# Dynamic Characterization of Cohesive Material Based on Wave Velocity Measurements
## 3 Results and Discussion  
### Frequency Dependency  
In Figures 8 and 9 the variation of the measured shear wave velocity (\(v_{\text{S}}\)) with a predetermined period of propagating wave at different specified stress levels is shown. In Figure 10, however, the equivalent plot for measured compression wave velocity (\(v_{\text{P}}\)) is illustrated. The presented results are related to both procedures adopted for determining the travel time. Inspection of Figures 8 and 9 indicates that, regardless of the method used to measure the travel time, the shear wave velocities decrease with the predetermined period (\(T\)), which transfers to frequency (\(f\)). Oscillation frequency is in fact the inverse of period. The linear approximation fits accurately most of the presented data obtained using the peak-to-peak and start-to-start methods. The coefficient of determination (\(R^{2}\)) reaches high values, from 70% upwards, which shows that for  
Figure 8: Variation of measured \(v_{S}\) with period: peak-to-peak method. (**a**) Period’s range from 0 to 1 ms; (**b**) Period’s range from 0.01 to 0.1 ms.  
the studied cohesive material, the linear regression functions explain the majority of \(\upsilon_{\text{S}}\) variation. In general, the differences in the shear wave velocities with different frequencies are not large, from 18 to 27 m\(\cdot\)s\({}^{-1}\) and from 32 to 38 m\(\cdot\)s\({}^{-1}\), depending on the technique used to measure the travel time. In said graphs, the influence of stress level (\(p^{\prime}\)) is visible, as well. The average effective stress level significantly affects \(\upsilon_{\text{S}}\). For the highest pre-set pressure, _i.e._, 410 kPa, the authors received the highest values of shear wave velocity, with average values around 307 m\(\cdot\)s\({}^{-1}\) (peak-to-peak method) and 319 m\(\cdot\)s\({}^{-1}\) (start-to-start method). The lowest values of \(\
u_{\text{S}}\) were recorded for the smallest given stress (30 or 180 kPa), their average being around 160 m\(\cdot\)s\({}^{-1}\) (peak-to-peak method) and 246 m\(\cdot\)s\({}^{-1}\) (start-to-start method). The difference in \(\upsilon_{\text{S}}\) decreases with the increasing pressure value. The results of the current study show also that for the period of propagation lasting up to 0.1 ms (_i.e._, \(f=10\) kHz), both selected procedures used in the time domain methods consistently produce the values of \(\upsilon_{\text{S}}\), which are less dependent on frequency than in the case of a larger period of wave (Figures 8 and 9 on the right). The frame in these figures indicates the highest values of shear wave velocity for each mean effective stress level. The authors suggest in this way certain frequencies at which the bender elements tests should be performed. Hence, the recommended frequency values, for which the highest shear wave velocities can be obtained, are in the range of 20 to 50 kHz. Certainly, it should be remembered that the optimum frequency of excitation depends on many factors, for  
Figure 10: Variation of measured \(\upsilon_{P}\) with period. (**a**) Peak-to-peak method; (**b**) start-to-start method. Figure 9: Variation of measured \(\upsilon_{S}\) with period: start-to-start method. (**a**) Period’s range from 0 to 1 ms; (**b**) Period’s range from 0.01 to 0.1 ms.  
example on the pressure applied in the cell during tests [46]. There is no rule at all for choosing the best wave [47]. In the literature, some authors have reported their experience with choosing a specific frequency. [PERSON] _et al._ in 1996 [38] noted that when bender elements are measuring the samples, whose height is between 100 and 140 mm, the most interpretable output waves occur in the range from 3 to 10 kHz. [PERSON] _et al._[33], however, observed that with a higher frequency, the near field effect is less discernible. For stiffer materials, the required frequency must be higher than the reported 10 kHz [33], although the problem of overshooting becomes more important.