# Evaluation of Unfilled Sheath in Concrete Structures Using Response Waveform in Time Domain
## 1 Introduction
### Impact Elastic Wave Method (IEW)  
Figure 1a,b show the measurement schematic of the IEW method. The simplest way to measure IEW is to use a steel ball for the excitation. Figure 1b shows the photo of the excitation; the steel ball is attached to a handle, and it is used like swinging a hammer. The elastic wave is emitted by hitting the surface of the structure with the steel ball, and the response is measured on the surface in the vicinity of the hitting point. From the response of the surface, it is possible to evaluate the thickness of the structure and the presence of internal defects on the plate-like structures [4; 5; 6; 7; 8; 9; 10; 21]. In addition, if excitation is given by the generally used steel ball size, the input frequency range is about 2 kHz to 20 kHz, which is lower than that of ultrasonic waves; input wavelength is long and the attenuation is small as a consequence. Furthermore, since a relatively low frequency band is used for the input, the effect of elastic waves due to dispersibility is very small [27]. This makes it suitable for the NDT of concrete structures with large thickness and composite materials. The elastic wave is repeatedly reflected on boundary surfaces of different acoustic impedances [4; 7; 28; 29]. The acoustic impedance is expressed as the product of the density and the elastic wave velocity in the medium, as shown in Equation (1):  
\[Z=\rho V, \tag{1}\]  
where \(Z\) is the acoustic impedance, \(\rho\) is the density of the medium and \(V\) is the elastic wave velocity of the medium. The reflectance of the elastic wave varies with the ratio of the acoustic impedance, which is calculated by Equation (2):  
\[\gamma_{p}=\frac{Z_{2}-Z_{1}}{Z_{2}{+}Z_{1}}, \tag{2}\]  
where \(\gamma_{p}\) is the reflectance, \(Z_{1}\) (\(Z_{1}\) = \(\rho_{1}V_{1}\)) is the acoustic impedance of the medium 1, and \(Z_{2}\) (\(Z_{2}\) = \(\rho_{2}V_{2}\)) is the acoustic impedance of the medium 2. Considering a model with a boundary surface as shown in Figure 2, if it is assumed that medium 1 is concrete and medium 2 is air and assuming the general physical properties are described as shown in Table 1, the calculated \(\gamma_{p}\) is about 1.0 from Equation (2), and it shows that the reflection is almost 100%. From such basic properties of the elastic waves, the thickness of a plate can be evaluated by using the characteristic of the elastic wave reflected repeatedly between the measuring and reflecting surfaces. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Medium** & \(\rho\) **(kg/m\({}^{3}\))** & \(V\) **(m/s)** & \(Z\) **(kg/s\(\cdot\)m\({}^{2}\))** \\ \hline
1: Concrete & 2400 & 4000 & \(9.60\times 10^{7}\) \\
2: Air & 1.29 & 332 & \(4.28\times 10^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Medium conditions. Figure 1: IEW method: (**a**) Schematic illustration of the IEW method; (**b**) Photograph of measurement status. Figure 2: Reflection and transmission of elastic waves. Figure 3a,b show an example of a schematic elastic wave path and its response waveform. The input elastic wave is reflected between the measuring surface and the reflecting surface. The natural frequency of the primary vibration mode by the multitude reflection of the elastic wave in the thickness direction is obtained from the round-trip distance and the primary elastic wave velocity of the thickness direction as follows:  
\[F_{0}=\frac{V_{P}}{2D_{0}}, \tag{3}\]  
where \(F_{0}\) is the natural frequency of the primary vibration mode, \(V_{P}\) is the primary elastic wave velocity in the thickness direction and is the phase velocity and \(D_{0}\) is the thickness. Conversely, the thickness is shown as Equation (4) on the basis of Equation (3):  
\[D_{0}=\frac{V_{P}}{2F_{0}}. \tag{4}\]  
In practice, a technique to evaluate the thickness of concrete structures from the dominant frequency obtained by Fast Fourier Transform (FFT) of the measured wave and primary elastic wave velocity is established [4]. In addition, [PERSON] et al.