## 2 Physics-Driven Models of Sound Transmission Loss
### Methodology of the STL Analyses  
The methodologies used to model the STL analyses described here follow the notation in Figure 1. The isotropic plate is in the plane \(z=0\), separating the source field \(P|_{z>0}\) and the transmission field \(P|_{z<0}\), both filled with a light fluid with density \(\rho_{0}\) and characteristic sound speed \(c_{0}\). Considering that a harmonic plane wave with wavenumber vector  
\[\mathbf{k}=\begin{pmatrix}k_{x}&k_{y}&k_{z}\end{pmatrix}^{T}=\frac{\omega}{c_{0}} \big{(}\sin\theta\cos\phi&\sin\theta\sin\phi&\cos\theta\big{)}^{T} \tag{4}\]  
Figure 2: Typical sound transmission loss curve as a function of frequency, showing the stiffness, resonance, mass and coincidence regions.  
impinges the plate with incident angle \(\theta\) and azimuth angle \(\phi\), the resultant incident acoustic pressure field is  
\[P|_{z>0}=P_{I}+P_{R}=p_{I}e^{i(\omega t-k_{x}x-k_{y}y-k_{z}z)}+p_{R}e^{i(\omega t -k_{x}x-k_{y}y+k_{z}z)}, \tag{5}\]  
where \(p_{I}\) is the amplitudes of the incident wave in the incident field \(P_{I}\), and \(p_{R}\) is the amplitude of the wave reflected by the plate, which forms the reflected field \(P_{R}\). The plate vibration also radiates a plane wave of magnitude \(p_{T}\) in the transmission field \(P_{T}\), for which acoustic pressure is given by:  
\[P|_{z<0}=P_{T}=p_{T}e^{i(\omega t-k_{x}x-k_{y}y-k_{z}z)}. \tag{6}\]  
The structural-acoustic coupling equation states that there must be continuity of acoustic and mechanics velocities at the interface of the plate with the acoustic fields:  
\[\frac{\partial P}{\partial z}\Big{|}_{z=0_{\pm}}=\rho_{0}\omega^{2}w, \tag{7}\]  
where \(w=W(x,y)e^{(i\omega t)}\) is the plate transverse displacement. For the sake of brevity, the factor \(e^{(i\omega t)}\) will be omitted from the equations hereafter. Therefore, the coupling equations in the source and transmission side are, respectively:  
\[ik_{z}(P_{R}-P_{I}) =\rho_{0}\omega^{2}w, \tag{8}\] \[-ik_{z}P_{T} =\rho_{0}\omega^{2}w, \tag{9}\]  
from which the following relationship between the pressure fields amplitudes is obtained:  
\[P_{R}=P_{I}-P_{T}. \tag{10}\]  
By definition, the acoustic transparency \(\tau\) is the ratio between transmitted and incident sound intensity, where the sound intensity can be written as:  
\[I_{i}=|P_{i}\cdot v_{i}|=\frac{|P_{i}|^{2}}{\rho_{0}c_{0}}, \tag{11}\]  
where the subindice \(i\) indicates the acoustic field, which can be the incident, transmitted or reflected field, and \(v_{i}\) is the local fluid velocity. Therefore, the acoustic transparency is:  
\[\tau(\phi,\theta,\omega)=\bigg{|}\frac{p_{T}}{P_{I}}\bigg{|}^{2}. \tag{12}\]  
The acoustic transparency for a diffuse field \(\tau_{d}(\omega)\) is the weighted average transmission coefficient of all possible incident and azimuth angles:  
\[\tau_{d}(\omega)=\frac{\int_{0}^{2\pi}\int_{0}^{\pi/2}\tau(\phi,\theta,\omega )\cos\theta\sin\theta d\theta d\phi}{\int_{0}^{2\pi}\int_{0}^{\pi/2}\cos\theta \sin\theta d\theta d\phi}. \tag{13}\]  
The STL is the inverse of the acoustic transparency measured on the dB scale.  
To find the relationship between the pressure field amplitudes in Equation (12), the constitutive equation of motion of the plate should be solved combined with Equation (10), which for an isotropic plate can be written as:  
\[D\
abla^{4}w-\omega^{2}mw=P|_{z=0^{-}}-P|_{z=0^{+}}=(P_{I}-P_{R})-(P_{T})=2(P_ {I}-P_{T}). \tag{14}\]  
The solution for each approach addressed in this paper is presented below.