## 2 Methods
### Measuring Principle
#### 2.2.1 Acoustic Evaluation Model
Equation (10) shows how the frequency-domain echo signal \(F(\omega,0)\) is related to the input signal \(E(\omega)\) and the material properties of the layers and interfaces and involves the Fourier transforms of the transmitted and reflected signals, the reflection coefficient \(R(\omega)\), and the Dirac delta function \(A\Delta(\omega)\). Its expression can be extended using the reflection coefficient formula for the liquid-solid interface as follows [22]:  
\[H_{0}^{r}(\theta,\omega)=\frac{Z_{l}(\omega)\text{cos}^{2}\,2\theta_{s}+Z_{s} (\omega)\text{sin}^{2}\,2\theta_{s}-Z}{Z_{l}(\omega)\text{cos}^{2}\,2\theta_{ s}+Z_{s}(\omega)\text{sin}^{2}\,2\theta_{s}+Z} \tag{11}\]  
where  
\[Z_{l}(\omega)=\frac{\rho c_{l}}{\text{cos}\,\theta_{l}},Z_{s}(\omega)=\frac{ \rho c_{s}}{\text{cos}\,\theta_{s}},Z=\frac{\rho_{0}c_{0}}{\text{cos}\,\theta}\]  
where \(\rho_{0}\) is the density of the couplant, \(c_{0}\) is the velocity of sound in the couplant, and \(c_{l}\) and \(c_{s}\) are the longitudinal and shear wave velocities in the fluororubber specimen, respectively. \(\theta\) is the incident angle of the sound wave from the coupling fluid to the liquid-solid interface, while \(\theta_{l}\) and \(\theta_{s}\) are the refracted angles of the longitudinal and shear waves, respectively, in the fluororubber specimen. Equation (11) provides the reflection coefficient spectrum \(H_{0}^{r}(\theta,\omega)\) for the fluororubber layer relative to the couplant and is a function of the acoustic impedances of the materials, the angle of incidence, and the frequency \(\omega\). Substituting Equation (6) into the above Equation, we obtain:  
\[Z_{l}(\omega)=\frac{\sqrt{\rho K_{s}}}{\text{cos}\,\theta_{l}}\left(\sqrt{ \frac{\sqrt{1+\omega^{2}H^{2}}+1}{2}}+\sqrt{\frac{\sqrt{1+\omega^{2}H^{2}}-1} {2}}i\right) \tag{12}\]  
Equation (12) describes the acoustic impedance \(Z_{l}(\omega)\) of the couplant layer at frequency \(\omega\), incorporating both the material properties (density \(\rho\) and stiffness \(K_{s}\)) and the characteristic parameter \(H\). By introducing the Poisson's ratio \(v\) of fluororubber, the shear wave velocity \(c_{s}\) can be obtained through the relationship between the bulk modulus \(K\) and the shear modulus \(G\). The detailed derivation is omitted here. \[F(\omega,1)=E(\omega)H_{0}^{a}(\omega)\big{[}H_{0}^{r}(\omega)+H_{0}^{s}(\omega) \big{]}H_{1}^{a}(\omega)\big{[}H_{1}^{r}(\omega)+H_{1}^{s}(\omega)\big{]}H_{1}^{a }(\omega)\big{[}U_{0}^{r}(\omega)+H_{1}^{s}(\omega)\big{]}H_{0}^{a}(\omega)H_{0}^ {s}(\omega)R(\omega)A\Delta(\omega) \tag{13}\]  
Equation (13) describes the frequency-domain echo signal \(F(\omega,1)\) as it propagates through multiple layers, reflecting and transmitting between each interface. Each term represents the wave's interaction with the material layers (coulplant and fluororubber), incorporating the reflection and scattering coefficients at each interface. The presence of Fourier transforms for all terms indicates that this is a model for analyzing the system in the frequency domain. The equation involves multiple convolutions of the signals, representing how the ultrasonic signal is modified by the material properties and interfaces it encounters. As seen in the propagation model shown in Figure 2, when only the case of the vertical incidence of sound waves \(\theta=0^{\circ}\) is considered, in the above expression:  
\[\left\{ \begin{aligned} H_{0}^{r}(\omega)+H_{1}^{r}(\omega)=1\\ U_{0}^{r}(\omega)=H_{1}^{r}(\omega)\end{aligned} \right. \tag{14}\]  
Equation (14) provides a relationship between the reflection coefficients at different interfaces in the layered medium. The sum of the reflection coefficients at the two interfaces must equal 1, and the reflected signal from the fluororubber layer is equal to the reflection coefficient at the fluororubber-medium interface.