# Study on Corrosion Monitoring of Reinforced Concrete Based on Longitudinal Guided Ultrasonic Waves
## 2 Theoretical Background
### Frequency Dispersion Characteristics  
Reinforcement as a wave conductor has multimodal and dispersive properties in terms of propagation, and longitudinal modal GUWs are widely used for damage detection in rod-like structures. The frequency dispersion curves describe an important feature of the propagation characteristics of GUWs in wave conductors. For a wave conductor with a regular cross-section, such as a circular rod, its frequency dispersion curve can be obtained by solving the characteristic equation. The frequency dispersion curve for a longitudinal GUW can be obtained by numerically solving the Pochhammer-Chree equation as shown in Equation (3).  
\[\frac{2\alpha}{a}(\beta^{2}+k^{2})J_{1}(\alpha a)J_{1}(\beta a)-(\beta^{2}-k^{ 2})\times J_{0}(\alpha a)J_{1}(\beta a)-4k^{2}\alpha\beta J_{1}(\alpha a)J_{0 }(\beta a)=0\, \tag{3}\]  
where \(\alpha^{2}=\omega^{2}/c_{L}{}^{2}-k^{2}\), \(\beta^{2}=\omega^{2}/c_{T}{}^{2}-k^{2}\). \(c_{L}\) and \(c_{T}\) are the longitudinal and transverse wave velocities, respectively. \(\omega\) is the angular frequency. \(k\) is the wave number. \(J_{0}\) and \(J_{1}\) are the corresponding Bessel functions, respectively.  
The specimens are of two types, reinforcement, and RC, where the RC consists of the reinforcement and covered concrete. The parameters of both materials are shown in Table 1. The frequency dispersion curves of the specimens were solved using the PCDISP program as a means of investigating the effect of the covered thickness on the propagation of GUWs in reinforcement.  
One of the unusual properties of GUWs are the infinite number of modes. Depending on the thickness of the object, the material under study, and the frequency, other higher-order dispersive modes appear (Figure 2). With a higher excitation frequency, depending on the thickness and material, higher-order modes are also excited, meaning separating the different modes and/or identifying the useful signal is a difficult task. Therefore, the first-order mode L(0, 1) is mainly utilized in the later sections of the finite element simulation and test study.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material** & **Type** & **MOE (GPa)** & **Poisson’s Ratio** &
\begin{tabular}{c} **Density** \\ **(kg/m\({}^{3}\))** \\ \end{tabular} & **Diameter (mm)** & **Length (mm)** \\ \hline Reinforcement & PSB500 & 208 & 0.3 & 7850 & 18 & 1100 \\ Covered concrete & C30 & 19 & 0.2 & 2066 & 75/110/160 & 700 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material parameters for reinforcement and covered concrete.  
Figure 2: Frequency dispersion curves of specimens with different parameters.  
The double-layer composite waveguide structure formed by the concrete and the reinforcement causes the GUW to be constantly refracted, scattered, and reflected in the contact area between the concrete and the reinforcement, which makes the modal components of the GUW more and more complex. Compared with Figure 2a, the group and phase velocities of the same mode L(0, n) at the same frequency shown in Figure 2b-d decrease with the increase in the covered diameter of the concrete. There are two attenuation forms of GUWs in RC: one is leakage and the other is absorption by the material. This leakage and absorption results in a decrease in the amplitude of the received GUW signal. The signal attenuation has two factors [28]: (1) The factors affecting the absorption of the GUW energy by the reinforcement are the material acoustic impedance and the GUW frequency. High-frequency signals have more energy attenuation caused by acoustic impedance. (2) The factors affecting energy leakage in concrete are the frequency, radius, and bond strength of RC. [PERSON] et al. [29] pointed out that the leaked energy would not provide enough information for debonding detection. Similarly, since this debonding detection is similar to corrosion detection, it is also suggested that this leaked energy will not have an impact on corrosion detection.