# Dynamic Pressure Analysis of Hemispherical Shell Vibrating in Unbounded Compressible Fluid
## 2 Control Equation and Solution
### Solution on the Compressible Fluid Case  
Using the Fourier series method, the velocity can be written in a harmonic function with a frequency of \(\omega\). This implies that:  
\[\frac{\partial^{2}u}{\partial t^{2}}=-\omega^{2}u \tag{9}\]  
For the physical property of the fluid, the density changes slightly, but the change of density over time and space cannot be neglected. Thus, in this paper, the density is assumed to be a constant as \(\rho_{o}\), but the changes of density over time and space have to be considered in the solution. From Equation (2), pressure can be obtained by the integration between limits from \(r\) to the unbounded position:  
\[\Delta p=p-p_{0}=-\rho_{o}\int_{r}^{\infty}\frac{\partial u}{\partial t}dr+ \frac{\rho_{o}}{2}u^{2}, \tag{10}\]  
From Equation (10), it is found that the pressure comes from two parts: the local acceleration, which is the velocity changes with respect to time, and the convective acceleration, which is the change in velocity with respect to space. It should be noted that there is only one boundary condition with respect to the pressure as prescribed in Equation (6). Therefore, the pressure at the position of \(r\) is an unknown value. This implies that there is an arbitrary constant that needs to be determined in Equation (10). For the assumption of \(u_{s}\ll c\), the second term of Equation (10) is very small. After neglecting the convective acceleration (the second term) and substituting the local acceleration (the first term) to the continuous equation of Equation (2), it can be obtained as follows:  
\[-\frac{\omega^{2}}{c^{2}}\int_{r}^{\infty}udr+\frac{\partial u}{\partial r}+ \frac{2u}{r}=0. \tag{11}\]  
Figure 2: The state of the vibration of a hemispherical shell.  
Equation (11) can be simplified as Equation (12) below after a derivative by \(r\). Having an arbitrary constant in pressure discussed above, two boundary conditions are necessary to solve Equation (11), although it is a first order differential equation. The two boundary conditions are the same as Equation (12):  
\[\frac{\partial^{2}u}{\partial r^{2}}+\frac{2}{r}\frac{\partial u}{\partial r}+( \frac{\omega^{2}}{c^{2}}-\frac{2}{r^{2}})u=0, \tag{12}\]  
Let \(s=\frac{\omega}{c}r\), and the above equation can be expressed in terms of the classical Bessel equation as follows:  
\[s^{2}\frac{\partial^{2}u}{\partial s^{2}}+2s\frac{\partial u}{\partial s}+(s^ {2}-1\times 2)u=0. \tag{13}\]  
Equation (13) is a classical Bessel equation, so the solution can be derived as:  
\[u=A\cdot j_{1}(s)+B\cdot y_{1}(s), \tag{14}\]  
where \(A\) and \(B\) are the arbitrary functions of the time, and \(j_{1}(s)\), \(y_{1}(s)\) are the first and the second-order Bessel function, respectively. Applying the boundary conditions into Equation (14), it can be rewritten as Equation (15) shows:  
\[\left\{\begin{array}{l}A\cdot j_{1}(s)+B\cdot y_{1}(s)\big{|}_{r=r_{s}}=u_{ s}(t)\\ A\cdot j_{1}(s)+B\cdot y_{1}(s)\big{|}_{r=R->\infty}=0\end{array}\right. \tag{15}\]  
Thus, to get the exact solution, there are only \(A\) and \(B\) left to be confirmed. It should be noted that when \(r=R\rightarrow\infty\), both \(j_{1}(s_{R})\) and \(y_{1}(s_{R})\) converge to zero. Therefore, the second term of the boundary condition is satisfied naturally. In such cases, taking into account that \(u\) is a harmonic function, it can be assumed that:  
\[A=C\cdot\cos(\omega t+\varphi),B=C\cdot\sin(\omega t+\varphi). \tag{16}\]  
According to the first term of the boundary conditions, the amplitude and phase angle will be equal to that of initial vibration, so that \(A\) and \(B\) can be confirmed. In addition, it is found that the solution can be determined totally by the boundary condition \(u_{s}(t)\).