\) as the density, and \(c\) as the speed of sound. With this knowledge, the reflection factor:  
\[R=\frac{Z_{2}-Z_{1}}{Z_{2}+Z_{1}} \tag{12}\]  
and the transmission factor:  
\[T=\frac{2Z_{2}}{Z_{2}+Z_{1}}=1+R \tag{13}\]  
are determined. Indices 1 and 2 mark Medium 1 and Medium 2. With the reflection and transmission factor, the sound wave pressure in both media can be calculated. Figure 1: Schematic representation of the incident, reflected, and transmitted waves at a fluid–fluid interface in a semi-infinite volume. Included are the speed of sound \(c_{1}\) and \(c_{2}\), the densities \(\rho_{1}\) and \(\rho_{2}\), and the wavelengths \(\lambda_{1}\) and \(\lambda_{2}\) of the two fluid media. If Equation (10) is set up for the spectrum, i.e., in the Fourier domain for a specific \(\omega\), it looks as follows:  
\[Z_{\mathfrak{F}}=\frac{P(f_{x},f_{y},z)}{V(f_{x},f_{y},z)}. \tag{14}\]  
Here, the index \(\mathfrak{F}\) at the specific impedance indicates that this is the specific impedance for the Fourier domain In the Fourier domain, the velocity is described by the pressure with the following equation:  
\[V(f_{x},f_{y},z)=\frac{P(f_{x},f_{y},z)k_{z}}{\omega\rho}, \tag{15}\]  
as can be found in [PERSON] [50]. This turns Equation (14) into:  
\[Z_{\mathfrak{F}}=\frac{c\rho}{\lambda f_{z}}. \tag{16}\]  
This is now an expression for the impedance that can be used in the spec-radiation method. When comparing Equations (11) and (16), the relation of Equation (6) is again recognizable. This means that if \(\lambda\) and \(f_{z}\) are known, the cosine of the angle of propagation is also known. With Equation (16), the reflection factor can be written as:  
\[R_{\mathfrak{F}}=\frac{Z_{\mathfrak{F},2}-Z_{\mathfrak{F},1}}{Z_{\mathfrak{F},2}+Z_{\mathfrak{F},1}}=\frac{\frac{c_{2}\rho_{2}}{\lambda_{2}f_{z,2}}-\frac {c_{1}\rho_{1}}{\lambda_{1}f_{z,1}}}{\frac{c_{2}\rho_{2}}{\lambda_{2}f_{z,2}} +\frac{c_{1}\rho_{1}}{\lambda_{1}f_{z,1}}} \tag{17}\]  
and the transmission factor as:  
\[Z=\frac{p(x,y,z)}{v_{z}(x,y,z)}, \tag{10}\]0  
The entire sound pressure in Fluid 1 according to Figure 1 is thus the sum of the incident and the reflected sound waves:  
\[ \begin{split} p_{f1}(x,y,0\leq z<h_{1},t)=&\mathfrak{F }_{f_{x}f_{y},\omega}^{-1}(H(f_{z,1},0\leq z^{+}<h_{1})\mathfrak{F}_{x,y}(p( x,y,z_{0},\omega)))+\\ &\mathfrak{F}_{f_{x}f_{y},\omega}^{-1}(H(f_{z,1},0\leq z^{-}<h_{ 1})R_{\mathfrak{F},12}\mathfrak{F}_{x,y}(p(x,y,z_{h1},\omega))).\end{split}  \tag{19}\]  
Here, \(z^{+}\) indicates that the sound propagation is calculated in the positive \(z\)-direction, and \(z^{-}\) indicates that the sound propagation is calculated in the negative \(z\)-direction. \(R_{\mathfrak{F},12}\) is the reflection factor in the Fourier domain for the interface of Fluids 1 and 2. The sound pressure in Fluid 2 is:  
\[p_{f2}(x,y,z\geq h_{1},t)= \mathfrak{F}_{f_{x}f_{y},\omega}^{-1}(H(f_{z,2},z^{+}\geq h_{1}) T_{\mathfrak{F},12}\mathfrak{F}_{x,y}(p(x,y,z_{h1},\omega))). \tag{20}\]  
Here, \(T_{\mathfrak{F},12}\) is the transmission factor at the interface between Fluid 1 and Fluid 2.