## 3 Results and Discussion
### Results for Attenuation  
Porous medium is dissipative in nature, and thus, the wave number \(k\) is complex. The waves generated obey a diffusion type process and therefore become attenuated. The imaginary part of the wave number corresponds to attenuation of waves [47]. A preliminary study of the effect of porosity on the dispersion characteristics and time-domain waveform was conducted by investigating the behavior of a liquid-filled pipe embedded in porous medium. There is a certain relationship between porosity and pipeline leakage. The results of this study are therefore presented to illustrate the nature of the attenuation, rather than as a rigorous validation. Nevertheless, fairly good agreement between the prediction and simulation was achieved.  
Attenuation as a function of frequency is depicted in Figure 11 for a water-filled pipe embedded in porous medium with different values of porosity, i.e., \(\beta\) = 0.1, 0.2, and 0.3. In Figure 11, we plot the attenuation curves of the \(\alpha\) and L(0,1) modes. Figure 11a shows that the attenuation of \(\alpha\) mode is zero; however, its mode shape is not suited to detection. At low frequencies, its axial displacement is dominant, but the actual set-up is not appropriate for imposing a displacement in the outer medium. At higher frequencies, its axial and radial displacements decrease exponentially from the outer wall, and again, the detection is not suitable to generate this mode. [PERSON] and [PERSON], in their study of fluid-filled pipes surrounded by different fluids [12], similarly found and discussed that the attenuation curve is zero. The attenuation of L(0,1) increases for higher values of frequency, first rapidly then almost linearly with frequency. The attenuation trends are practically identical for the different values of the porosity. Again, there is very good agreement between the dispersion characteristics and the attenuation. As investigated by [PERSON] et al. [48], for a fluid-filled pipe surrounded by a fluid medium, the attenuation increases as a result of the added radiation damping. As shown in Figure 11a, the radiation damping effect increases with porosity (the water in the porous medium increases with the porosity). At very low frequencies (below 200 Hz), this effect is relatively small, whereas it increases distinguishably at higher frequencies relative to the value at low porosity. A similar phenomenon is observed in the case of the water-filled pipe embedded in finite porous medium from Figure 11b. It can be seen that a finite medium reduces radiation damping to the pipe wall, causing smaller losses [49]. It is observed that the attenuation in the water-filled pipe embedded in a finite porous medium decreases significantly compared to that in an infinite medium,  
Figure 10: Time-domain waveforms with respect to time for various thicknesses of porous medium layer (\(\beta\) = 0.1). Black, red and blue lines represent layer thickness \(d\) = 0.025, 0.05, and 0.10 m, respectively.  
this value being 1.79 dB/m for a finite porous medium and 103.17 dB/m for an infinite porous medium at 0.045 MHz (\(\beta=0.1\)).  
By comparing Figure 11a,b, it can be seen that the attenuation taking place in a water-filled pipe when it is embedded in a finite porous medium is significantly less than that in a water-filled pipe embedded in an infinite porous medium. In both the cases, attenuation increases with porosity and wave frequency. This indicates that, in practice, a pipeline leak can be indirectly reflected by the value of porosity.