## 3 Calculation Results and Discussions
Previously reported measured mechanical properties of 7% Agar are used in the model, instead of the properties from Table 1. We selected a data set for 7% Agar silica gel presented in experiments by [PERSON] [13], which were carried out via the simulation calculation with the method presented in this paper, and chose the group velocity and phase velocity of the L (0,1) mode at 900 Hz for comparison. After sorting according to [PERSON]'s modulus, the data of the phase velocity and group velocity obtained by different methods are shown in Figure 5.  
From the comparative results shown in Figure 5, it can be seen that the proposed method can accurately obtain the phase velocity and group velocity of the L (0,1) mode at 900 Hz under the condition of the vascular mechanical properties selected in [10]. It is found that the propagation characteristics at the frequency of 900 Hz are seriously affected by the geometric shape. In contrast, the thin-layer-based method has a severe deviation in the low-frequency area, so the phase velocity obtained by the thin-layer-based method in [15] is very low. The method in this paper again reflects the advantages of the analysis based on the hollow cylinder model over that which is based on the thin-layer soft-plate model. Figure 4: Comparison of phase velocity (**a**) and group velocity (**b**) dispersion-curve results between the proposed method and the thin-layer-based method in [15]. Figure 5: Dispersion curves of phase velocity (**a**) and group velocity (**b**) of L (0,1) mode at 900 Hz between the proposed method, the thin-layer-based method in [15], and the actual experiment. Note: The experimental data were from the measurements results for the urethane tube in [13]. In addition to the thin-plate-based model mentioned above, the global matrix method is also an effective method to solve the conduction properties of isotropic tubes. However, the global matrix method sometimes failed to trace a curve (the cross between adjacent branches) for the complicated material, such as a vascular model with relatively low Young modulus values. A comparison of the proposed method and global matrix method in mode tracing is clearly shown in Figure 6 for the arterial vessel model with characteristics in Figure 3.0 The curve of the global matrix method comes from Disperse, published by Imperial College London [21]. The most important thing shown in Figure 6 is that the proposed Legendre-polynomial-based method could avoid tracing a wrong mode, so that it can reduce or eliminate the mode-jumping (mode-crossing) phenomenon, thus making numerical algorithms much more robust and efficient in comparison to the global-matrix method. After the effectiveness of this method being verified, other characteristics of guided wave propagation in arterial vessels can be explored by using this method. The first is the influence of [PERSON]'s modulus of blood vessels on the dispersion curve. Figure 7 shows the dispersion curves (red) in the blood vessel with mechanical properties shown in Table 1, together with the corresponding dispersion curves of [PERSON]'s modulus of 80% (400 kPa) and 120% (600 kPa). According to the dispersion curves shown in Figure 7, the phase velocity of each mode of the same frequency increases, and all modes of the guided wave, together with the cutoff frequency, move towards high frequency when [PERSON]'s modulus increases. For the L (0,1) mode, the maximum of the phase velocity for low frequencies also rise with the increase of frequency. Then, to study the relationship between [PERSON]'s modulus and the propagation characteristics, three frequencies, 1 kHz, 2 kHz, and 5 kHz, were selected for scanning to find out the sensitivities of different frequencies to [PERSON]'s modulus of blood vessels.