## 4 Results and Discussion
### BG of an Infinite Pipe  
In this part, the substructure length is taken as \(l_{\mathrm{A}}=l_{\mathrm{B}}=0.5\) m and the lattice constant is \(l_{\mathrm{PC}}=1\) m, if not specified. In order to validate the accuracy of SEM in calculating the periodic pipe, a literature comparison is first carried out for the band structure, as shown in Figure 2, in which the grey regions indicate the BG regions of the system. It is noted that due to lack of available research on a fluid-conveying pipe with periodically varying cross-section, the SEM herein is applied in a static-water-filled periodic pipe composed of different materials, and the results obtained are compared with those in Ref. [21] by the TMM under the same parameter values (for the specific parameter values employed, see the literature). It can be seen that the two results are in good agreement, indicating the accuracy of the present SE procedure. Moreover, it is observed that under the specified range of the incident frequency, there exist two BG regions: 50 Hz-76 Hz, 228 Hz-396 Hz, and the higher region is broader. Figure 3 depicts the imaginary part of the wave vector as function of the incident frequency for the present fluid-conveying pipe with periodically varying cross-section by the SEM and TMM. Actually, Figure 2 has indicated the fact that the BG of the pipe can  
Figure 2: Band structure of a static-water-filled periodic pipe: (**a**) real part of the wave vector, (**b**) absolute imaginary part of the wave vector. be determined by the existence of the imaginary part of \(q\). In Figure 3 and what follows, therefore, the BG regions will be measured directly by this rule. It can be seen in Figure 3 that the results of SEM and TMM are in good agreement, showing that the two methods are both effective in calculating the band structure of the periodic pipe conveying fluid. In addition, under the specified range of the incident frequency, there exist four BG regions: 47 Hz-67 Hz, 213 Hz-315 Hz, 573 Hz-623 Hz, 983 Hz-1133 Hz. The lowest BG region is the narrowest, and the highest BG is the broadest. Figure 4 illustrates the effect of the ratio of substructure outer radius on the BG of the present periodic pipe. It is clear that not any BG is present when the two segments hold the same outer radius, and it occurs only in the case of difference in radius. This demonstrates that the geometrical difference in cross-section of the two segments is the essence causing BG. In addition, it is found that the lowest BG region, BG1, is the narrowest one. With increasing the ratio, BG1 has little change, BG2 and BG3 get broader greatly, while the highest BG region, BG4, gets broader first, and then becomes narrower until vanished, showing that it is not always a monotonic evolution of BG with the ratio of substructure outer radius. Moreover, the regions of BG1, BG2 and BG4 start around the ratio \(r_{\text{o,B}}/r_{\text{o,A}}=1.05\), while that of BG3 starts around \(r_{\text{o,B}}/r_{\text{o,A}}=1.3\). Figure 5 displays the effect of the ratio of substructure length on the BG. A complicated impact of the substructure lengths can be immediately found. It is observed that with increasing the ratio, the BG regions basically move to higher frequencies, especially for higher BGs. However, the evolution of each BG region with the ratio is different. Within the specified range of the ratio, with increasing \(l_{\text{B}}/l_{\text{A}}\), BG1 becomes a little broader, BG2 experiences an evolution of 'broader-narrower', BG3 'broader-narrower-broader' and BC4 'broader-narrower-broader-narrower-broader'. BG4 holds the leading variation. Some peculiar points of ratio resulting in local contraction of the BG region are present in BG3 and BG4. It is noted that according to the reports of most related literatures, the excitation frequency acting on the present slender pipes conveying fluid is usually below 1200 Hz.