## 2 Theory of the Non-Detection Zone
### The Equivalent Model of Low-Frequency Ultrasonic Guided Wave in a Square Steel Bar
#### 2.1.2 Theory on the Equivalent Model in Low-Frequency  
The three-dimensional model of the steel square bar is set in the Cartesian coordinate system \((x,y,z)\), as shown in Figure 1a. The independent of \(z\) deformation is specified by the displacement vector \(\overrightarrow{u}=(u_{1},u_{2},0)\) lying in the plane \((x,y)\), where \(u_{1}\), \(u_{2}\), \(u_{3}\) represent the displacement variable in the \(x\), \(y\), \(z\) directions respectively, and the model has a similar 2D plane-strain deformation in the plate-structure. The similar theoretical analysis on the equivalent model for low frequency ultrasonic wave has been proposed by [PERSON] et al. [18]. We can see the similar low-frequency phase velocities of \(S0\) and \(A0\) modes provided by the beam and plate models for an aluminium sample in the reference [18], as shown in Figure 1b. In low frequency, this conclusion, that the axisymmetric longitudinal mode in the rod with diameter d is very similar to the symmetric mode distribution in the plate with the same thickness d, is proposed by [PERSON] [2]. In this paper, when we focused on vibration mode \(S0/A0\), without regard to other vibration modes, the ultrasonic wave propagation analysis is simplified and equivalent in low frequency. In essence, the three-dimensional (3D) ultrasonic wave in the rod is equivalent to the two-dimensional (2D) Lamb wave in this study, which is used as the equivalent model of the low-frequency ultrasonic guided waves in a square steel bar. Both ultrasonic guided wave models are based on the same acoustic propagation control equations [1]. When low frequency ultrasonic wave excitation covers the entire incident surface, as shown in Figure 1a, the rectangular cross-section of ABCD in the plate-like structure is used to'replace' the rectangular cross section 'A' 'B' C 'D' in the rod-like structure for the vibration mode analysis, and the vibration of ultrasonic wave between the front and back surfaces in rod-like structure can be equivalent to the vibration of ultrasonic wave between the front and back surfaces in the plate-like structure. A similar ultrasonic guided wave mode is obtained [3], the longitudinal wave L (0, 1) and the flexural wave F (1, 1) are similar to the \(S0\) and \(A0\) waves respectively in a 2D plate-like ultrasonic propagation. The simulations of the pulse waveform and the actual waveform exhibit good consistency. Compared with the 3D model, the estimation accuracy is slightly lower but the equivalent model reduces the simulation hardware requirements and computational costs, thereby improving the simulation efficiency. In solid matter, as shown in Figure 1a, the vector form of the wave equation for the ultrasonic wave can be represented as follows [1]:  
\[(\lambda+\mu)\
abla(\
abla\bullet\overrightarrow{u})+\mu\
abla^{2} \overrightarrow{u}+(\frac{\lambda^{\prime}+\mu^{\prime}}{\omega})\
abla( \
abla\bullet\frac{\partial\overrightarrow{u}}{\partial t})+(\frac{\mu^{ \prime}}{\omega})\
abla^{2}\frac{\partial\overrightarrow{u}}{\partial t}=\rho \frac{\partial\overrightarrow{u}}{\partial t} \tag{1}\]  
where \(\lambda\) and \(\mu\) are the Lame constants; \(\lambda^{\prime}\) and \(\mu^{\prime}\) denote the attenuation factor in the medium.