Acoustics of gas-bearing sediments. II. Measurements and
models
Aubrey L. Anderson a) and Loyd D. Hampton
Applied Research Laboratories, The University of Texas at Austin, Austin, Texas 78712
(Received 28 January 1980; accepted for publication 14 March 1980)
Acoustical properties of water saturated and gassy sediments are observed to be significantly different.
The present state of knowledge of the acoustical properties of saturated sediments, gassy water, and gassy
sediments is reviewed in a companion paper. The dynamics of bubbles in water and in various solid
materials, including sediments, are experimentally examined here. Pulsation resonance is exhibited by the
bubbles in all materials examined. Predictions of bubble resonance frequency and damping are shown to
agree with the measurements. Equations for sound speed and attenuation, based on the model of
resonating gas bubbles, are shown to agree with published measurements in gassy sediments. Parameters
required for predicting gassy sediment acoustical properties are identified. Ranges of values of these
parameters for various sediments are discussed.
PACS numbers: 43.10. Ln, 43.35.Bf, 43.30.- k
CONTENTS
I. Introduction
II. Measurement procedures
A. Impedance tube description
B. Sample preparation, bubbles in water
C. Sample preparation, bubbles in Agar
D. Sample preparation, bubbles in Kaolinitc clay
E. Sample preparation, bubbles in polyurethane
III. Measurement results
IV. Bubble dynamics
V. Acoustical properties of gassy sediments
A. Sound speed
B. Attenuation
VI. Conclusions
References
1890
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1902
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1903
I. INTRODUCTION
The acoustical properties of saturated and of gassy
sediments are significantly different. In order to pre-
dict the acoustical properties of gassy sediments, the
dynamics of gas bubbles in sediments must be under-
stood. For gas bubbles significantly larger than the in-
dividual sediment particles, the average elastic pro-
perties of the surrounding sediment particle-water
mixture will cause the bubble to behave as if it were
immersed in a continuous homogeneous elastic solid.
In the preceding paper (Part I) expressions for re-
sonance frequency and damping of bubbles in fish tissue
were shown to combine the elastic and inertial pro-
perties of the gas and the tissue. If the shear rigidity
vanishes, the resonance frequency becomes that for a
bubble, in a fluid. If the shear rigidity becomes very
large, the resonance frequency becomes that for a
cavity in a solid. If the elastic properties and the loss
mechanisms are appropriately identified, then the re-
sonance frequency and bubble motion damping for bub-
bles in sediments will be given by Eqs. (38), (43), (44), and
(46) of the preceding paper. The resonance frequency
is then
0 4c) + (1)
where
a)Present address: Naval Ocean Research and Development
Activity NSTL Station, MS 39529.
r = bubble radius,
y =ratio of specific heats of the gas,
Po = ambient hydrostatic pressure,
Ps =bulk density of the sediment,
G = sediment dynamic shear modulus, and
A -gas polytropic coefficient given by Eq. (26)
(Part I).
Because of the significant difference between the
acoustical properties of gassy materials below and
above the bubble resonance frequency, behavior of gas
bubbles in various materials was investigated experi-
mentally. Measurements were made in an impedance
tube. The acoustical impedance of a column of pure
water, versus frequency, was compared with that of
(1) a column of water with bubbles, (2) a cylinder of
Agar gel containing bubbles, (3) a column of kaolinRe
clay and water containing bubbles, and (4) a cylinder
of polyurethane compound containing bubbles. The im-
pedance tube, sample preparation, and measurement
techniques are described in Sec. II below. Results are
given in Sec. III. Based on these results, the dynamics
of bubbles in sediments is discussed in Sec. IV. In
Sec. V the acoustical properties of gassy sediments
are predicted and compared with observations.
II. MEASUREMENT PROCEDURES
A. Impedance tube description
The impedance tube consisted of a driver assembly
and the tube containing a sample column of length l•.
The tube was stainless steel 30.5 cm (1 ft) long, 7.6
cm (3 in.) i.d., with a 0.6 cm (0.25 in.) wall thickness.
The bottom of the sample column was driven by a
piston sealed into the cylindrical impedance head with
double O-rings. The piston was driven by an electro-
magnetic shaker (driver)ø An accelerometer was
mounted on the piston to measure its acceleration;
velocity at the lower end of the water column was
determined by integrating the accelerometer output.
The sound pressure p at the lower end of the water
1890 J. Acoust. Soc. Am. 67(6), June 1980 0001-4966/80/061890-14500.80 ¸ 1980 Acoustical Society of America 1890
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column was measured by a probe hydrophone mounted
in a short reservoir between the piston and the tube.
Voltage outputs of the accelerometer (E•) and of the
hydrophone (Ep) were input to an automatic impedance
computer. The acoustical impedance Z a presented
to the driver at the bottom of the water column is
given by
, (2)
where S c is the piston surface area and xp, k• are cali-
bration constants relating the hydrophone and accelero-
meter output voltages to the applied pressures and ac-
celerations. Complete descriptions of the impedance
tube and of the calibration procedures are given by
Behrens x and by Hixson et al. 2
B. Sample preparation, bubbles in water
The bubbles used in the impedance tube measure-
ments were air sacs with thin polyethylene walls,
manufactured in sheets for use as a cushioning ma-
terial for packing. The sacs were cylindrical in
shape. The volumes of the three sizes were 0.26 cm s,
5.6 cm •, and 9.3 cm s. For purposes of later discus-
sions, they will be called bubble types 1,2, and 3,
respectively. Measurements were made with from 1
to 20 of the smallest bubbles (type 1); only single large
(type 2 and 3) bubbles were used. In each case, the
air sac(s) was attached with rubber cement to a loop
in the lower end of a 0.25-cm-diam copper rod.
C. Sample preparation, bubbles in Agar
For one series of measurements, a large air sac
(type 3) was placed inside a cylinder of Agar gel. To
support the gel and hold the air sac in place as the
cylinder was being formed, a wire cage was first
made. To form the Agar cylinder, the wire cage and
bubble were placed inside a cylindrical mold 7.5 cm
i.d. A 5% by weight Agar solution was prepared by
mixing the Agar with deionized water and heating the
mixture to 90øC to form a sol. The resulting Agar cy-
linder, containing the large bubble, was 7.5 cm diam
and 9.1 cm high. Impedance measurements were made
with the Agar cylinder and bubble immersed to dif-
ferent depths in the water column.
D. Sample preparation, bubbles in kaolinite clay
Both types 1 and 2 single bubbles were measured with
the impedance tube filled with a mixture of water and
kaolinire clay. The sample was prepared by mixing the
dry clay with deionized water and then placing the mix-
ture under vacuum for one week. This degassing pro-
cedure was performed to ensure that there were no
air bubbles in the clay-water mixture.
The clay-water mixture was removed from the va-
cuum and transferred under water to the impedance
tube. For the impedance measurements, bubbles (air
sacs) were pushed into the clay on the end of small
copper rods, as they had been for the measurements
in water.
E. Sample preparation, bubbles in polyurethane
A single type 3 bubble and a single type 1 bubble were
encased in separate cylinders of Scotchcast 221, a
commercially available polyurethane potting compound.
The cylinders were 5.4 cm diam and 4 cm high. The
bubbles were held in place with thread when the Scotch-
cast was poured. After curing, the samples were im-
mersed to various depths in water in the impedance
tube to measure their resonance characteristics.
III. MEASUREMENT RESULTS
To verify the impedance tube capabilities, measure-
ments were first made using only pure water. The
measured acoustic impedance of various water column
lengths versus frequency is shown in Fig. 1. As the
water column length decreases, the acoustic impe-
dance decreases (the mass decreases) and the quar-
ter-wavelength antiresonance frequency, fmax, in-
creases. The compressional wave sound speed of
the water in the tube, ce, can be computed using
Ce= /maxXre, (3)
where fm• is the quarter-wavelength antiresonance
frequency, and h m is four times the water column
length.
Values calculated with Eq. (3), using data from the
impedance tube, are shown in Fig. 2. Measured sound
speed approaches a constant value of about 1300 m/s
for column lengths in excess of 10 cm. This is 88% of
the sound speed in pure water (1483 m/s) at 20.1øC,
the temperature for these data. The value is below
that for water because of the compliance of the tube
wall. Values of acoustic impedance at 500 Hz were
computed for the water column lengths shown in Fig.
1. Measured and computed values are compared in
Table I. For columns longer than 5 cm, the values
agree within 1 dB.
Acoustic impedance measurements were also made
with a single small bubble (type 1 air sac)and with a
single large bubble (type 2 air sac) immersed to
various depths in a column of water. Examples of the
data are shown in Figs. 3 and 4. An additional series
of antiresonance peaks, •+, and resonance minima,
•., are exhibited by these data.
The antiresonance maximum impedance occurs at the
frequency f,•a•, for which the water column length, to
the pressure release (air) interface, is one quarter-
wavelength. Thus, the increase in frequencyfm• with
increasing bubble immersion d•epth in a constant length
water column is associated with an apparent decrease
in column-length--the bubble is, in effect, reducing
the water column length.
In Fig. 5, frequency fma, is plotted versus water
column length, lw, for the pure water data. On the
same plot, fma, data are plotted versus the distance
between the piston and bubble (lw-d b) for several sets
of bubble-in-water impedance measurements. The data
indicate that a large bubble provides an effective pres-
sure release at its immersion depth because the water
column appears to be only as long as the distance from
1891 J. Acoust. Soc. Am., Vol. 67, No. 6, June 1980 A.L. Anderson and L. D. Hampton: Gas-bearing sediments. II 1891
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lOO
90
80
e• 70
i
z
tu •0
o
'< 50
40
30
O. 1
NO BUBBLE
.... ,,"X/'" /
, I I
0.2
fmax
WATER COLUMN
LENGTH,•w, cm
15.9
./
./
13.2 i I I I I
lO.2 8•.1
.":
ß - •,• 5.4
:: ; i i'!
•' :; i ' ß
.
::, /':
.-' .• './ \
• • ':
i \ /'..
',
.* / 0.7
/
,.., .qk /./
.
/
,
.
0.4 0.6
I I
0.8 1
FREQUENCY - kHz
FIG. 1. Acoustic impedance versus frequency.
2 4 6 8 10
piston to bubble. The small bubble reduces the apparent
column length, but does not completely relieve the
pressure at its immersion depth.
Next, consider the additional antiresonance (f+) peaks
and resonance (f.) dips caused by the bubbles. The
bubble acts as a compliance, dividing the water column
into two masses, one above the bubble, of length d•,
and one below the bubble, of length (l w -d•). The an-
tiresonance (f+) occurs for vibrations of the top mass
and bubble compliance as a two-element resonator.
The resonance dip (f.) occurs for vibrations of the
two masses and the bubble compliance as a three-ele-
ment resonator. Baird s has shown that the frequency
f+ is related to the fundamental pulsation resonance
frequency of the bubble immersed in an infinite body
of water (i.e., to the frequency predicted by Minnaert's
equation in Table IV: Part I). The following equation
relates the frequency f+ to the free field resonanCefo.
/. =fo[1 +(ro/•,)(4d•/rt-1)]'•/•', (4)
15oo
i 1000--
-
500 -
I I I I
0 5 10 15 20
WATER COLUMN LENGTH- crn
i I
25 30 35
FIG. 2. Water column sound speed
versus total water column length.
1892 J. Acoust. Soc. Am., Vol. 67, No. 6, June 1980 A.L. Anderson and L. D. Hampton: Gas-bearing sediments. II 1892
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TABLE I. Comparison of measured and computed acoustic
impedance for water column (500 Hz).
Column
length
(crn)
20 log I Z•[, dB re 1 g/(cm 4 x s)
Measured Computed
15.9 62.9 62.4
13.2 61.1 60.7
10.2 59.1 58.5
8.1 57.2 56.5
5.4 54.0 53.0
2.9 5O .6 47.6
0.7 44.O 41.9
where
r o = bubble radius,
r t = tube radius,
d b = bubble immersion depth,
fo =free field resonance of bubble in water, and
f. =frequency of bubble antiresonance impedance
peak.
The usefulness of impedance tube measurements and
of Eq. (4) in determining free field bubble resonance
was tested by several measurements off. versus bub-
ble immersion depth. Examples of the data are shown
for the small (type 1) bubbles in Fig. 6 and for the large
(type 2) bubbles in Fig. 7; predicted antiresonance peak
frequencies are also shown. The predicted values were
calculated using Eq. (4); the bubble radius was taken
to be that of a spherical bubble of the same volume as
the bubble (air sac) used in the experiment. The work
of S•rasberg 4' 5 indicates that this is valid for non-
spherical bubbles of approximately equal orthogonal
dimensions. Free field resonance frequenciesfo were
computed using Eq. (28) of the preceding paper.
The measurements agree with predictions well within
the repeatability of the data, especially for the large
bubbles. These sets of measurements thus verify the
utility of the measurement technique for predicting the
free field resonance of a bubble.
Measurements off+ were also made on different num-
bers of small (type 1) bubbles immersed to various
depths in the impedance tube. The resulting data are
plotted versus total air volume in the bubbles (air sacs)
in Fig. 8. These data were obtained for center to cen-
ter separations of 1.5 diameters for the type 1 air sacs.
Figure 8 also shows antiresonance frequency, f+, ver-
sus immersion depth for a single type 2 air sac. As the
total air volume in the small bubbles approaches the
air volume in the large bubble, the measured resonance
of the small bubble cluster approaches the resonance
of the single large bubble. The data indicate that, for
this close (1.5 diam) proximity of the bubbles, the re-
sonance frequency is primarily a volumetric effect,
i.e., the small bubble collection resonates as if the
total ai r volume were contained in a single larger bub-
lOO
8o
i•1
z
,, 60
SMALL BUBBLE
90 (TYPE 1 AIR SAC)
15.9 cm WATER
COLUMN ONLY
fmax
I I I
/- 2.5 BUBBLE IMMERSION DEPTH - cm
/--5.1
•-7.6
• •/-,O.2
12.7
40-
30 I I I
O.I 0.2 0.4
FIG. 3. Acoustic impedance versus frequency.
i i i i
0.8 1 2
FREQUENCY - kHz
I I I I
i i i I i i i
4 6 8
1893 J. Acoust. Soc. Am., Vol. 67, No. 6, June 1980 A.L. Anderson and L. D. Hampton: Gas-bearing sediments. II 1893
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100 t I • t t • i I I
90
80
LARGE BUBBLE
(TYPE 2 AIR SAC)
I I ' I I I I I I
2.5 BUBBLE IMMERSION
fmax ja• 5.1 Z6 DEPTH - cm
....:. •. 1o. 2
:.
ß %. / ! ß .
15.9 cm WATER
COLUMN ONLY
FIG. 4. Acoustic impedance versus frequency.
I I I I I I I
4 6 8 10
ble. This is in contrast to the measurements of
MacPherson, e where the widely dispersed bubbles re-
sonated at the free field frequencies of the individual
bubbles.
The resonance characteristics of bubbles in solids
were tested by measurements in Agar gel, in kaolinitc
clay, and in Scotchcast 221 potting compound. Bubble
antiresonance frequency, f+, data were obtained for a
single type 3 bubble in an Agar cylinder immersed to
various depths in the water column. These data are
compared in Fig. 9 with calculations made using Eq. (1)
(for the free field resonance frequency of a bubble in
a medium having nonzero shear modulus) and Eq. (4)
>-
U
Z
:3 3
U
Z
Z
0 2
ß
&
ß
-i
o& ß
ß
&
&
&
ß WATER COLUMN
& SINGLE LARGE BUBBLE (TYPE 2)
ß SINGLE SMALL BUBBLE, (TYPE 1) 15.9 cm WATER COLUMN
ß SINGLE SMALL BUBBLE, (TYPE 1) 30.5 cm WATER COLUMN
& ß
0 0 5 10 15 20 25
WATER COLUMN LENGTH BETWEEN PISTON AND AIR INTERFACE (Jw)or BUBBLE (.Jw-db) -½m
FIG. 5. Column length resonance frequency, fmax, versus column length to air interface or bubble.
I I I I
1894 J. Acoust. Soc. Am., Vol. 67, No. 6, June 1980 A. L. Anderson and L. D. Hampton' Gas-bearing sediments. II 1894
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800
700 -
i
u 600-
z
500 -
TYPE 1 AIR SAC
PREDICTED
ß MEASURED
400 I I I I I I I I I I I
4 5 6 7 8 9 10 11 12 13 14 15
BUBBLE IMMERSION DEPTH, (Jb- cm
FIG. 6. Antiresonance frequency, f+, of column and bubble
versus bubble immersion depth.
(to correct the free field resonance to the resonance,
in a tube). A value of 10 6 dyn/cm •' was used for the
shear modulus of Agar gel in applying Eq. (1). This
value was obtained using a combina[ion of new mea-
surements and values in the literature. ?
On completing the measurements for the bubble in
Agar gel, the gel was stripped from the wire cage and
bubble. Measurements of the resonance frequency of
the bubble, still in the wire cage, were made as the
bubble was immersed to different depths in the water
column. The results are also shown in Fig. 9. The-
oretical values were computed using Eq. (28) (Part I)
and Eq. (4). Another series of measurements was
made by immersing the wire cage, without the bubble,
in a column of water. The wire cage did not change
the measurements from those obtained with only a
column of water. Thus, the wire cage was acoustically
invisible at the frequencies of these measurements.
The data shown in Fig. 9 indicate that Eq. (1) predicts
the bubble resonance frequency in a medium with shear
modulus on the order' of 106 dyn/cm •', and with density
close to that of water.
Column antiresonance frequency data were obtained
for a single small (type 1) bubble and for a single large
(type 2) bubble immersed to various depths in a kaoli-
nite clay and water mixture. The small bubble results
are shown in Fig. 10, the large bubble results, in Fig.
11. In both figures, the data are compared to predic-
tions obtained using Eqs. (1) and (4). A 10-ml sample of
the clay-water mixture was used to determine its bulk
density, 1.42 g/cm s. The shear modulus, 2 x 10 s
dyn/cm •' was taken from Cohen. s These values were
used to obtain the theoretical lines in Figs. 10 and 11.
Data and predictions are also shown in these figures
for the same bubbles immersed in water. Note that
the antiresonance frequency of the bubbles in the clay-
water mixture is less than in water. This occurs be-'
cause the density of the medium increases significantly
while the mixture shear modulus is not large enough to
significantly modify the numerator in Eq. (1). This is
in contrast to the results for a bubble in Aga