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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 1890 1890 1891 1891 1891 1891 1891 1897 1899 1899 1902 1902 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 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.132.123.28 On: Thu, 13 Feb 2014 02:16:58 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 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.132.123.28 On: Thu, 13 Feb 2014 02:16:58 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 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.132.123.28 On: Thu, 13 Feb 2014 02:16:58 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 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.132.123.28 On: Thu, 13 Feb 2014 02:16:58 100 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 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.132.123.28 On: Thu, 13 Feb 2014 02:16:58 800 700 - 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