## 3 Phase Characteristics of the Guided Wave Reflected from a Reflector  
Based on reference [18], the guided waves propagating in the axial direction in pipes involve torsional waves T(N,_m_) and longitudinal waves L(_N,m_), where the integer \(N\) denotes the circumferential order and \(m\) represents the group order of a mode. Because the fundamental torsional guided wave mode T(0,1) has the attracting characteristics of no dispersion, low probability of mode conversion, uniform wave-structures in the thickness direction and no response to liquids that may exist at the pipe surface, T(0,1) is most commonly used for pipe inspection [19] and was selected in this work for investigation. This paper considers a case in which T(0,1) is incident on a reflector and only T(0,1) could potentially propagate in the pipe (T(0,1) excited at a frequency lower than the T(0,2) cut-off  
Figure 3: Numerical simulation results of guided-wave A-scan and circumferential scanning in pipes by finite element software: (**a**) an undamaged pipe; (**b**) a pipe with a girth weld; (**c**) A-scan signal of the pipe; (**d**) A-scan signal of the girth weld; (**e**) circumferential scanned image of the pipe; (**f**) circumferential scanned image of the girth weld. frequency). The presence of the reflector causes guided-wave scattering, consequent reflection and transmission of the incident wave (IN), as shown in Figure 4a. The chosen axial direction of the pipe was the \(x\)-axis, and the coordinate origin is located at the reflector. The acoustic pressures \(p_{i}\), \(p_{r}\) and \(p_{t}\) of IN, reflected wave (RE) and transmission wave (TR) are defined as the following form of simple harmonic waves respectively:  
\[ \begin{array}{l}p_{i}=P_{i0}\exp(i(\omega t-kx))\ (\text{propagation along the positive direction of the x-axis})\\ p_{r}=P_{r0}\exp(i(\omega t+kx))\ (\text{propagation along the negative direction of the x-axis})\\ p_{t}=P_{r0}\exp(i(\omega t-kx))\end{array}  \tag{1}\]  
where \(P_{i0}\), \(P_{r0}\) and \(P_{r0}\) are the maximum acoustic pressures; \(k\) and \(\omega\) are the wavenumber and angular frequency, respectively. According to the relationship between acoustic pressure and acoustic impedance [20], the corresponding particle velocities can be depicted as:  
\[ \begin{array}{l}v_{i}=\frac{P_{i0}}{\rho c}\exp(i(\omega t-kx))\\ v_{r}=-\frac{P_{r0}}{\rho c}\exp(i(\omega t+kx))\\ v_{t}=\frac{P_{r0}}{\rho c}\exp(i(\omega t-kx))\end{array}  \tag{2}\]  
where \(\rho\) is the density of the pipe; \(c\) is guided wave velocity. Considering that the boundary conditions should be continuous at the reflector, the acoustic pressure and volume velocity are continuous:  
\[ \begin{array}{l}p_{i}(0)+p_{r}(0)=p_{t}(0)\\ \delta_{1}\times(v_{i}(0)+v_{r}(0))=\delta_{2}\times v_{t}(0)\end{array}  \tag{3}\]  
where \(\delta_{1}\) and \(\delta_{2}\) are the cross-sectional areas of the pipe and the reflector respectively. Substituting Equations (1) and (2) into Equation (3), we obtain the relationship:  
\[\delta_{1}\times(P_{i0}-P_{r0})=\delta_{2}\times(P_{r0}+P_{r0}) \tag{4}\]  
Further assuming \(\xi\), \(=\frac{\delta_{1}}{\delta_{2}}\), the reflection coefficient (RC) is obtained according to Equation (4):  
\[\text{RC}=\frac{P_{r0}}{P_{i0}}=\frac{\xi-1}{\xi+1}\text{=R}_{\text{e}}+\text{ jI}_{\text{m}} \tag{5}\]  
where \(\text{R}_{\text{e}}\) and \(\text{I}_{\text{m}}\) are the real and imaginary parts of the RC, which are illustrated in the complex plane in Figure 4b. Therefore, the impedance angle and module of RC shown in Figure 4b are calculated respectively as:  
\[ \begin{array}{l}|\theta|=\left|\arctan(\text{I}_{\text{m}}/\text{R}_{\text {e}})\right|=|\theta_{\text{Re}}-\theta_{\text{In}}|\\ |\text{RC}|=\sqrt{\text{I}_{\text{m}}{}^{2}+\text{R}_{\text{e}}{}^{2}}\end{array}  \tag{6}\]  
Figure 4: Guided wave propagation in a pipe and reflection coefficient (RC) in the complex plane: (**a**) guided wave propagation in a pipe with a reflector; (**b**) RC in the complex plane.