## 2 Theory
### Acoustic Reflection of Bubbly Liquid in Multilayer Media
In the reflection coefficient curve for a given gas-content value, the frequency interval \(\Delta f\) between two adjacent valleys is constant, and \(\Delta f\propto c/d\). The value of \(\Delta f\) is proportional to the sound velocity of the mixed medium and inversely proportional to the thickness of the bubbly liquid layer. Therefore, in the case where d is known, the sound velocity of the mixed medium can be developed by measuring the value of \(\Delta f\). Based on the above frequency response curve of the reflection coefficient, the change in valley frequency interval with gas content can be calculated. Figure 4 depicts the relationship between gas concentration and the maximum reflection coefficient and frequency interval. The figure shows that when the gas concentration increases, the \(\Delta f\) drops. The value of \(\Delta f\) ranges from 67 kHz to 15 kHz in the range of 0.05-1% of the gas concentration. Additionally, the variation in \(\Delta f\) for various gas contents is clear. When the gas content is  
Figure 3: Frequency response of acoustic reflection coefficients in multilayer media containing bubbly liquid layers at different gas contents. Figure 2: Model of acoustic normal incidence in a multilayer medium containing a bubbly liquid layer. small, \(\Delta f\) is especially sensitive to the change of the gas content, and the 0.01% increase of the gas content can lead to the \(\Delta f\) about 5 kHz change. Therefore, by measuring the frequency response curve of the reflection coefficient, the value of the valley frequency spacing \(\Delta f\) can be obtained, and the gas content of the intermediate bubbly liquid layer can be further deduced. For human tissues, the relevant parameters can be substituted into a three-layer medium model containing a bubbly liquid layer for approximate calculation. The calculated reflection coefficient curve with frequency is shown in Figure 5. The model parameters employed are \(\rho_{1}\) = 1090 kg/m\({}^{3}\), \(c_{1}\) = 1590 m/s, \(\rho_{3}\) = 1200 kg/m\({}^{3}\), \(c_{3}\) = 3400 m/s, \(d\) = 0.004 m, \(\rho_{2}^{l}\) = 1050 kg/m\({}^{3}\), \(c_{2}^{l}\) = 1570 m/s, \(\rho_{2}^{S}\) = 1.29 kg/m\({}^{3}\), and \(c_{2}^{S}\) = 340 m/s. The subscripts 1, 2, and 3 represent the parameter values of the first, second and third layers respectively. The superscripts \(g\) and \(l\) represent the parameter values of gas and liquid, and \(d\) is the thickness of the second layer medium. Compared with the reflection curve of water containing gas bubbles shown in Figure 3, the change of the acoustic impedance of the medium also makes the value taken at the valley of the curve increase, and the valleys are not as prominent as in Figure 3. Nevertheless, the curves in Figures 3 and 5 show the same patterns of changes, i.e., the valley frequency spacing varies with the change of gas content in the intermediate bubbly liquid layer. Therefore, the magnitude of the gas content in the intermediate blood layer can be evaluated by the valley interval on the reflection curve. This method has potential applications in human body bubble detection. \[v_{in}+v_{rn}=v_{i(n+1)}+v_{r(n+1)} \tag{22}\]0 Variation of the maximum value of reflection coefficient and valley frequency intervals with gas content. \[v_{in}+v_{rn}=v_{i(n+1)}+v_{r(n+1)} \tag{22}\]1 Frequency response of acoustic reflection coefficients in multilayer media of human tissue containing bubbly liquid layers at different gas contents. To compare with the subsequent experimental results, acrylic plates are added on both sides of the bubbly liquid layer. The thickness of the acrylic plate is 3 mm. The specific distribution of the model is pure liquid- acrylic plate-bubbly liquid- acrylic plate-pure liquid. The parameters of the plating layer are \(\rho\) = 1300 kg/m\({}^{3}\) and \(c\) = 3000 m/s. Additionally, the density and the sound velocity of water are also substituted into the calculation.