## 2 Theory
### Acoustic Reflection of Bubbly Liquid in Multilayer Media  
Sound waves are incident on the boundary surface of the two media, due to the difference in acoustic impedance, sound waves are reflected and transmitted. The boundary conditions of acoustic pressure continuity and normal velocity continuity are satisfied on the boundary surface. The equations are as follows [7]:  
\[p_{in}+p_{rn}=p_{i(n+1)}+p_{r(n+1)} \tag{21}\]  
\[v_{in}+v_{rn}=v_{i(n+1)}+v_{r(n+1)} \tag{22}\]Based on the boundary conditions and the acoustic properties of the liquid containing bubbles, the reflection coefficient of the acoustic wave in the multilayer medium containing the bubbly liquid layer can be calculated. The reflection coefficient \(R\) of the vertical incidence on the infinite interface of pure liquid and bubbly liquid is:  
\[R=\frac{Z_{2}-Z_{1}}{Z_{2}+Z_{1}}=\frac{\sqrt{\frac{\rho_{2}(1+\beta)^{2}}{(\rho _{1}+\beta\rho_{2})\beta}}(\rho_{1}+\beta\rho_{2})c_{2}-(1+\beta)\rho_{1}c_{1}}{ \sqrt{\frac{\rho_{2}(1+\beta)^{2}}{(\rho_{1}+\beta\rho_{2})\beta}}(\rho_{1}+ \beta\rho_{2})c_{2}+(1+\beta)\rho_{1}c_{1}} \tag{23}\]  
\(Z_{1}\) and \(Z_{2}\) are the acoustic impedance of pure liquid and bubbly liquid, respectively. The constants employed in the calculation are \(\rho_{1}\) = 1000 kg/m\({}^{3}\), \(c_{1}\) = 1540 m/s, \(\rho_{2}\) = 1.29 kg/m\({}^{3}\), and \(c_{2}\) = 340 m/s. Figure 1 shows the reflection coefficient as a function of gas content when the plane acoustic wave is incident vertically to the interface of water-bubbly water. It can be seen that the reflection coefficient tends to increase rapidly in the range of gas content from 0.05% to 1%, with a difference of about 0.35 between the maximum and minimum values. While the variation of the reflection coefficient is no longer severe after the gas content is greater than 2%. It is concluded that the change of reflection coefficient is very sensitive when the gas content is small. However, as the gas content further increases, the curve of reflection coefficient gradually flattens out and the change with the rise of gas content is no longer noticeable. Human tissue containing bubbles can be approximated as a stratified medium containing bubbly liquid. The reflection and transmission of sound waves become very complex due to the sound propagation properties of gas-liquid mixed media. Considering the physical model shown in Figure 2, the three layers of media are water-bubbly water-water and bubbly liquid layer thickness d is 0.004 m. In this model (see Figure 2), \(Z_{1}\) and \(Z_{2}\) are the acoustic impedance of pure liquid and bubbly liquid, respectively, and the reflection coefficient \(R\) for vertical incidence is derived as:  
\[R=\frac{Z_{2}^{2}-Z_{1}^{2}}{Z_{2}^{2}+Z_{1}^{2}-2 jZ_{1}Z_{2}\cot(\frac{2\pi fd }{c_{2}})} \tag{24}\]  
Figure 1: Reflection coefficient vs. gas content at the vertical incidence of acoustic waves at the infinity interface of liquid-bubbly liquid. \(f\) is the frequency of the incident acoustic wave. Then, the reflection coefficient is obtained by substituting Equations (17)-(24):  
\[R=\frac{((\rho_{1}^{2}c_{1}^{2}-\rho_{2}^{2}c_{2}^{2})\beta-c_{2}^{2}\rho_{1} \rho_{2})(1+\beta)}{2\sqrt{\frac{\rho_{2}(1+\beta)^{2}}{(\rho_{1}+\beta\rho_{2 })\beta}}\beta c_{1}c_{2}j\rho_{1}(\rho_{1}+\beta\rho_{2})\cot(\frac{2\pi f \delta}{\sqrt{\frac{\rho_{2}(1+\beta)^{2}}{(\rho_{1}+\beta\rho_{2})\delta^{2}}} })-((\rho_{1}^{2}c_{1}^{2}+\rho_{2}^{2}c_{2}^{2})\beta+c_{2}^{2}\rho_{1}\rho_{2 })(1+\beta)} \tag{25}\]  
Figure 3 shows the three reflection coefficient curves for the gas content being 0.05%, 0.25%, and 0.5%. It can be seen from Figure 3 that the changes of reflection coefficients are all periodic and undulating. There are a series of resonance peaks and valleys in the frequency range. At the valley frequencies, the incident acoustic energy is almost completely transmitted, while the reflection coefficient tends to zero. The magnitudes of the reflection coefficients at different gas content are different.