# Evaluation of Unfilled Sheath in Concrete Structures Using Response Waveform in Time Domain
## 4 Results
### Distribution and Magnitude of Difference Values  
Figure 16 shows a cross-section view of the specimen and configuration around the sheath of specimen_2 and specimen_4. Figure 17a,b show the comparison of the distribution of the difference values of specimen_2 and specimen_4 to consider the influence of the depth and diameter of the sheath. The X-coordinate of the input point for this result is 800 mm and \(N\) is 1000, respectively. In Figure 17a, the horizontal axis shows the Difference value, and the vertical axis shows the input point on the Y-coordinate, and the legend shows the depth. According to Figure 17a, the difference value increases between 550 mm to 650 mm in the coordinate of the input point, and the maximum Difference value is found at the 600 mm of the sheath position if the depth is 50 mm. However, if the depth is 137 mm, the Difference value increases between 500 mm and 700 mm. Compared to the depth of 50 mm, the range is enlarged 100 mm, and it demonstrates that the effect of the sheath is expanded. Elastic waves three-dimensionally propagate in the specimen, and it spreads concentrically [7; 8; 9; 10; 11; 12]. Thus, the propagation area of elastic waves increases with time. Because of this, if the location of the sheath is deep, the reflected waves from the sheath are expected to reach more extensive area. Figure 18a,b briefly illustrate the propagation and reflection of the elastic waves in a specimen with a void. In the case of Figure 18a, the propagation area is narrow because the distance between the measurement surface and the void is short. In contrast, in the case of Figure 18b, the distance between the measurement surface and sheath is longer, and the reflected wave from the sheath spreads wider. Finally, the response is affected widely on the measurement surface if the location of the void is deep. Therefore, if the location of the void is deeper, the Difference value increases over a wider  
Figure 16: Cross-section view of specimen surfaces and configuration and the sheath. area due to the influence on the reflected wave from the void. This suggests that DVA is capable to evaluate the depth of the void on the basis of the distribution of Difference value. In Figure 17b, the horizontal axis shows the Difference value, the vertical axis shows the position of the input point on the Y-axis, and the legend shows the diameter of the sheath. According to Figure 17b, the Difference value increases between 550 mm and 750 mm and the maximum Difference value is observed at 650 mm in the case of \(\varphi_{\text{S}}=33\) mm. Similarly, a high Difference value is shown at 600 mm in the sheath position. These results indicate that DVA is capable to evaluate a small void as well. From this result, it is possible to be understand that the size of the void has a great influence on the Difference value. Figure 19 shows the magnitude of the average Difference values of five measurement points on the input line (Line_06: Y = 600 mm) over the sheath. The horizontal axis shows the measurement case, and the vertical axis shows the Difference value. Comparing Case_1 and Case_2, the relative difference of the average error in the Difference value is 5.9% to the averaged Difference value of Case_1, and this implies that the sensitivity of the depth to Difference value is low. On the other hand, if Case_1 and Case_3 are compared, it shows  
Figure 17: DVA results (X-axis of point = 800 mm): (a) Comparison by depth; (b) Comparison by sheath diameter. Figure 18: Schematic illustration of the propagation of the reflected waves due to change in void depth: (a) Shallow void; (b) Deep void. Figure 21a,b, Figure 22a,b and Figure 23a,b illustrate the contours of the Difference value and dominant frequency that are calculated from the results of the experiments. The horizontal and vertical axes of the figure are the XY coordinates of input point on the measurement surface. The legend in (a) shows the Difference value normalized by the maximum Difference value of each experiment.