## 2 Methods
### Viscoelastic Model for Acoustic Wave Propagation
#### 2.1.2 Physical Interpretation of the Model  
The ratio \(\hat{\sigma}/\hat{\varepsilon}\) is treated as the Hookean elastic modulus \(C_{11}\). From Equation (1), it is easy to know that \(C_{11}=K+\frac{4}{3}G\). This leads to the derivation of the quasi-longitudinal wave velocity in fluororubber as follows:  
\[c_{l}=\sqrt{\frac{K+\frac{4}{3}G}{\rho}}=\sqrt{\frac{(k_{1}+k_{2})\tau^{2} \omega^{2}+k_{2}}{(\tau^{2}\omega^{2}+1)\rho}}\sqrt{1+\frac{\eta_{0}\tau^{2} \omega^{2}-(\eta_{0}+\eta_{1})}{(k_{1}+k_{2})\tau^{2}\omega^{2}+k_{2}}}\omega i \tag{5}\]  
where \(\rho\) is the density of fluororubber. Let \(K_{s}=\frac{(k_{1}+k_{2})\tau^{2}\omega^{2}+k_{2}}{(\tau^{2}\omega^{2}+1)}\) and \(H=\frac{\eta_{0}\tau^{2}\omega^{2}-(\eta_{0}+\eta_{1})}{(k_{1}+k_{2})\tau^{2} \omega^{2}+k_{2}}\), then the complex form of the quasi-longitudinal wave velocity is given by:  
\[c_{l}=\sqrt{\frac{K_{s}}{\rho}}\Bigg{(}\sqrt{\frac{\sqrt{1+\omega^{2}H^{2}+1} }{2}}+\sqrt{\frac{\sqrt{1+\omega^{2}H^{2}}-1}{2}}i\Bigg{)} \tag{6}\]  
The term \(\sqrt{\frac{K_{s}}{\rho}}\) is the initial phase velocity based on elastic properties (like in an ideal elastic material). The complex part inside the parentheses, involving the frequency \(\omega\) and the characteristic parameter \(H\), accounts for the viscoelastic nature of the medium, influencing both the wave's speed and how it is attenuated over time. This equation is useful for analyzing wave propagation in viscoelastic structures, like fluororubber in this paper. The quasi-longitudinal wave number \(k_{l}=\omega/c_{l}\), which implies that the wave number in fluororubber material also exists in a complex form. Assuming a plane wave propagates along the depth direction in the fluororubber specimen, the sound pressure of this plane wave can be simply described as [21]:  
\[p(t,z)=p_{a}e^{i(\omega t-k_{2}z)}=p_{a}e^{i\omega(t-z\sqrt{\frac{\rho(\sqrt{ 1+\omega^{2}H^{2}}-1)}{2K_{s}(1+\omega^{2}H^{2})}})}e^{-\omega z\sqrt{\frac{e^ {i\sqrt{1+\omega^{2}H^{2}}-1}}{2K_{s}(1+\omega^{2}H^{2})}}} \tag{7}\]  
When \(\beta=-\omega\sqrt{\frac{\rho(\sqrt{1+\omega^{2}H^{2}}-1)}{2K_{s}(1+\omega^{2 }H^{2})}}\), it can be seen that the amplitude of ultrasonic wave propagation in the fluororubber material decays according to \(e^{-\beta z}\) from Equation (7). The above model starts with the \"stress-strain\" relationship derived from mechanical tests such as tension, compression, and torsion, and extends this to the domain of simple harmonic vibration. In this model, \(\eta_{0}\) is temperature sensitive and is used to describe the equilibrium stress-temperature behavior of fluororubber. In ultrasonic propagation, \(\eta_{0}\) is used to characterize the thermal conduction absorption component of fluororubber, \(\eta_{1}\) represents the viscosity coefficient of fluororubber, \(\tau\) is used to describe the relaxation time, \(k_{1}\) is used to describe the elastic stress resulting from the crosslinking degree and network integrity of the macromolecular chains within the rubber, \(k_{2}\) represents the effect of filler particles in rubber on the elastic modulus, and \(k_{2}/(k_{1}+k_{2})\) represents the reinforcement factor of the filler. According to the equation of \(K_{s}\), it is known that when the angular frequency \(\omega\) is sufficiently large, \((k_{1}+k_{2})\) represents the magnitude of the stiffness coefficient and can be used to describe the reference sound velocity. From Equation (6), it can be seen that the sound velocity depends on the frequency, which explains the phenomenon of ultrasonic dispersion in fluororubber. Therefore, the model can handle viscoelastic problems such as creep and relaxation, and it is also capable of explaining characteristics such as ultrasonic attenuation and frequency shift. Equations (1)-(7) unify acoustic parameters such as sound velocity and sound attenuation with the mechanical properties. This forms the fundamental principle through whichthis paper achieved the characterization of the mechanical properties of fluororubber using ultrasonic measurement methods.