## 2 3D Analytical Solutions to the Wave Equations of a RDGC
### 3D Solutions for a RDGC  
A RDGC with either expanding (positive \(\theta\)) or shrinking (negative \(\theta\)) sections is shown in Figure 3. \(\theta\) is the angle between the bevel edge and the \(z\) axis. \(b_{i}\) and \(b_{0}\) are the widths of the inlet and the outlet, respectively.  
Figure 3: (**a**) A rectangular duct with expanding sections, (**b**) a rectangular duct with shrinking sections.  
Figure 2: A straight rectangular duct.  
Since the cross-sectional area \(S\) changes along the \(z\) direction, the 1D sound wave equation in the \(z\) direction is obtained with modifying Equation (1) as  
\[\frac{1}{S}\frac{\partial}{\partial z}\left(S\frac{\partial p}{\partial z} \right)=\frac{1}{c^{2}}\frac{\partial^{2}p}{\partial t^{2}}. \tag{6}\]  
For an expanding duct, \(S\) is given by  
\[S=(z\tan\theta+b_{i})h \tag{7}\]  
Substituting Equation (7) into Equation (6) yields a homogeneous second-order variable coefficient differential equation as follows:  
\[\frac{\partial^{2}p}{\partial z^{2}}+\frac{\tan\theta}{z\tan\theta+b_{i}} \frac{\partial p}{\partial z}=\frac{1}{c^{2}}\frac{\partial^{2}p}{\partial t^ {2}}. \tag{8}\]  
The solution of Equation (8) is given by [26]  
\[p(z,t)=\{B_{+}J_{0}(k_{z}\alpha)+B_{-}J_{0}(-k_{z}\alpha)+C_{+}K_{0}(-jk_{z} \alpha)+C_{-}K_{0}(jk_{z}\alpha)\}e^{i\omega t} \tag{9}\]  
where \(\alpha=z+b_{i}/\tan\theta\). The quantities \(B_{+}\), \(B_{-}\), \(C_{+}\) and \(C_{-}\) are the corresponding amplitudes. \(J_{0}(.)\) is the zeroth order of the Bessel function of the first kind, and \(K_{0}(.)\) is the zeroth order of the modified Bessel function of the second kind [27]. When \(k_{z}\alpha\) becomes imaginary, the values of \(J_{0}(k_{z}\alpha)\) and \(J_{0}(-k_{z}\alpha)\) are possible to be infinite. To converge the solution, \(B_{+}\) and \(B_{-}\) are set to zero. As a result, Equation (9) is simplified as  
\[p(z,t)=\{C_{+}K_{0}(-jk_{z}\alpha)+C_{-}K_{0}(jk_{z}\alpha)\}e^{i\omega t}. \tag{10}\]  
Substituting Equation (10) into Equation (5), with \(b\) replaced by \(b_{i}+z\tan\theta\), the 3D solution of an expanding RDGC is derived as  
\[p(x,y,z,t)=\sum\limits_{m=0}^{\infty}\sum\limits_{n=0}^{\infty}\cos\left( \frac{m\pi x}{\alpha\tan\theta}\right)\cos\left(\frac{n\pi y}{h}\right)\{C_{+,mn}K_{0}(-jk_{z,mn}\alpha)+C_{-,mn}K_{0}(jk_{z,mn}\alpha)\}e^{i\omega t} \tag{11}\]  
\[k_{z,mn}=\sqrt{k^{2}-\left(\frac{m\pi}{\alpha\tan\theta}\right)^{2}-\left(n \pi/h\right)^{2}} \tag{12}\]  
where \(C_{+,mn}\) and \(C_{-,mn}\) are the coefficients to be determined with boundary conditions.  
Solve the momentum equation [25]  
\[\frac{\partial v}{\partial t}=-\frac{1}{\rho_{0}}\
abla\cdot p \tag{13}\]  
to give the particle velocity \(v\) of an expanding RDGC as follows:  
\[\begin{array}{l}v(x,y,z,t)=-\frac{1}{\rho_{0}\omega}\sum\limits_{n=0}^{ \infty}\sum\limits_{m=0}^{\infty}k_{z,mn}\cos\left(\frac{m\pi x}{\alpha\tan \theta}\right)\cos\left(\frac{n\pi y}{h}\right)\{C_{+,mn}K_{1}(-jk_{z,mn} \alpha)-C_{-,mn}K_{1}(jk_{z,mn}\alpha)\}e^{i\omega t}\\ +\underbrace{\left(-\frac{1}{j\rho_{0}\omega}\right)\sum\limits_{m=0}^{\infty }\sum\limits_{n=0}^{\infty}\sum\limits_{m=0}^{\infty}\frac{m\pi x}{\alpha^{2 }\tan\theta}\sin\left(\frac{m\pi x}{\alpha\tan\theta}\right)\cos\left(\frac{n \pi y}{h}\right)\{C_{+,mn}K_{0}(-jk_{z,mn}\alpha)+C_{-,mn}K_{0}(jk_{z,mn} \alpha)\}e^{i\omega t}}_{X}\end{array} \tag{14}\]  
where \(\rho_{0}\) is the ambient air density and \(K_{1}(.)\) is the first order of the modified Bessel function of the second kind.  
The derivation of the solutions for a shrinking RDGC is similar to that of an expanding RDGC and gives the same equations of Equations (11) and (14) with a negative \(\theta\).