## 3 Results and Discussion  
### Dispersion Curves of Different Models  
For this part of the study, the nearest offset (i.e., the distance between the impact and the near receiver) was set to 80 cm, and the receiver spacing was 40 cm, which is the same as the thickness of the top concrete layer. The impact duration was set to 250 us, which is roughly the time caused by an instantaneous knock of a typical 30 mm-diameter solid steel ball-hammer at the concrete surface. The purpose was to investigate how different model schemes, both uniform and layered, as shown in Table 2, affect the shapes of phase spectra and dispersion curves.  
Figures 6-9 show the phase spectra and dispersion curves of the four models. For the uniform model, the phase spectrum shows several repeated cycles, which exhibit a relatively regular and smooth pattern. The first four cycles are quite consistent and have no obvious disturbances. As a result, the R-wave velocity dispersion curve obtained from unwrapping the phase angles on the phase spectrum is generally flat, especially when the R-wave length is larger than 0.2 m, averaging around 2050 m/s, which is quite close to the theoretical value of concrete calculated from Equation (7). It is worth noticing that when the R-wave length is smaller than 0.2 m, the R-wave velocity slightly drops. This can be attributed to the element size effect of the model [17]. It has been shown that the model element size should be less than one-tenth of the shortest R-wave length of interest in order to get a satisfactory analysis result. The element size used in this study is 2 cm; therefore, the R-wave velocity after the R-wave length of 0.2 m tends to be more accurate.  
Compared to the uniform model, the first phase cycle at the low frequency of the phase spectrum of the concrete-bedrock model seems to have more interference, thus yielding a more scattered point distribution on the dispersion curve. This scattering starts right from the R-wave length of 40 cm, which coincides with the thickness of the concrete. Except for this scattered portion, the other part of the dispersion curve is almost the same as that of the uniform model.  
For the concrete-weak interlayer-bedrock model, unlike the above two, a noticeable discontinuity occurs inside the first cycle of the phase spectrum, generating a more scattered dispersion curve, and the scattering also seems to begin at the R-wave length of 40 cm. For the layered model 3 (concrete-weak layer), the discontinuity inside the first phase cycle becomes more obvious, and this creates an abrupt drop of R-wave velocity at an R-wave length of about 40 cm on the dispersion curve.  
It should be noted that for all those four dispersion curves, the R-wave velocity distribution before the R-wave length of 40 cm essentially follows the same trend, and the average (neglecting the element size effect) well approaches the theoretical R-wave velocity of concrete, 2058 m/s.  
From Figures 6-9, it can be seen that the interface contact condition may be evaluated by inspecting both phase spectra and dispersion curves. When the material properties on both sides of the interface are far-off from each other, like the case of the layered model 3, there will be very clear discontinuities both inside the first phase cycle and on the dispersion curve at the R-wave length roughly equal to the concrete layer thickness. When a weak interface exists between the two layers, which is quite common in the hydraulic tunnel construction, discontinuity will occur in the first phase cycle, but the dispersion curve may not display the abrupt drop of R-wave velocity at the same wave-length as that of the layered model 3.