## 1 Introduction  
There are several places where bubbly liquids exist in nature and industrial settings, including near-bottom and subsurface media in the ocean, mixed oil-gas-water media in petroleum reservoirs, and even within the human body, such as bubble-containing blood when divers experience decompression sickness [1; 2; 3; 4]. Trace bubbles in liquids can lead to dramatic changes in the propagation characteristics of acoustic waves, and thus acoustic methods have the applicable potential for bubble detection, especially when the gas content is relatively tiny [5; 6]. When using acoustic methods to detect the bubbles of decompression sickness, human tissues can also be approximated as a stratified medium if the acoustic wavelength is small enough. Therefore, the bubbly liquids mentioned above can be reduced to a stratified medium model to be processed and discussed. The propagation of acoustic waves in complex, layered media has been widely investigated for many years, and layered media containing bubbly liquid have also attractedmuch attention [7; 8; 9]. Research on the reflection and transmission of acoustic waves in these complex media is of great significance. Accurate measurements of their gas content can be applied for hydroacoustic physics, hydrocarbon resources exploration, and the fast diagnosis of decompression sickness [1; 2; 3]. To study the propagation of sound waves in multilayer media containing bubbly liquid, we first must understand the acoustical properties of gas-liquid media. [PERSON] [10] proposed that the speed of sound in bubbly liquid is equal to that of a homogeneous medium with the same density and elastic properties. [PERSON] et al. [11; 12] considered the nonlinear effect of bubbles and calculated the scattering of acoustic waves by bubbles. [PERSON] [13] further derived a theoretical model of sound propagation in bubbly liquid and obtained a formula for the speed of sound that is only affected by the gas content. [PERSON] et al. [14] established a linearized model of plane waves through gas-liquid mixtures. They investigated the compressibility of individual bubbles under the influence of pressure pulses and explored the impact of different kinetic effects on the speed. [PERSON] et al. [15; 16] constructed an effective model for the propagation of acoustic waves in bubbly liquid and derived an effective sound speed equation. [PERSON] et al. [17; 18] rigorously deduced a nonlinear acoustic wave propagation model in gas-liquid media, which used the asymptotic method to derive an approximate analytical solution for the sound speed under nonlinear oscillations. In recent years, [PERSON] [19] compared different theoretical models in the low-frequency range. [PERSON] et al. [20; 21] performed simulations for linear and nonlinear propagation models of acoustic waves in bubbly liquids and obtained the sound speed curves and attenuation coefficient curves under different parameters. [PERSON] et al. [22; 23; 24] numerically simulated the propagation of acoustic waves in bubbly liquids and analyzed their nonlinear characteristics. [PERSON] et al. [25; 26] experimentally analyzed the acoustical characterization of microbubble clouds. The acoustic properties of bubbly liquid have been extensively studied for many years. Based on the deep understanding of the acoustic properties in gas-liquid media, the laws of sound propagation in multilayer media containing a bubbly liquid layer have also received much attention [7; 8; 9]. [PERSON] [27] and [PERSON] et al. [28] investigated the differential frequency sound generated by the interaction of two high-frequency harmonics in a bubbly liquid layer based on the strong nonlinearity of bubbles. [PERSON] et al. [29] conducted experimental measurements of the scattering intensity of the bubble-containing layer. For the case of oblique incidence, [PERSON] and [PERSON] [7] examined the reflection and refraction of acoustic waves at the interface between pure water and bubbly water. [PERSON] et al.