# Wavenumber-Frequency Analysis of Internal Aerodynamic Noise in Constriction-Expansion Pipe
## 2 Theoretical Background
### Wavenumber-Frequency Analysis for Internal Aerodynamic Noise  
AIV is known to originate with the high acoustic energy that is generated by pressure-reducing devices, which causes pipe shell vibration and thus induces excessive dynamic stress. Acoustic energy propagates at the speed of sound, i.e., the energy travels at the same speed as the compressible pressure waves. The total pressure fluctuation induced by valve flow in a pipe consists of compressible and incompressible pressure fluctuations. The incompressible part is the hydrodynamic pressure of a turbulent flow field with a relatively high energy content, whereas the compressible pressure fluctuation propagates at the speed of sound, which is determined by the compressibility of the fluid. The incompressible pressure fluctuation propagates at the convection speed of the mean flow. Therefore, it is essential to separate the compressible and incompressible acoustic pressure fluctuations to accurately assess the AIV. This can be achieved using a wavenumber-frequency analysis, which is described in detail below [13].  
Because the cross-sectional shape of a pipe is circular, it is better to perform a wavenumber-frequency analysis in a cylindrical coordinate system. The analytical solution of the three-dimensional (3D) wave equation in a circular duct can be written in the following form:  
\[p(r,\theta,z,t)=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}J_{m}(k_{r,m,n}r)e^{im \theta}e^{i\omega t}\times\left\{C_{+,m,n}e^{-jk_{z,m,n}^{+}z}+C_{-,m,n}e^{+jk_ {z,m,n}^{-}z}\right\} \tag{6}\]  
where \(r\), \(\theta\), and \(z\) are the coordinates in the radial, circumferential, and axial directions, respectively, and \(t\) is time. \(J_{m}\) is the 1 st kind of Bessel function of the \(m\)-th order in the circumferential direction and \(\omega\) is the angular frequency, \(k_{z}\) is the wavenumber in \(z\)-direction. \(C\) is the unknown constant and the subscript \(+\) and \(-\) mean the upstream and downstream parts, respectively.  
Using the Fourier transform for the \(z\)-direction propagated wave, the spectral power density is expressed as follows:  
\[S(m,k_{z},f)=\sum_{n=0}^{\infty}\frac{1}{J_{m}(k_{r,m,n}r_{0})}\int_{t=-\infty }^{\infty}\int_{z=-\infty}^{\infty}\int_{\theta=0}^{2\pi}p(r_{0},\theta,z,t) e^{-i(\omega t+m\theta-k_{z}z)}d\theta dzdt \tag{7}\]where \(k_{r,m,n}\) is the radial wavenumber of the \(m\)-th circumferential mode and \(n\)-th radial mode, and \(k_{z}\) is the wavenumber in the axial direction. The dispersion relation of the internal acoustic waves to a constant mean flow of Mach number \(M\) can be expressed in the following form:  
\[k_{z,m,n}^{2}+k_{r,m,n}^{2}=\left(k_{0}+Mk_{z,m,n}\right)^{2} \tag{8}\]  
Thus, the axial wavenumber in the positive and negative axial directions can expressed as follows:  
\[k_{z,m,n}^{\pm}=\frac{\mp Mk_{0}+\left[k_{0}^{2}-\left(1-M^{2}\right)k_{r,m,n }^{2}\right]^{1/2}}{1-M^{2}} \tag{9}\]  
Because the acoustic modes affecting the pipe wall vibration are the \(\theta\) and \(z\) modes, the Fourier transform is considered only in the \(\theta\) and \(z\) directions, whereas the coordinate in the \(r\) direction is treated as a constant. For the cut-on higher-order mode, the following relation must be satisfied.  
\[k_{0}^{2}-\left(1-M^{2}\right)k_{r,m,n}^{2}>0 \tag{10}\]  
In this paper, the discrete Fourier transform is performed using the Hanning window, which can be written in the following form:  
\[S(m,p\Delta k_{z},o\Delta f)=\frac{1}{N_{t}N_{z}N_{\theta}}\sum_{n=0}^{\infty }\frac{1}{J_{m}(k_{r_{0},m,n}r_{0})}\sum_{k=1}^{N_{t}}\sum_{l=0}^{1}\sum_{j=1 }^{N_{t}}w_{klj}p_{klj}e^{2\pi i\left(\frac{b_{t}}{N_{t}}+\frac{m\theta}{N_{ \theta}}\right)\frac{l\gamma}{N_{z}}} \tag{11}\]  
\[w_{klj}=\frac{1}{2}\left[1-\cos\left(\frac{2\pi\sqrt{2^{l}+l^{2}+k^{2}}}{\sqrt {\left(N_{\theta}-1\right)^{2}+\left(N_{z}-1\right)^{2}+\left(N_{t}-1\right)^ {2}}}\right)\right]:\text{Hanning window}\]