**Near-field effects**

Distinguishing S-wave arrivals in the near-field is difficult since the arrival of the S-wave may be obscured by other wavefronts that propagate at higher velocities (e.g. P-wave velocity - Sanchez-Salinero et al. 1986, Lee and Santamarina 2005). This effect becomes significant especially at closer distances between sources and receivers. Sanchez-Salinero et al. (1986) developed a closed-form solution for longitudinal and transverse motions in linear-elastic and homogeneous media. Fig. 7a shows the arrivals for distances between source and receiver (L) equal to 1, 2, 3, 4, 6, and 8 shear-wave wavelengths (λs). Note that for greater relative source-receiver distances (L/λs), the separation between P- and S-wave arrivals increases, the amplitude of the transverse displacement caused by the P-wave decreases and the S-wave arrival is better evaluated. To clearly separate the P- and S-wave arrivals, L/λs must be greater than 4. However, this ratio also depends on the material damping. Understanding of the near-field effects may help interpreting both P- and S-waves arrival times which can also be used to evaluate the elastic modulus and Poisson's ratio. According to the wave propagation experiments by Sawangsuriya et al. (2006), wave propagation results were obtained using long cylindrical Kaolinite specimens (77.5 mm diameter) that were trimmed to different heights and hence different source-receiver distances. Since the specimens were identical, the slope of the measured travel time versus distance was used to calculate the P- and S-wave velocities (see Fig. 7b). The calculated Poison's ratio (ν) is 0.17. As the source-receiver distance increases, the P-wave arrival is more difficult to determine (Fig. 7a), for this reason there are less P-wave arrival time data points at greater source-receiver distances in Fig. 7b. It is therefore recommended that the minimum source-receiver distance (L) must be at least three times the wavelength (λs) (greater for large damping ratios) to facilitate the interpretation of the S-wave arrivals.

### **Boundary effects - reflection**

When a bender element deflects, it generates S-waves in the direction of the bender element plane and P-waves in the direction normal to the bender element plane (Santamarina et al. 2001). Therefore, the presence of lateral boundaries may cause the reflected P-waves to arrive faster than direct S-waves hindering the proper evaluation of S-wave velocities. The difference in travel time arrivals depends on the Poisson's ratio (ν) of the geomaterial (Lee and Santamarina 2005). In 3-D arrangements of sources and receivers, the limiting separations can be expressed as a function of dimensionless ratios H/L and R/L and the Poisson's ratio as illustrated in Fig.8. Note that the normalized distances below the curves are the acceptable distances where the S-wave arrives earlier than the reflected P-wave. It is therefore important that depending on the Poisson's ratio of the geomaterial, the relative distance between sources and receivers and to the boundaries (H/L and R/L) must be limited to avoid possible P-wave reflection interferences.

**Figure 7.** Near-field effects: (a) normalized receiver response (transverse particle motion) with increasing source-receiver separation (model parameters: Vp=288 m/s, Vs=175 m/s, damping D=0.01 – closed-form solution by Sanchez-Salinero et al. 1986), and (b) measured P- and S-waves travel times in Kaolinite specimens (Sawangsuriya et al. 2006). Note: M = constraint modulus, G = shear modulus, E = Young's modulus, ρ = mass density.

#### **Boundary effects - refraction**

168 Wave Processes in Classical and New Solids

following:

**Near-field effects** 

**Boundary effects - reflection** 

important role in the evaluation of wave propagation test data. For example when S-waves are created in a bender element wave propagation test, reflection and refraction from boundaries may produce waves other than direct S-waves; near-field effect may obscure the S-wave arrival; or wave attenuation and dispersion may prevent the proper identification of multiple reflections (Sanchez-Salinero et al. 1986, Fratta and Santamarina 1996). Sawangsuriya et al. (2006) performed a series of experimental studies to explore and to establish the limits to minimize the boundary effects in the wave propagation experiments. The experimental results are presented as simple dimensionless ratios that can provide guidelines for the design and interpretation of the wave propagation experiments as the

Distinguishing S-wave arrivals in the near-field is difficult since the arrival of the S-wave may be obscured by other wavefronts that propagate at higher velocities (e.g. P-wave velocity - Sanchez-Salinero et al. 1986, Lee and Santamarina 2005). This effect becomes significant especially at closer distances between sources and receivers. Sanchez-Salinero et al. (1986) developed a closed-form solution for longitudinal and transverse motions in linear-elastic and homogeneous media. Fig. 7a shows the arrivals for distances between source and receiver (L) equal to 1, 2, 3, 4, 6, and 8 shear-wave wavelengths (λs). Note that for greater relative source-receiver distances (L/λs), the separation between P- and S-wave arrivals increases, the amplitude of the transverse displacement caused by the P-wave decreases and the S-wave arrival is better evaluated. To clearly separate the P- and S-wave arrivals, L/λs must be greater than 4. However, this ratio also depends on the material damping. Understanding of the near-field effects may help interpreting both P- and S-waves arrival times which can also be used to evaluate the elastic modulus and Poisson's ratio. According to the wave propagation experiments by Sawangsuriya et al. (2006), wave propagation results were obtained using long cylindrical Kaolinite specimens (77.5 mm diameter) that were trimmed to different heights and hence different source-receiver distances. Since the specimens were identical, the slope of the measured travel time versus distance was used to calculate the P- and S-wave velocities (see Fig. 7b). The calculated Poison's ratio (ν) is 0.17. As the source-receiver distance increases, the P-wave arrival is more difficult to determine (Fig. 7a), for this reason there are less P-wave arrival time data points at greater source-receiver distances in Fig. 7b. It is therefore recommended that the minimum source-receiver distance (L) must be at least three times the wavelength (λs) (greater for large damping ratios) to facilitate the interpretation of the S-wave arrivals.

When a bender element deflects, it generates S-waves in the direction of the bender element plane and P-waves in the direction normal to the bender element plane (Santamarina et al. 2001). Therefore, the presence of lateral boundaries may cause the reflected P-waves to arrive faster than direct S-waves hindering the proper evaluation of S-wave velocities. The difference in travel time arrivals depends on the Poisson's ratio (ν) of the geomaterial (Lee As the separation between source and receiver increases, refracted waves may travel along rigid boundaries (having higher velocities) masking the S-wave arrivals. This phenomenon can be observed when the test specimen is confined within a rigid wall container. When monitoring S-waves, the velocity is calculated by an assumed straight ray travel length and the first arrival time. However, the presence of the rigid boundary may mask the arrival of the direct S-wave if a refracted wave arrives first. This phenomenon was observed by using specimens prepared inside the cylindrical rigid wall container where the source-receiver distance and the distance to the wall varied (Sawangsuriya et al. 2006). Results matched well with a simple refraction model used to determine the shortest travel time from the two possible wave paths. A sketch of the model and a summary of experimental results are shown in Fig. 9. Results suggest that for a given medium and PVC wall having finite wave velocities, the bender elements need to be placed at a minimum distance (d) greater than 0.4 L to avoid receiving refracted signals. As a consequence, it is recommended that with the presence of rigid boundaries, the distance of bender elements to the boundary (d) shall be greater than approximately 0.4 L to avoid refracting wave from arriving first.

Wave Propagation Methods for Determining Stiffness of Geomaterials 171

time domain or in the frequency domain. In the time-domain method, the travel time of the S-wave is monitored by plotting recorded signal with time. The S-wave velocity is then calculated if the tip-to-tip distance between the transmitting and receiving bender elements is known (Dyvik and Madshus 1985; Viggiani and Atkinson 1995). In general, the signals may be different from those in Figs. 4 and 5, possibly due to the stiffness of the sample, the boundary conditions, the test apparatus, the degree of fixity of the bender element into the platen or housing, the elastic properties of the bimorph, the cantilever length, the support properties (i.e., fixity), and the elastic properties of the surrounding medium as aforementioned. Once these boundary and scale effects are evaluated, and their effects are considered, the travel time between the transmitter and the receiver benders can be determined. The S-wave arrival corresponds to point C (Fig. 4) is typically chosen for the determination of the travel time in order to avoid the near-field effect caused by the arrival of energy with P-wave velocity (point A in Fig. 4 – Sanchez-Salinero et al. 1986; Kawaguchi

In the frequency-domain method, the transmitter sends a continuous sinusoidal wave at a single frequency through the sample to the receiver, and the phase shift between the two waves is measured. By varying the frequency, the phase delay of the transmitted and received signals can be observed, and the frequencies may be found when they are exactly in-phase and exactly out-of-phase when the phase differences are exact multiple N of . The gradient of the plot between wave number (N) against frequency gives the group travel time. This is also called as π-point method. Similar results may be obtained using a sine sweep instead of a continuous wave of a single frequency. The Automatic Bender Elements Testing System (ABETS) program has been developed and uses phase-delay method to determine the shear wave velocities from bender element tests (Greening and Nash 2004). The principle of the ABETS is to send a sine wave pulse of continuously varying frequency generated by a soundcard in a computer through the sample. The signals from the transmitter and the receiver are collected and analysed by embedded Fast Fourier Transform (FFT) analysis. By selecting the range of frequencies for which the coherence is close to unity which indicates high quality signal that there is no interference at that transmitted frequency, a linear fit to the unwrapped phase data is used to determine the

The S-wave velocity in particulate geomaterials depends on the state and history of the effective stresses, void ratio, degree of saturation, and type of particles. Based on experimental evidence and analytical studies, the velocity-effective stress relationship for granular geomaterials is expressed as a power function having two physically-meaningful parameters: a coefficient α and an exponent β. These parameters represent the S-wave velocity at a given state of stress and its variation with stress changes. The velocity-stress power relationship for granular media under isotropic loading is expressed as (Roesler 1979,

et al. 2001; Lee and Santamarina 2005).

group's travel time within selected frequency range.

**5. Evaluation of S-wave velocity** 

White 1983):

**Figure 8.** Limiting distances to avoid reflected P-wave arrivals when testing with boundaries. The lines in the plot represent the upper limits, the recommended test dimensions should fall under these lines (Sawangsuriya et al. 2006).

**Figure 9.** (a) Sketch of the test setup and the model for data evaluation and (b) experimental results from Kaolinite specimens (water content = 22%, unit weight = 9.7 kN/m3) (Sawangsuriya et al. 2006).

## **Identifying wave travel time arrivals**

A selection of the travel time for a wave travelling from a transmitter bender to a receiver bender has been controversial. Generally, the travel time can be determined either in the time domain or in the frequency domain. In the time-domain method, the travel time of the S-wave is monitored by plotting recorded signal with time. The S-wave velocity is then calculated if the tip-to-tip distance between the transmitting and receiving bender elements is known (Dyvik and Madshus 1985; Viggiani and Atkinson 1995). In general, the signals may be different from those in Figs. 4 and 5, possibly due to the stiffness of the sample, the boundary conditions, the test apparatus, the degree of fixity of the bender element into the platen or housing, the elastic properties of the bimorph, the cantilever length, the support properties (i.e., fixity), and the elastic properties of the surrounding medium as aforementioned. Once these boundary and scale effects are evaluated, and their effects are considered, the travel time between the transmitter and the receiver benders can be determined. The S-wave arrival corresponds to point C (Fig. 4) is typically chosen for the determination of the travel time in order to avoid the near-field effect caused by the arrival of energy with P-wave velocity (point A in Fig. 4 – Sanchez-Salinero et al. 1986; Kawaguchi et al. 2001; Lee and Santamarina 2005).

In the frequency-domain method, the transmitter sends a continuous sinusoidal wave at a single frequency through the sample to the receiver, and the phase shift between the two waves is measured. By varying the frequency, the phase delay of the transmitted and received signals can be observed, and the frequencies may be found when they are exactly in-phase and exactly out-of-phase when the phase differences are exact multiple N of . The gradient of the plot between wave number (N) against frequency gives the group travel time. This is also called as π-point method. Similar results may be obtained using a sine sweep instead of a continuous wave of a single frequency. The Automatic Bender Elements Testing System (ABETS) program has been developed and uses phase-delay method to determine the shear wave velocities from bender element tests (Greening and Nash 2004). The principle of the ABETS is to send a sine wave pulse of continuously varying frequency generated by a soundcard in a computer through the sample. The signals from the transmitter and the receiver are collected and analysed by embedded Fast Fourier Transform (FFT) analysis. By selecting the range of frequencies for which the coherence is close to unity which indicates high quality signal that there is no interference at that transmitted frequency, a linear fit to the unwrapped phase data is used to determine the group's travel time within selected frequency range.
