Evolution of the Surface Wave Test
The existence of surface waves was reported by Rayleigh in 1887. It was not however until 1938 that his theoretical and experimental work found an application in the studies of foundation vibrations conducted by the German Society of Soil Mechanics (DEGEBO). Applications to pavements were performed next by Bergerstorm and Linderholm (1946), Van der Poel (1951) and Nijboer and Van der Poel (1953). Further studies of the dispersion curves for soil and pavement profiles were conducted by Jones (1958, 1962), Heukelom and Foster (1960), Ballard (1964) and Fry (1965). In nearly all of these studies the excitation consisted of a steady state harmonic vertical force and the method became known as the steady state Rayleigh wave technique. In this early work the equipment was bulky and the interpretation of the data was based on relatively simple empirical rules which could result in erroneous results for complicated but realistic material profiles. As a consequence, the method failed to gain widespread acceptance. With the development of portable and sophisticated electronic equipment capable of performing accurate high frequency data acquisition and complex mathematical manipulations rapidly in the field and with the establishment of a theoretically sound basis for data analysis the surface wave test has been improved and simplified as the Spectral Analysis of Surface Waves (SASW) method (Heisey, et. al. 1982, Stokoe and Nazarian 1983). In recent years a considerable amount of theoretical and experimental research work has been conducted at the University of Texas at Austin in order to understand better and improve the applicability of the method (Shao 1985, Sanchez Salinero 1987, Sheu 1987, Rix 1988, Roesset et. al. 1990, Kang 1990).
Equipment and Field Testing
The general arrangement of the source, receivers (accelerometers), and recording equipment in a SASW test is shown schematically in Fig. 1. No boreholes are required because both the source and receivers are placed on the surface. A piezoelectric shaker can be used effectively as a source to generate surface waves over frequencies ranging from about 1kHz to 50kHz. The high frequencies are necessary to sample the surface layer of stiff pavements. A digital waveform analyzer coupled with a microcomputer is used to capture and process the output from the receivers.
The vertical accelerometers and source are arranged in a linear array. The distance, D, between receivers (see Fig. 1) may be varied by the operator to optimize the test results for a particular site. The distance between the source and the first receiver, d1, is usually kept equal to D but may also be increased by the operator to minimize destructive
interference from body wave reflections. However, d1/D=1.0 is normally a good arrangement, as shown in a number of analytical studies.
Surface Wave Dispersion
In the original technique, a steady-state vibrator acting vertically on the surface of the soil produced a harmonic
excitation at a known frequency. A vertically oriented sensor was moved away from the source until the recorded motion was in phase with the excitation. The distance between any two of these successive positions was assumed to correspond to one wavelength L of a Rayleigh wave propagating along the surface. So, for a frequency ω (in radians/sec), or f (in Hz), the phase velocity of the surface wave would be
ν=Lf=Lω/2π (1)
Repeating this process for different excitation frequencies f a plot of velocity versus frequency (or wavelength) was obtained. Such a plot is known as a dispersion curve. In the SASW method, instead of using a steady state vibrator at a fixed frequency, an impulsive or random-noise load is applied at the surface of the soil deposit. A variety of sources can be used to generate the impact, from hand held hammers of different sizes (small hammers are sufficient for high frequency excitation), to drop weights (heavier weights for low frequency excitation). The passage of the wave train generated by the impact is monitored by two vertical receivers, located also on the surface, as shown in Figure 1. The electrical signals recorded by the receivers are digitized and transformed to the frequency domain, using a Fast Fourier Transform algorithm, by a dynamic spectral analyzer, which provides also automatically the cross spectrum and the coherence function of the two records. The phase difference between the signals is obtained directly from the cross spectrum as a function of frequency. If is the phase difference in radians at a frequency ω, and D is the distance between the two receivers, the travel time is
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(2) and the velocity of propagation is
(3) The corresponding wavelength is then
(4)
The shape of the dispersion curve depends on the variation of soil properties with depth. For a given frequency and wavelength the particle motion is restricted to a soil depth of the same order of the wavelength and the velocity of
propagation of the surface waves depends almost exclusively on the soil properties over that depth. For high frequencies, and short wavelengths, the phase velocity reflects thus the properties of the soil near the surface and as the frequency decreases the properties of deeper and deeper layers get into play (Fig. 2).
To estimate the soil properties from the experimental dispersion curve the original steady state Rayleigh wave method assumed that the measured propagation velocity was equal to the shear wave velocity of the soil deposit at a depth of one or half a wavelength. In the past decade this procedure has been modified to account for the relationship between the shear wave velocity and the Rayleigh wave velocity for a half space. The Rayleigh wave velocity Vr varies from 0.874 to 0.955 Vs depending on Poisson’s ratio. For values of Poisson’s ratio ν larger than 0.1 we can write approximately
Vs=C Vr (5)
with
C=1.135–0.182v (6)
then
(7) where G=shear modulus, E=Young’s modulus, γ=total unit weight, and g =acceleration of gravity.
With the development of the SASW method, an additional modification was made to this procedure by Heisey et. al. (1982) who considered that the propagation velocity was the Rayleigh wave velocity of the material at a depth of 1/3 of the wavelength. This approach has been used recently by Vrettos and Prange (1990) in the study of dispersion curves at various sites where the material stiffness increased gradually with depth with excellent results.
A more accurate and more sophisticated procedure to backcalculate the material properties from the experimental dispersion curves is to assume a given soil profile, to conduct an analytical study to obtain the dispersion curve corresponding to that profile, to compare this theoretical curve with the experimental one, to introduce appropriate modifications to the profile and to repeat the process until a satisfactory agreement is reached. The main objective of this paper is to discuss two alternative ways in which the analytical determination of the dispersion curve can be performed.