## 2 Underwater Acoustic Channel Transmission Model  
Underwater communication based on sonar without exception should use the research results of underwater acoustic channels as its solid foundation. Therefore, to achieve the desired effect of advanced underwater acoustic signal processing, a deep grasp of the effects of underwater acoustic channels on the attenuation, distortion, and fluctuation of underwater acoustic signals to achieve channel matching is necessary [29]. Some physical effects in the hydroacoustic channel can have a large impact on the communication performance, such as Rayleigh and Rice fading, time delay expansion, and Doppler expansion. There are many factors that produce these effects, and the regularity is complex; these are the elements that need extra attention when the hydroacoustic channel is simulated.  
The process of sound wave propagation is actually a fluctuation process which can be described by the fluctuation equation. The fluctuation equation is as follows [30]:  
\[\Delta^{2}\mathrm{P}=\frac{1}{\mathrm{C}^{2}}\frac{\partial^{2}\mathrm{P}}{ \partial\mathrm{t}^{2}} \tag{1}\]  
\(\Delta^{2}\) is the Laplace operator, \(\mathrm{P}\) is the sound pressure, and \(\mathrm{C}\) is the speed of sound. The propagation process of sound waves is described by the sound field model; sound field analysis essentially solves the fluctuation equation to finally obtain the law of sound wave propagation. Because this fluctuation equation is for an ideal homogeneous medium with small-amplitude acoustic waves, in the actual ocean channel, the approximate solution is generally derived based on different classes of sound field models.  
Commonly used sound field models are as follows: (1) Ray theory model: The sound waves are used to propagate the sound energy. The sound waves from the sound source follow different paths to the receiving point, so the received sound energy is the superposition of all the sound waves. (2) Multipath expansion model: The whole channel is considered to be a network system in which the sound source and the receiver are in a stationary state, and the sound velocity distribution is vertically stratified. (3) Normal mode model: When a normal mode is used to describe sound propagation, each eigenfunction corresponds to one solution of the fluctuation equation. The advantages are high accuracy and low computation for low-frequency applications of the model. The remaining models are (4) the fast field model and (5) the parabolic model.  
The transmitted signal will undergo multiple reflections in seawater and finally reach the receiver. Figure 1 shows the acoustic wave transmission model, which is based on the ray theory model, demonstrating several different paths of acoustic wave propagation: the direct path and the propagation path that undergoes multiple reflections. Moreover, these propagation paths are the key to calculating the final received signal.  
However, the channel is complex and variable, not only with ocean noise, but also with the attenuation, propagation delay, and Doppler effect. Therefore, the simulation of a hydroacoustic channel needs to consider various factors. The acoustic propagation model is as follows:  
Figure 2 shows the intrinsic path model schematic. The transmission signal is subject to a different attenuation and propagation delay on different propagation paths. The interference received by the signal in the channel, such as delay attenuation and the Doppler effect, is simulated as a random component, taken as obeying a Gaussian distribution, and  
Figure 1: The acoustic propagation model.  
superimposed with the principal component, and it finally constitutes the channel output. Table 1 summarizes the main notation that is used in this paper.