# Research on the Impact of a Fluid Field on an Acoustic Field in Herschel-Quincke Tube
## 2 Governing Equations and Solution Method  
### Governing Equations  
#### 2.1.1 Acoustic Formulation  
With the continuous growth of computing resources, researchers are able to study the phenomenon of flow-induced noise or sound propagation effects related to any velocity field. To describe the physical process of the influence of any velocity field on sound propagation, [PERSON] employed an Euler-Lagrangian mixed framework to describe fluid dynamics disturbances. Although few people know about this method, the Galrun equation provided the original framework, and many researchers have since further developed this equation. Aeroacoustics mainly involves the generation and propagation of sound in moving media and there are many publications about it. The key idea of this research method is to re-derive the Navier-Stokes equation to obtain the wave operator of a stationary fluid as an equivalent source on the right side of the equation. These sound sources generate an equivalent sound field in a stationary fluid. In a sense, this method describes the sound source and provides accurate analogies. The physical properties of sound sources in fluids have not been studied deeply [23]. The Galrun equation can be used to describe the evolution of Lagrangian displacement and study the acoustic with flow. It is a second-order partial differential equation derived from a first-order system of partial differential equations:  
\[ \begin{cases}\dfrac{d\rho}{dt}+\rho\
abla\cdot\mathbf{v}=0\\ \rho\dfrac{d\mathbf{v}}{dt}+\
abla p=\mathbf{0}\\ \dfrac{d\rho}{dt}-c^{2}\dfrac{d\rho}{dt}=0,\end{cases} \]  
which is called [PERSON]'s equations. \(\{\rho,\mathbf{v},p\}\) is the density, particle velocity and pressure, respectively. \(d/dt=\partial/\partial t+\mathbf{v}\cdot\
abla\) is the substantial derivative. The two terms to the right of the equal sign are both Euler descriptions, where \(\partial/\partial t\) is the local derivative and \(\mathbf{v}\cdot\
abla\) is the convective derivative. \(c\) is the speed of sound. In the case of perfect gas isentropy, the relationship is as follows:  
\[\dfrac{p}{\rho^{\gamma}}=\text{const},\qquad c^{2}=\gamma\dfrac{p}{\rho} \tag{1}\]  
where \(\gamma\) is the heat capacity ratio. The Euler perturbation of any physical quantity is represented by the exponent E and can be defined as the difference between the total disturbance \(\
abla\) and driving configuration \(\Psi_{0}\) at the geometric point \(\mathbf{x_{0}}\):  
\[\epsilon\Psi^{E}(\mathbf{x_{0}},t)=\Psi(\mathbf{x_{0}},t)-\Psi_{0}(\mathbf{x_ {0}},t) \tag{2}\]  
where \(\epsilon\) is the dimensionless coefficient that defines the disturbance amplitude. In the case of a perfect flow isentropic, the linearized Euler equation (LEE) system takes the following form:  
\[ \begin{cases}\dfrac{\text{d}_{0}p^{E}}{dt}+\mathbf{v}^{E}\cdot\
abla p_{0}+ \gamma p_{0}\
abla\cdot\mathbf{v}^{E}+\gamma p^{E}\
abla\cdot\mathbf{v_{0}}=0 \\ \dfrac{\text{d}_{0}\mathbf{v}^{E}}{dt}+\mathbf{v}^{E}\cdot\
abla\mathbf{v_{0}} +\dfrac{1}{\rho_{0}}\
abla p^{E}+\dfrac{p^{E}}{\rho_{0}c_{0}^{2}}\dfrac{\text {d}_{0}\mathbf{v_{0}}}{dt}=\mathbf{0}\end{cases}  \tag{3}\]  
where \(p^{E}\), \(\mathbf{v}^{E}\), \(\mathbf{v_{0}}\) and \(\rho_{0}\) are the Euler perturbation of pressure, the average density, the flow speed and the average density, respectively. The Galrun equation is derived from a mixed Euler-Lagrange perturbation of the Euler equation. It has the form of second-order differential equations in space and time, and is written as a Lagrange displacement:  
\[\rho_{0}\dfrac{\text{d}_{0}^{2}\mathbf{w}^{L}}{dt^{2}}-\
abla(\rho_{0}c_{0}^{2 }\
abla\cdot\mathbf{w}^{L})-(\
abla\cdot\mathbf{w}^{L})\rho_{0}\dfrac{\text{d }_{0}\mathbf{v_{0}}}{\text{d}t}-^{T}\
abla\mathbf{w}^{L}\cdot\
abla p_{0}=0 \tag{4}\]  
where \(\text{d}_{0}/dt=\partial/\partial t+(\mathbf{v_{0}}\cdot\
abla)\) is the convective derivative, and \(\mathbf{w}^{L}\) is the Lagrangian perturbation of the particle displacement. The Galrun equation can also be written in mixed form, as follows:  
\[