## 2 Method  
### Shear Horizontal Waves  
As shown in Figure 1, in circumferential scanning using guided waves, the transducer to generate and receive guided waves moves around the origin point \(O_{2}\) at the radius \(R_{1}\). The wave front of the incident guided waves expands during the propagation. The reflection waves are generated if the defects are covered by the wave front of the incident waves. The flight time and peak values of the reflection waves are used to evaluate the location and size of the defects, respectively. Shear horizontal waves (SH waves) and Lamb waves are two kinds of mechanical waves propagating in a solid plate. For waves propagating along the direction \(u_{1}\) as in Figure 1, displacements along the direction \(u_{3}\) occur in Lamb waves, which are perpendicular to the plane of the plate, but the displacements of SH waves are along the direction \(u_{2}\) within the plane of the plate. Considering that the tank floors in service are in contact with liquids (storage materials, such as different kinds of oil) and that the attenuation of shear waves in liquids is much higher than that of longitudinal waves, SH waves may be better candidates for an inspection with less energy leakage. Researches of torsional waves in pipes filled with liquids and bars immersed in liquids can be found in [34; 35], respectively, as references. Quantitative comparisons of the SH and Lamb waves will be carried out further under the conditions of different kinds of petroleum products with various volumes. And affecting factors such as the curvatures of the tank floors caused by the gravities of the stored materials and sludge deposition under the tank floors will be investigated as well. The SH waves are hardly affected by liquids [36]. Therefore, in this study, with the purpose of verification of the proposed imaging method, the model is simplified as an isotropic plate with the thickness \(d\) without contact of liquids, the dispersion equation of SH waves can be expressed as [37]  
\[\frac{\omega^{2}}{c_{T}^{2}}-\frac{\omega^{2}}{c_{p}^{2}}\ =\left(\frac{n\pi}{d} \right)^{2} \tag{1}\]  
\[c_{T}=\sqrt{\frac{E}{2(1+\mu)\rho}} \tag{2}\]  
where \(\omega=2\pi f\), \(f\), \(c_{p}\), and \(n\) are frequency, phase velocity, and order, respectively, and \(c_{T}\) is the fundamental shear wave velocity, which is determined by mechanical properties of the plate such as density \(\rho\), Young's modulus \(E\), and Poisson's ratio \(\mu\). Combining the definition of the wave number \(k=\frac{\omega}{c_{p}}\), the group velocity \(c_{g}=\frac{da\
u}{dk}\), and Equations (1) and (2), dispersion curves of phase velocity and group velocity can be calculated:  
\[c_{p}(fd)=2c_{T}\frac{fd}{\sqrt{\left(2 fd\right)^{2}-\left(nc_{T}\right)^{2}}} \tag{3}\]  
Figure 1: The diagram of the circumferential scanning using guided waves. \[c_{g}(fd)=c_{T}\sqrt{1-\frac{(n/2)^{2}}{\left(fd/c_{T}\right)^{2}}} \tag{4}\]  
where \(fd\) is the frequency-thickness product. Dispersion curves of phase velocity and group velocity for the iron plates with mechanical properties \(\rho=7850\) kg/m\({}^{3}\), \(E=210\) GPa, and \(\mu=0.3\) are shown in Figure 2. All the modes are named by their orders, and for the wave mode SH\(n\), a frequency lower limit exists when \(n\) is no less than 1, which is called the cutoff frequency. \[\left(fd\right)_{n}=\frac{nc_{T}}{2} \tag{5}\]  
The thickness of the tank floor commonly used is between 6 and 19 mm [38]. In this study, the thickness \(d\) of the iron plate used for the verification in numerical simulations and experiments is selected as 4 mm for the easy operations in the laboratory. The wave structures of SH0, SH1, SH2, and SH3 modes in the direction \(u_{2}\) are shown in Figure 3. It can be observed that the pure SH0 mode without dispersion can be generated when the excitation frequency is lower than the cutoff frequency of the SH1 mode.