## 3 Calculation Results and Discussions  
We solve Equation (11) by Matlab and calculate the dispersion curve of guided waves in blood vessels. In all simulation operations in this paper, the Legendre polynomial number introduced is \(N=10\), and the convergence of solutions will be shown below. In the simulation process of the Matlab program, the differential process in Equation (13) is replaced by difference. Under the vascular mechanical characteristics shown in Table 1, the dispersion curves can be obtained, as shown in Figure 2:  
The longitudinal guided waves in the arterial wall can be divided into three modes: axial-symmetry longitudinal mode, axial-symmetry torsional mode, and non-axisymmetric flexural mode, which are labeled by L (0, m), T (0, m) and F (n, m) [27]. As the frequency increases, various high-order flexural modes will nearly approach the corresponding longitudinal modes [23]. It can be found in Figure 2 that there is a peak in the low-frequency region of the phase velocity of the L (0,1) mode, which can also be clearly seen in the measured data of [14]. This is because the wavelength of the guided wave is longer in the low-frequency region and is comparable to the diameter of the blood vessel. So the geometric shape of the blood vessel has an impact on the propagation characteristics of the guided wave. In addition, except for L (0,1) mode, all other modes have cutoff frequency, near which the I havephase velocity and group velocity of the corresponding mode changes significantly, and the phase velocity tends to infinity while the group velocity tends to 0. Then, the convergence of solutions under the Legendre polynomial number \(N=10\) is worth discussing. After setting the Legendre polynomial number to 2\(\sim\)10, respectively, part of the dispersion curve of the phase velocity under the mechanical property of Table 1 is shown as Figure 3.  
As shown in Figure 3a,b, there were obvious errors and zero values that are not expected in the curve. In contrast, the dispersion curves under \(N\geq 6\) tend to be stable. Considering that the frequency-thickness of the guided wave in blood vessels does not change dramatically in the proposed method, it is feasible and effective to select a parameter \(N=10\) with redundancy. Figure 2: Dispersion curves of phase velocity (**a**) and group velocity (**b**) in simulation operation. Then, to illustrate the accuracy and advantages of the proposed theoretical model, we use Matlab to conduct simulation calculations of the guided wave dispersion curves in hollow cylinders, and compare these results with those obtained by the thin-layer-based method. We use [PERSON]. [PERSON]'s method [15] to obtain the result by the equation of antisymmetric Lamb waves for thin-elastic layers and compare it with the method proposed in this paper. In the comparison experiment, the thickness of the thin-layer model is 0.2 mm, which is the same as the wall thickness of the vascular model. [PERSON]'s modulus, [PERSON]'s ratio, and density of blood vessels are shown in Table 1. The dispersion curves obtained by the thin-layer-based method in [15] are shown together with the results of our method in Figure 4.  
Figure 3: Dispersion curves of phase velocity under different Legendre polynomial numbers. When the Legendre polynomial number \(N=2\), the result does not converge and return the solution of NaNs. It can be observed from the results of Figure 4 that the L (0,1) mode in the cylindrical model approaches the antisymmetric mode A0 in the thin plate for high frequencies, which is consistent with the existing research results [28]. In addition, the method in this paper can obtain not only the L (0,1) mode but also a variety of other modes, such as T (0,1) and L (0,2). According to the comparison with the measured dispersion curve in [14], the cylindrical model in this paper can better match the results of the reported experiments, while the results provided by the two methods converge for high frequencies.