## 2 Governing Equations
### Solution of the Compressive Fluid  
It can be assumed that the velocity can be a harmonic function with a frequency of \(\omega\), so it can be written as:  
\[\frac{\partial^{2}u}{\partial t^{2}}=-\omega^{2}u \tag{8}\]  
Meanwhile, based on the physical property of the fluid, the density change is very small, but the rate of the density with respect to time or respect to space may not be small. As a matter of fact, the smaller the change of the density, the larger the change of the pressure according to the state equation of fluid. Therefore, in this paper, the density is assumed to be a constant value, \(\rho_{o}\), and the rate of the density with respect to time or to space are variables. From the second term of Equation (2), it can be transformed to Equation (9):  
\[\Delta\rho=\rho-\rho_{0}=-\frac{\rho_{o}}{c^{2}}\int_{r}^{\infty}\frac{ \partial u}{\partial t}dr+\frac{\rho_{o}}{2}\frac{u^{2}}{c^{2}} \tag{9}\]  
From Equation (3), it can be seen that the changes in density or pressure come from two sources: The change of the velocity with respect to time or to space. By neglecting the second term of Equation (3) and substituting another term into the continuous equation, the following equation can be obtained (Equation (10)):  
\[-\frac{\rho_{o}}{c^{2}}\int_{r}^{\infty}\frac{\partial^{2}u}{\partial t^{2}} dr+\rho_{o}\frac{\partial u}{\partial r}+\frac{\rho_{o}u}{r}=0 \tag{10}\]  
Figure 1: The schematic states of the vibration of the cylindrical shell.  
Afterwards, Equation (3) can be transformed after being divided by the density, \(\rho_{o}\):  
\[-\frac{\omega^{2}}{c^{2}}\int_{r}^{\infty}udr+\frac{\partial u}{\partial r}+\frac {u}{r}=0 \tag{11}\]  
Furthermore, the equation can be simplified by taking the derivative on both sides of Equation (11):  
\[\frac{\partial^{2}u}{\partial r^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}+ (\frac{\omega^{2}}{c^{2}}-\frac{1}{r^{2}})u=0 \tag{12}\]  
By letting \(s=\omega\cdot r/c\), the above equation can be simplified as Equation (13), finally:  
\[s^{2}\frac{\partial^{2}u}{\partial s^{2}}+s\frac{\partial u}{\partial s}+(s^{ 2}-1)u=0 \tag{13}\]  
Equation (13) is the classical Bessel function [38; 39]; the general solution can be derived as:  
\[u=A\cdot J_{1}(s)+B\cdot Y_{1}(s) \tag{14}\]  
where \(A\), \(B\) is an arbitrary function with respect to time. \(J_{1}(s)\), \(Y_{1}(s)\) is the first and second order Bessel function, respectively. The coefficient of \(A\), \(B\) can be confirmed via the boundary condition:  
\[\left\{\begin{array}{l}\left.u_{s}(t)\right|_{r=r_{s}}=A\cdot J_{1}(s_{r_{s }})+B\cdot Y_{1}(s_{r_{s}})\\ 0=A\cdot J_{1}(s_{R})+B\cdot Y_{1}(s_{R})\end{array}\right. \tag{15}\]  
If the radius of the boundary is \(R\rightarrow\infty\), the second boundary condition will be satisfied naturally as \(J_{1}(s_{R})\), \(Y_{1}(s_{R})\) tends to 0. In this case, it is reasonable to assume \(u\) as a harmonic function with a frequency of \(\omega\) as well, so that:  
\[A=C\cdot\cos(\omega t+\varphi),\ B=C\cdot\sin(\omega t+\varphi) \tag{16}\]  
where, \(C\), \(\varphi\) are the coefficients which express the amplitude and phase angle, respectively. By considering the velocity as a harmonic function, \(u_{s}(t)=D\cdot\sin(\omega t)\) can be obtained, where \(D\) is the amplitude of the vibration. Additionally, the boundary condition can be written as Equation (17) below by submitting Equation (16) to Equation (15). The coefficients of \(C\) and \(\varphi\) can thus be confirmed:  
\[C\cdot(\cos(\omega t+\varphi)J_{1}(s_{r})+\sin(\omega t+\varphi))Y_{1}(s_{r} )=D\cdot\sin(\omega t) \tag{17}\]