## 2 Theory and Derivation  
The wave equation of a perfect fluid can be decomposed into three constituent elements: the mass conservation equation, the Euler equation, and the adiabatic equation of state. Moreover, these elements encapsulate the adiabatic relationship between pressure and density, as follows [31]:  
\[\frac{\partial\rho}{\partial t}=-\
abla\cdot\rho v \tag{1}\]  
\[\frac{\partial v}{\partial t}+(v\cdot\
abla)v=-\rho^{-1}\
abla p(\rho) \tag{2}\]  
\[p=p_{0}+p^{\prime}\bigg{[}\frac{\partial p}{\partial\rho}\bigg{]}_{S}+\frac{1 }{2}\big{(}p^{\prime}\big{)}^{2}\bigg{[}\frac{\partial^{2}p}{\partial\rho^{2} }\bigg{]}_{S} \tag{3}\]  
Here,  
\[c^{2}=\bigg{[}\frac{\partial p}{\partial\rho}\bigg{]}_{S} \tag{4}\]  
where, \(\rho\) denotes the density, \(v\) denotes particle velocity, and \(p\) denotes the pressure, and \(p^{\prime}\) denotes the pressure fluctuation. The subscript \(S\) indicates the requirement to take the thermodynamic partial derivative under isentropic conditions. Combing Equations (1) and (4), the relationship can be shown as follows [31]:  
\[\frac{\partial\rho^{\prime}}{\partial t}=-\
abla\cdot(\rho_{0}v) \tag{5}\]  
\[\frac{\partial v}{\partial t}=-\frac{1}{\rho_{0}}\
abla p^{\prime}(\rho) \tag{6}\]  
\[\frac{\partial p^{\prime}}{\partial t}=c^{2}\bigg{(}\frac{\partial\rho^{ \prime}}{\partial t}+v\cdot\
abla\rho_{0}\bigg{)} \tag{7}\]  
If \(\rho_{0}\) is constant, Equation (7) can be rewritten as:  
\[p^{\prime}=\rho^{\prime}c^{2} \tag{8}\]  
With real-world conditions, the hydrological environment exhibits significant variability. Consequently, it is posited that liquid density \(\rho_{0}\) and sound velocity \(c\) are as temporally invariant as temperature. Furthermore, liquid density and sound velocity do not fluctuate over \(t\). Equation (5) and Equation (6) are combined to derive the wave equation. By altering the order of operations in the derivation and applying Equation (7), sound pressure can be written as follows:  
\[\rho\
abla\cdot\bigg{(}\frac{\
abla p}{\rho}\bigg{)}-c^{-2}\frac{\partial^{2 }p}{\partial t^{2}}=0 \tag{9}\]  
According to Equation (9), if \(\rho\) is constant in time and space, the standard sound pressure wave equation is shown as follows:  
\[\
abla^{2}p-c^{-2}\frac{\partial^{2}p}{\partial t^{2}}=0 \tag{10}\]Equations (5) and (6) can be taken to derive the wave equation of particle velocity under the unidirectional propagation of sound waves, is shown as follows [32]:  
\[\frac{\partial v}{\partial t}+(v\cdot\
abla)v=-\rho^{-1}\
abla p(\rho) \tag{2}\]0  
Equation (11) can be transformed into a scalar wave equation by introducing \(\phi\):  
\[\frac{\partial v}{\partial t}+(v\cdot\
abla)v=-\rho^{-1}\
abla p(\rho) \tag{2}\]1  
By incorporating the density invariant condition (\(\
abla\rho=0\)) in Equation (12) and the sound pressure solution process into Equation (11), the following relationship can be obtained as follows:  
\[\frac{\partial v}{\partial t}+(v\cdot\
abla)v=-\rho^{-1}\
abla p(\rho) \tag{2}\]2  
In addition, the \(\phi\) meet the following conditions:  
\[\frac{\partial v}{\partial t}+(v\cdot\
abla)v=-\rho^{-1}\
abla p(\rho) \tag{2}\]3  
Using the relationship between velocity and displacement \(v=\partial u/\partial t\), the displacement potential \(\psi\) is defined as follows:  
\[\frac{\partial v}{\partial t}+(v\cdot\
abla)v=-\rho^{-1}\
abla p(\rho) \tag{2}\]4  
Combined with Equations (10) and (14), the wave equation of the displacement potential can be written as follows:  
\[\frac{\partial v}{\partial t}+(v\cdot\
abla)v=-\rho^{-1}\
abla p(\rho) \tag{2}\]5  
The kinematic relationships of Equations (5), (7), and (15) can show the sound pressure equation in terms of displacement potential, which is as follows:  
\[\frac{\partial v}{\partial t}+(v\cdot\
abla)v=-\rho^{-1}\
abla p(\rho) \tag{2}\]6  
In Equation (17), \(K\) is defined as the bulk elastic modulus. Here:  
\[\frac{\partial v}{\partial t}+(v\cdot\
abla)v=-\rho^{-1}\
abla p(\rho) \tag{2}\]7  
In addition, Equation (18) is the basic equation for ideal linear elastic fluids.