# Enhancing Impact Localization from Fluid-Pipe Coupled Vibration under Noisy Environment
## 2 Materials and Methods
### Fluid-Cylindrical Structure Coupled Wave Propagation  
The characteristics are not the same as those of waves propagating through fluidic media in a free field. The interaction between the cylindrical shell and liquid makes the propagation through pipe structures distinctive. The vibration mode of a cross-section is closely associated with propagation along an axial direction. In other words, the wavenumber characteristics of the cross-section is related to the wavenumber in the axial direction, and the wavenumbers of a cross-section inside the radiation circle can make a relevant wavenumber for the axial direction propagate well through the shell of a cylinder [11]. Some of these wavenumbers have been confirmed by wavenumber spectra obtained from pipe surface acceleration measurements [12]. Much work has been done on energy distribution among the various wave types and they suggest that, well below the pipe-ring frequency, four wave types are responsible for most of the energy transfer: three axisymmetric waves (n = 0) and a wave (n = 1) related to beam bending [9; 12]. Of the n = 0 waves, the first (termed s = 1) is a predominantly fluid-borne wave [32] and involves motion of the shell and fluid. The first wave (s = 1) is explicitly appropriate for measuring an impact signal because it is fluid-based and has lower attenuation than the others.  
The wave speed of the first wave is defined as shown below:  
\[c=c_{l}\left[1+\frac{2\frac{B_{f}}{a}}{\frac{Et_{l}}{a^{2}}-ph\omega^{2}}\right]^{ -1/2} \tag{1}\]  
where \(c_{l}(=\sqrt{B_{f}/\rho_{f}})\) is the longitudinal wave speed in free space, \(B_{f}\) is the fluid bulk modulus, \(\rho_{f}\) is the density of the fluid, \(a\) is the mean radius, \(E\) is [PERSON]'s modulus of the cylinder, \(h\) is the thickness of the cylinder, \(\rho\) is the material density of the cylinder, and \(\omega\) stands for angular frequency. Equation (1) shows that the wave speed is less than the longitudinal wave of the liquid in free space. This is because of the interaction between an elastic material and an incompressible liquid.  
Wave speeds for a steel pipe of three typical nominal diameters with respect to frequency are given in Figure 1. The material properties were chosen as the nominal values of steel and water and are listed in Table 1. The thicknesses were determined by including coating layers in and out of the cylinder. As the diameter increases, the maximum speed around zero frequency goes down, and the frequency that can propagate through the pipe also decreases. The maximum speed obtained for a pipe with 1200 mm nominal diameter at 0 Hz is about 1050 m/s. It is noteworthy that the wave that propagates in a larger pipeline has lower frequency components and is highly dispersive above a certain higher cut-off frequency. It is, therefore, absolutely necessary to make use of an appropriate non-dispersive frequency band to localize the impact source well, using the coupled wave. Moreover, as shown in Figure 1, the coupled wave is always less dispersive at a low frequency. Furthermore, the frequency characteristics of the piping system have to be taken into account to improve the detectability of the coupled wave because every system has its own frequency characteristics. These play the role of a filter that causes enhancement or reduction of the wave energy with respect to the frequency features.