Frequency dependencies of phase velocity and attenuation
coefficient in a water-saturated sandy sediment from
0.3 to 1.0 MHz
Kang Il Lee
Department of Physics, Kangwon National University, Chuncheon 200–701, Republic of Korea
Victor F. Humphrey
Institute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ,
United Kingdom
Byoung-Nam Kim and Suk Wang Yoona兲
Department of Physics and Institute of Basic Science, Sungkyunkwan University, Suwon 440-746,
Republic of Korea
共Received 24 March 2006; revised 19 January 2007; accepted 9 February 2007兲
The frequency-dependent phase velocity and attenuation coefficient for the fast longitudinal wave in
a water-saturated sandy sediment were measured over the frequency range from 0.3 to 1.0 MHz.
The experimental data of phase velocity exhibited the significant negative dispersion, with the mean
rate of decline of 120± 20 m / s / MHz. The Biot model predicted the approximately nondispersive
phase velocity and the grain-shearing 共GS兲 model exhibited the slightly positive dispersion. In
contrast, the predictions of the multiple scattering models for the negative dispersion in the
glass-grain composite were in general agreement with the experimental data for the water-saturated
sandy sediment measured here. The experimental data of attenuation coefficient was found to
increase nonlinearly with frequency from 0.3 to 1.0 MHz. However, both the Biot and the GS
models yielded the attenuation coefficient increasing almost linearly with frequency. The total
attenuation coefficient given by the algebraic sum of absorption and scattering components showed
a reasonable agreement with the experimental data for overall frequencies. This study suggests that
the scattering is the principal mechanism responsible for the variations of phase velocity and
attenuation coefficient with frequency in water-saturated sandy sediments at high frequencies.
© 2007 Acoustical Society of America. 关DOI: 10.1121/1.2713690兴
PACS number共s兲: 43.30.Ma 关RAS兴
Pages: 2553–2558
I. INTRODUCTION
The Biot model 共Biot, 1956a, b, 1962兲 for elastic wave
propagation in porous media has been applied to sediment
acoustics with varying degrees of success. It was originally
applied to the analysis of ultrasound geophysical test data for
porous rock samples. In 1970, the model was applied by
Stoll and Bryan 共1970兲 in a form more suitable for watersaturated sediments. Recently, Buckingham 共1997, 2000,
2005兲 developed the grain-shearing 共GS兲 model for wave
propagation in saturated, unconsolidated granular materials,
including marine sediments. Since the mineral grains are unbonded, it is assumed that the shear rigidity modulus of the
medium is zero, implying the absence of a skeletal elastic
frame. The GS model has been successfully applied to predict the relationship between mechanical properties, such as
grain size and porosity, and acoustic properties of marine
sediments. Chotiros and Isakson 共2004兲 also proposed a new
theoretical model based on the grain-to-grain contacts in unconsolidated granular materials.
Dispersion of phase velocity has been extensively investigated in marine sediments by a number of authors 共Hamp-
ton, 1967; Wingham, 1985; Turgut and Yamamoto, 1990;
Stoll, 2002; Tang et al., 2002; Williams et al., 2002; Chotiros
and Isakson, 2004兲. Most of the studies have been performed
over various frequency ranges from a few hundreds of hertz
up to a few hundreds of kilohertz, where the propagation of
interest occurs over ranges that are sufficiently long to include bottom interactions. The fast longitudinal wave in marine sediments has been known to exhibit the positive dispersion of phase velocity at frequencies lower than about
200 kHz, which is predicted by theoretical models such as
the Biot and the GS models. In contrast, an interesting highfrequency experimental study demonstrated that the fast
wave velocity in water-saturated sandy sediments shows the
negative dispersion from 0.2 to 1.2 MHz 共Moussatov et al.,
1998兲. However, the velocity dispersion predicted by the
Biot model has been reported to be very weak, but positive at
frequencies higher than 0.2 MHz 共Moussatov et al., 1998兲,
and the GS model also tends to predict the slightly positive
dispersion in this frequency range 共Buckingham, 1997兲.
These results may imply that the underlying mechanism responsible for the velocity dispersion in the high frequency
range is different from that in the low frequency range. Thus,
a兲
Electronic mail: [email protected]
J. Acoust. Soc. Am. 121 共5兲, May 2007
0001-4966/2007/121共5兲/2553/6/$23.00
© 2007 Acoustical Society of America
2553
both the Biot and the GS models may only have validity in a
limited range of frequency for the dispersion of phase velocity in water-saturated sediments.
As extensively discussed by Buckingham 共2005兲 and
Ohkawa 共2006兲, the frequency dependence of attenuation coefficient in marine sediments has been argued for many
years. The Biot model predicts an attenuation coefficient increasing as the half power of frequency, which is due to the
fluid viscosity only. Nolle et al. 共1963兲 found the f 1/2 dependence of attenuation coefficient in water-saturated sands at
frequencies of 0.2, 0.5, and 1.0 MHz. In contrast, most data
exhibit a linear scaling of attenuation coefficient with frequency over a wide range of frequencies 共Wingham, 1985;
Hamilton, 1987兲, which is consistent with the prediction of
the GS model based on grain-to-grain shearing mechanisms
共Buckingham, 1997兲. In fact, the GS model yields an attenuation coefficient that is almost but not quite linear in frequency. On the other hand, Ohkawa 共2006兲 demonstrated
that the frequency dependence of attenuation data collected
during the sediment acoustics experiment in 1999 共SAX99兲
follows f 1/2 at frequencies below 50 kHz, as the Biot model
predicts, and the deviation of attenuation from the Biot
model at frequencies higher than 200 kHz is due to effects of
scattering. This suggests that the scattering is likely to be the
principal mechanism responsible for the attenuation in the
ultrahigh frequency range.
The present study aims to provide an insight into the
frequency dependencies of phase velocity and attenuation
coefficient for the fast longitudinal wave in a water-saturated
sandy sediment over the frequency range from 0.3 to 1.0
MHz. The frequency-dependent phase velocity and attenuation coefficient in the water-saturated sandy sediment were
measured using two different matched pairs of transducers
with a diameter of 25.4 mm and center frequencies of 0.5
and 1.0 MHz. They were compared with the predictions obtained from the Biot and the GS models.
FIG. 1. Schematic diagram of the experimental setup for ultrasonic measurements in through-transmission geometry.
B. Ultrasonic measurements
Figure 1 illustrates the schematic diagram of the experimental setup for ultrasonic measurements in throughtransmission geometry. A sediment specimen and transducers
were immersed in a water bath with the dimensions of
650 mm⫻ 750 mm⫻ 1500 mm at room temperature. Two
different matched pairs of unfocused, broadband transducers
with a diameter 共D兲 of 25.4 mm and center frequencies of
0.5 MHz 共Panametrics V301兲 and 1.0 MHz 共Panametrics
V302兲 were used in order to cover the wide range of frequencies of interest. The opposing faces of coaxially aligned
transducers were separated by a distance of 150 mm, greater
than the near-field distances 共D2 / 4␭兲 of 53 and 108 mm for
the 0.5 and the 1.0 MHz transducers. A 200 MHz pulser/
receiver 共Panametrics 5900PR兲 was used to generate pulses
and to receive signals. Received signals were averaged over
100 pulses in time domain using a 500 MHz digital storage
oscilloscope 共LeCroy LT342兲 and stored on a computer for
off-line analysis.
In order to measure the phase velocity, the received signals were recorded with and without the sediment specimen
in the acoustic path. The frequency-dependent phase velocity
c p共␻兲 was determined by
II. MATERIALS AND METHODS
A. Sandy sediment
The sandy sediment used in the present study was composed of clean medium sand with a porosity of 0.408± 0.013
and a mean grain diameter of 425± 84 ␮m spanned from
250 to 500 ␮m. The grain sizes of the sediment were much
smaller than the ultrasonic wavelength 共␭兲 over the bandwidth investigated. The sediment was boiled for 1 h to remove air bubbles and to saturate it with water. The watersaturated sediment was then poured into a small rectangular
container immersed in water with one aperture at its top. The
sediment container has the dimensions of 100 mm
⫻ 100 mm⫻ 50 mm. The thickness of 50 mm was chosen to
optimize the measurement of velocity dispersion and to
avoid excessive attenuation of ultrasonic pulses. In order to
minimize the transmission losses at the surfaces of the container perpendicular to wave propagation, its front and back
walls 共i.e., the faces which measured 100⫻ 100 mm2兲 were
made from thin plastic films 100 ␮m thick.
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J. Acoust. Soc. Am., Vol. 121, No. 5, May 2007
c p共 ␻ 兲 =
cw
,
1 − 关cw⌬␾共␻兲/␻d兴
共1兲
where ␻ is the angular frequency of the wave, d is the thickness of the specimen, and ⌬␾共␻兲 is the difference in unwrapped phases of the received signals with and without the
specimen. The unwrapped phase difference ⌬␾共␻兲 was calculated by taking the fast Fourier transform 共FFT兲 of the
digitized received signal, as follows 共Wear, 2000兲. The phase
of the signal at each frequency was taken to be the inverse
tangent of the ratio of the imaginary to real part of the FFT at
that frequency. Since the inverse tangent function yields
principal values between −␲ and ␲, the phase had to be
unwrapped by adding an integer multiple of 2␲ to all frequencies above each frequency where a discontinuity appeared. The temperature-dependent speed of sound in distilled water, cw, is given by 共Kaye and Laby, 1995兲
Lee et al.: Acoustic properties of a water-saturated sediment
cw = 1402.9 + 4.835T − 0.047016T2 + 0.00012725T3 ,
共2兲
where T is the temperature in °C. The estimates of Eq. 共2兲
were consistent with the measured values in the water bath
given the precision of the digital thermometer used. Measurements of phase velocity were also repeated with the
specimen reversed. A total of 10 measurements 共5 measurements in each direction兲 were obtained for a mean value,
repositioning the specimen after each measurement. The
sound speed in the sediment, ranging approximately from
1590 to 1680 m / s, is sufficiently close to that in fresh water at room temperature, 1483 m / s, that potential
diffraction-related errors in this substitution method
caused by the disparity in speeds between the two media
may be ignored 共Verhoef et al., 1985; Xu and Kaufman,
1993兲.
The attenuation coefficient was determined by using the
same signal acquired for phase velocity measurements, as
follows. The FFT was taken to obtain the amplitude spectrum of the signal. The signal loss as a function of frequency
was obtained by subtracting the logarithm of the amplitude
spectrum of the signal through the sediment specimen from
that through water only, and divided by the thickness of the
specimen to determine the attenuation coefficient in units of
dB/cm. Finally, the attenuation coefficient was corrected by
taking into account the transmission losses at each interface,
water/sediment and sediment/water, over the range of frequencies of interest. Ten measurements of attenuation coefficient in the specimen were averaged to obtain a mean
value.
FIG. 2. 共a兲 Temporal signals and 共b兲 pressure spectra measured with and
without the sediment in the acoustic path for the pair of 0.5 MHz transducers.
III. RESULTS AND DISCUSSION
A. Temporal signals and pressure spectra
B. Frequency dependence of phase velocity
Figures 2 and 3 show the temporal signals and their
corresponding pressure spectra measured with and without
the sediment in the acoustic path, using the two different
pairs of transducers with the center frequencies of 0.5 and
1.0 MHz. The temporal signals observed in the sediment correspond to the fast longitudinal wave. In the present study, no
attempt was made to measure the slow wave in the sediment.
As shown in Figs. 2 and 3, each sediment signal exhibits an
earlier arrival time than that observed in water. This is because the sound speed in the sediment is greater than that in
water. It is notable that the sediment signal for the pair of
1.0 MHz transducers in Fig. 3共a兲 exhibits pulse elongation
with the higher frequencies arriving later, characteristic of
negative dispersion. As can be seen in spectra, the center
frequencies of the pulses were clearly shifted to lower frequencies, resulting from increasing attenuation with frequency. This is more pronounced for the pulse centered at
1.0 MHz than that at 0.5 MHz. The usable frequency bandwidths were taken to be from 0.3 to 0.7 MHz for the pair of
0.5 MHz transducers and from 0.5 to 1.0 MHz for the pair
of 1.0 MHz transducers.
The frequency-dependent phase velocity in the watersaturated sandy sediment was experimentally measured and
theoretically predicted by the Biot and the GS models. The
experimental results were plotted as a function of frequency
in Fig. 4, over the usable frequency bandwidths of the transducers, i.e., from 0.3 to 0.7 MHz for the 0.5 MHz transducer
measurements 共circles兲 and from 0.5 to 1.0 MHz for the
1.0 MHz transducer measurements 共asterisks兲. The error bars
denote the standard deviations of ten measurements and represent the random uncertainties of the measurement procedure. It should be noted that there is some overlap of the two
frequency bandwidths, i.e., from 0.5 to 0.7 MHz, where the
phase velocities for the two sets of transducers agree within
the experimental uncertainty. As clearly seen in Fig. 4, the
average of measurements of phase velocity exhibits the significant negative dispersion, with the velocity ranging from
1680± 5 m / s at 0.3 MHz to 1590± 4 m / s at 1.0 MHz. The
mean rate of decline in phase velocity was
120± 20 m / s / MHz between 0.3 and 1.0 MHz. This negatively sloped trend is similar to that measured in watersaturated sandy sediments from 0.2 to 1.2 MHz by Moussatov et al. 共1998兲 共1760– 1680 m / s兲.
J. Acoust. Soc. Am., Vol. 121, No. 5, May 2007
Lee et al.: Acoustic properties of a water-saturated sediment
2555
TABLE I. Input parameters of the Biot model for a water-saturated sandy
sediment.
FIG. 3. 共a兲 Temporal signals and 共b兲 pressure spectra measured with and
without the sediment in the acoustic path for the pair of 1.0 MHz transducers.
The solid and the dashed curves in Fig. 4 represent the
phase velocities predicted by the Biot and the GS models,
respectively. The input parameters of the Biot and the GS
models for a water-saturated sandy sediment are summarized
Parameter
Value
Density of sand grain 共␳s兲
Density of fluid 共␳ f 兲
Bulk modulus of sand grain 共Ks兲
Bulk modulus of fluid 共K f 兲
Bulk modulus of frame 共Kb兲
Shear modulus of frame 共␮b兲
Bulk log decrement 共␦k兲
Shear log decrement 共␦␮兲
Viscosity of fluid 共␩兲
Permeability 共␬兲
Pore size parameter 共a兲
Tortuosity 共␣兲
Porosity 共␤兲
2650 kg/ m3
1000 kg/ m3
36.9 GPa
2.2 GPa
66.3 MPa
24.9 MPa
0.15
0.2
0.001 Pa s
1 ⫻ 10−10 m2
58.2 ␮m
1.73
0.408
in Tables I and II. A large amount of information can be
found in the literature about the determination of the parameters required for these two models 共Chotiros, 1995; Buckingham, 1997, 2000, 2005; Williams et al., 2002兲. The sediment depth parameter, d = 0.063 m, in Table II was
determined by curve fitting the prediction of the GS model to
the experimental data of phase velocity at 0.3 MHz. As seen
in Fig. 4, the Biot model predicts the approximately nondispersive phase velocity over the frequency range from
0.3 to 1.0 MHz and the GS model exhibits the slightly positive dispersion. This suggests that both of these models may
not be suitable at very high frequencies for the dispersion of
phase velocity in water-saturated sediments.
Multiple scattering models have shown to be useful in
gaining an insight into wave propagation in unconsolidated
granular composites saturated with fluid. Schwartz and Plona
共1984兲 reported success in applying multiple scattering techniques 共Nicholson and Schwartz, 1982; Tsang et al., 1982兲 to
the dispersion of phase velocity and attenuation in the glassand the Plexiglas™-grain composites from 0.3 to 2.0 MHz.
They used suspensions comprised of water-saturated,
roughly monosized spherical grains with a porosity of 0.380.
Especially, the glass grain had similar acoustic properties 共
TABLE II. Input parameters of the GS model for a water-saturated sandy
sediment.
FIG. 4. Experimental and theoretical phase velocities plotted as a function
of frequency for the fast longitudinal wave in the water-saturated sandy
sediment.
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J. Acoust. Soc. Am., Vol. 121, No. 5, May 2007
Parameter
Value
Density of grain 共␳s兲
Density of fluid 共␳ f 兲
Bulk modulus of grain 共Ks兲
Bulk modulus of fluid 共K f 兲
Longitudinal coefficient 共␥ p0兲
Shear coefficient 共␥s0兲
Strain-hardening index 共n兲
rms grain roughness 共⌬兲
Reference depth in sediment 共d0兲
Reference grain diameter 共u0兲
Reference porosity 共␤0兲
Depth in sediment 共d兲
Grain diameter 共ug兲
Porosity 共␤兲
2650 kg/ m3
1000 kg/ m3
36.9 GPa
2.2 GPa
388.8 MPa
45.9 MPa
0.0851
4.55 ␮m
0.3 m
1000 ␮m
0.377
0.063 m
425 ␮m
0.408
Lee et al.: Acoustic properties of a water-saturated sediment
FIG. 5. Experimental and theoretical velocity shifts plotted as a function of
frequency for the fast longitudinal wave in the water-saturated sandy sediment and in the glass-grain composite.
␳s = 2480 kg/ m3, cs = 5850 m / s兲 and a mean diameter
共545 ␮m兲 to those of the sand grain of the sediment used
here. They compared the model predictions with the experimental data and concluded that, at high frequencies, the
Plexiglas-grain composite exhibits the stronger scattering effects than the glass. It is notable that both unconsolidated
composites showed the significant negative dispersion of
phase velocity and the multiple scattering models, such as
the familiar quasicrystalline approximation 共QCA兲 and the
self-consistent effective medium approximation 共EMA兲,
could successfully explain the negative dispersion 共Schwartz
and Plona, 1984兲. Figure 5 compares the predictions of these
multiple scattering models for the glass-grain composite
against the experimental data for the water-saturated sandy
sediment measured here. It can be seen that the experimental
results for the water-saturated sandy sediment presented in
this study are also in general agreement with their predictions for the negative dispersion in the glass-grain composite.
Future studies will apply the multiple scattering techniques
to elucidate the characteristics of dispersion in the watersaturated sediment in more depth.
C. Frequency dependence of attenuation coefficient
Figure 6 shows the experimental and theoretical attenuation coefficients plotted as a function of frequency for the
fast longitudinal wave in the water-saturated sandy sediment
over the same frequency range as shown in Fig. 4. The error
bars denote the standard deviations of ten measurements. It
is shown that the experimental measurements are in good
agreement with each other over the overlapped frequency
bandwidth 共from 0.5 to 0.7 MHz兲. The solid and the dashed
curves in Fig. 6 represent the attenuation coefficients predicted by the Biot and the GS models, using the input parameters in Tables I and II, respectively. As seen in Fig. 6,
the attenuation data increase nonlinearly with frequency
whereas both the Biot and the GS models yield the attenuation coefficient that is almost linear in frequency from
0.3 to 1.0 MHz. It can also be seen that the Biot model consistently underestimates the experimental measurement of atJ. Acoust. Soc. Am., Vol. 121, No. 5, May 2007
FIG. 6. Experimental and theoretical attenuation coefficients plotted as a
function of frequency for the fast longitudinal wave in the water-saturated
sandy sediment.
tenuation. This is because the Biot model predicts absorption
due to the viscous dissipation of the pore fluid only. In contrast, the prediction obtained from the GS model is larger
than the measurement over most of the frequency range.
Ohkawa 共2006兲 found an excellent agreement between
the SAX99 attenuation data and the calculated attenuation
coefficient given by the algebraic sum of absorption and
scattering components. The absorption attenuation corresponds to the viscous dissipation of the pore fluid following
the f 1/2 dependence, as the Biot model predicts, and the scattering attenuation is given by the empirical f 2 dependence
obtained by curve fitting to the experimental data for watersaturated fine sands with a mean grain diameter of 230 ␮m
measured by Seifert et al. 共1999兲. The dotted curve in Fig. 6
represents the total attenuation coefficient, ␣, in units of dB/
cm, calculated using
␣ = ␣a + ␣s ,
共3兲
where ␣a is the absorption of the Biot model and ␣s is the
scattering obtained by curve fitting to the experimental data
of attenuation coefficient for the water-saturated sandy sediment measured here, expressed as
␣s = 0.28 ⫻ 10−10 ⫻ 20 log共e兲f 2 ,
共4兲
where f is the frequency of the wave in hertz. As found in
Fig. 6, the total attenuation coefficient shows a reasonable
agreement with the experimental data for overall frequencies.
These findings underpin the fact that the scattering is the
dominant cause of attenuation in the water-saturated sandy
sediment with a mean grain diameter of 425 ␮m from
0.3 to 1.0 MHz. However, the scattering attenuation becomes weaker at lower frequencies and effects of the fluid
viscosity may become more important. The role of absorption and scattering in attenuation at low and high frequencies provides an explanation for the difference observed
when comparing in situ and laboratory measurements.
Lee et al.: Acoustic properties of a water-saturated sediment
2557
IV. CONCLUSIONS
We have investigated the frequency dependencies of
phase velocity and attenuation coefficient for the fast longitudinal wave in a water-saturated sandy sediment over the
frequency range from 0.3 to 1.0 MHz. The experimental
data of phase velocity exhibited the significant negative dispersion, with the mean rate of decline of 120± 20 m / s / MHz.
The Biot model predicted the approximately nondispersive
phase velocity and the GS model exhibited the slightly positive dispersion. In contrast, the predictions of the multiple
scattering models, such as the familiar QCA and the selfconsistent EMA, for the negative dispersion in the glassgrain composite were in general agreement with the experimental data for the water-saturated sandy sediment measured
here. The experimental data of attenuation coefficient was
found to increase nonlinearly with frequency from
0.3 to 1.0 MHz. However, both the Biot and the GS models
yielded the attenuation coefficient increasing almost linearly
with frequency. The total attenuation coefficient given by the
algebraic sum of absorption and scattering components
showed a reasonable agreement with the experimental data
for overall frequencies. This study suggests that the scattering is the principal mechanism responsible for the variations
of phase velocity and attenuation coefficient with frequency
in water-saturated sandy sediments at high frequencies.
ACKNOWLEDGMENTS
This work was supported by the Korea Research Foundation Grant funded by the Korean Government 共MOEHRD兲
共KRF-2005-214-C00060兲. The authors thank the referees for
their cogent and insightful comments.
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Lee et al.: Acoustic properties of a water-saturated sediment