# Evaluation of Unfilled Sheath in Concrete Structures Using Response Waveform in Time Domain
## 4 Results
### Influence of the Number of Data on the Calculation of Difference Value
#### 4.1.2 Distribution of the Difference Value  
The distribution of the difference values in Figure 13a-f show that difference value increases relatively in the vicinity of the sheath position in all the results. In addition, in accordance with Figure 13a, it shows that the difference value increases at the sheath position even if the time in which the reflected wave returns from the reflecting surface is larger than the propagation time. In Case_1, 50 us (\(N=100\)) is greater than the arrival time of the reflected wave from the sheath of 25 us. Thus, it is possible to detect the defect with smaller \(N\) if the depth of the defect is shallow. However, the location and size of the internal defects are generally unknown. Hence, if \(N\) is set not to cover the time in which the elastic waves return from the reflecting surface, it is not guaranteed that the soundness of thickness regions is evaluated correctly. Due to this, the tendency is different from Figure 13a in Figure 14a. In the case of Figure 14a, the arrival time of the reflected wave is 69 us due to the change of sheath depth, which exceeds 50 us of \(N=100\). As a result, the difference value does not increase at the sheath position because the reflected wave does not arrive until 50 us from the sheath. Furthermore, Figure 15a shows that the increase of the difference value is not clearly observed although 50 us satisfies the arrival condition of the reflected wave from the sheath. The input frequency of the IEW requires a sufficient wavelength to resonate the target. In this case, the elastic wave is not reflected sufficiently due to the wavelength, which is determined to give enough input wavelength for detecting the thickness. Therefore, the input frequency based on the thickness of the target might be inevitably low relative to the frequency to detect the defect location. In such input conditions, if the defect is small, it is difficult to establish a reflected wave between the defect and the measuring surface. For this reason, [PERSON] et al. advocated that a high-frequency excitation is necessary to detect the small defects [4]. Because of these facts, it is concluded that it is difficult to capture the characteristic increase of the difference value with \(N=100\) in this case. Similarly, it is difficult to evaluate the sheath position from  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Number of Data:**\(N\) & **Propagation Time (\#s)** & **Propagation Distance (mm)** \\ \hline
100 & 50 & 200 \\
250 & 125 & 500 \\
500 & 250 & 1000 \\
1000 & 500 & 2000 \\
1500 & 750 & 3000 \\
2000 & 1000 & 4000 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Propagation time and distance.  
Figure 15: DVA results for Case_3 (\(\varphi_{\text{S}}\): 33 mm, Depth: 50 mm): (a) \(N=100\); (b) \(N=250\); (c) \(N=500\); (d) \(N=1000\); (e) \(N=1500\); (f) \(N=2000\).  
the distribution of the difference values if \(N\) is less than 1000 as shown in Figure 14b,c and Figure 15b,c. In contrast, the results for \(N\) = 1000 and above show the increase of the difference values at the sheath position. These results are obtained because of multiple reflections of the elastic wave between the measurement surface and the defect, and the increase of the different value is emphasized. In particular, the difference value clearly increases at the sheath location with \(N\) = 1000. As the introduced N should be adequately short, it is supposed that 1000 is the appropriate value of \(N\) to detect the sheath in this study. Hereafter, 1000 is adopted as \(N\) for further discussion.  
The results indicate that \(N\) should be chosen to cover the time in which the multiple reflection is sufficiently formed between the sheath position subsequent. Based on the above discussion, it is revealed that DVA evaluates the position of the sheath correctly by choosing the appropriate \(N\).