A review of theoretical models for attenuation of sound by bubble clouds

Transcription

1 Acoustical Oceanography: Paper ICA A review of theoretical models for attenuation of sound by bubble clouds Maria Paz Raveau (a) (a) Pontificia Universidad Católica de Chile, Santiago, Chile, mpraveau@uc.cl Abstract Attenuation of sound propagating through a bubble cloud has been primarily investigated at high frequencies, far above the bubble resonance. Near the resonance, the effective medium method has been used to study sound attenuation, by considering the bubble cloud as a single scattering object, whose internal acoustic properties are determined by a propagation wave number. This approach, however, do not consider the acoustic interaction between bubbles, which may be significant for dense clusters. Recently, the optical theory has been used to calculate the total power loss from the incident wave due to scattering and absorption by bubble clouds. This loss is directly related to the behavior of the scattered wave in the forward direction, and can be estimated utilizing a scattering model which incorporates multiple scattering effects between bubbles. However, two problems arises with this model: (1) it is unclear how to relate the sound extinction with the attenuation coefficient when the interaction between bubble is significant, and (2) the optical theory assumes that the receiver is away from the source, which may not be true for noise mitigation applications. This work aims to explore the limitations of these two models, in regard to the frequency range of applicability, density of bubbles in the cloud, and distances among source, bubbles and receiver. A review of published experimental data is included. Keywords: bubbles, scattering, attenuation

2 A review of theoretical models for attenuation of sound by bubble clouds 1 Introduction Sound propagation in the ocean has been a topic of great interest since the middle of the last century. While light is absorbed within a few meters in the water column, sound can propagate long distances with little attenuation. Therefore, acoustic methods have been widely used as a tool to investigate the ocean. Recently, anthropogenic underwater noise has concerned biologists and ecologists, due to its effects in marine environments. It is known that low-frequency noise, for example, can interfere with marine mammal echo-location systems. [1]. Therefore, it is necessary to develop accurate sound attenuation models to improve the design of noise mitigation systems. A well known strategy is to use air bubbles in water, ([2, 3, 4]) which reduce the amplitude of sound propagated through them, specially in the vicinity of the resonance frequency. The purpose of the present work is to review the existing sound attenuation models in bubbly water and explore their limitations, in particular regarding the effects of multiple scattering near the resonance frequency. 2 Review of attenuation models 2.1 Clay and Medwin (1977) The methodology described by Clay and Medwin assumes that the water contains n bubbles of radius a per cubic meter, and that the bubbles are far enough apart to prevent interaction effects. Let us assume an incident plane wave facing a single bubble of extinction cross section σ e, in [m 2 ]. The attenuation per unit distance is: [5] α b [ db m ] = 10log 10 e σ e n (1) According to the extinction theorem, also called the forward scattering theorem or the optical theorem,[6] the extinction cross section (σ e ) is directly related to the behavior of the scattered wave in the forward direction: σ e = 4π k Im{f b(0,0)}; (2) where Im denotes the imaginary part of, and k is the propagation wave number for water (without bubbles). f b = a/{(ω 2 0 /ω2 ) 1 + iδ B }[m] is the scattering amplitude of a single bubble, where ω 0 = 2π f 0 and f 0 is the Minnaert resonance frequency. [7] f b (0,0) means that f b must be evaluated in the forward direction. The response of a single bubble is omnidirectional, but for an arbitrary object the directional scattering response must be considered. Replacing eq.(2) into eq.(1) yields to: 2

3 α b [ db m ] = 10log 10 e 4πn k Im{f b(0,0)} (3) 2.2 The effective medium approach In 1945, Foldy published his pioneering work in multiple scattering of waves, which derived into a new approach to the problem of ensemble scattering by bubbles, known as the efective medium method. The idea is to consider the entire bubble cloud as a second ( effective") fluid, for which the acoustic properties are described by an effective" wave number k e. The acoustic field due to a wave propagating through the cloud may be determined by solving the Helmholtz equation ( 2 + k 2 e)p = 0. If the bubbles are all identical, k e is given by [8] k 2 e = k 2 + 4πnf b (4) Later on, Commander and Prosperetti performed a model examination through a comparison with available experimental data on linear pressure wave propagation in bubbly liquids.[9] Even when his focus was on non-linear models, the linear case of these theoretical formulations reduces to the results of Foldy. They derived a procedure to obtain the attenuation from the effective wave number. Setting k e /k = u iv, the traveling wave passing through this medium is: e iωt ik ex = e iωt ik(u iv)x The real part of the propagation wave number (i.e., e kvx ) correspond to the decay, therefore, the attenuation coefficient A, in db per unit length is: Rearrangement of eq.(4) yields to: A [ ] db = 20kv log m 10 e = 20Im(k e ) log 10 e (5) k 2 e k 2 = 1 + 4πnf b k 2, (6) Using the two first terms of Taylor s expansion 1 + x 1 + x 2 (valid for x 1), eq.(6) becomes: k e k + 2πnf b k, (7) Substituting eq.(7) into eq.(5) reduces to eq.(3). This means that when k 2 4πnf b (the contribution of multiple scattering to the effective wave number is much less than the water s contribution), the attenuation coefficient proposed by Commander and Prosperetti (eq.(5)) is equivalent to the one found in Clay and Medwin (eq.(3)). 3

4 2.3 When to neglect multiple scattering In Ref.[5], a criterion for using eq.(1) is introduced: σe d < 1 (8) where d is the average interbubble distance (commonly calculated as n 1/3 ). This criterion imposes a limitation on the bubble separation, such that the bubbles are far enough apart to prevent interaction effects. From the attenuation expression derived from the effective wave number (eq.(5)), a limitation arises regarding the interaction among bubbles, even when Foldy s theory was developed to address multiple scattering of waves. As was noticed by Waterman and Truell,[10] Foldy assumed a priori that the exciting field acting on a scatter at a point, is equal to the total field which would exist at that point if the scatterer were not there. This assumption seems not to be always true, as Commander and Prosperetti realized when they tested the theory against the data sets of Silberman and Kol tsova et al. They found that the agreement between theory and experiments deteriorates significantly in the neighborhood of the bubbles resonance frequency. Considering that the average pressure field exciting a bubble should be considerably greater than the pressure wave scattered by a neighboring bubble, Commander and Prosperetti proposed the following criterion: n 2/3 σ s 1 (9) where σ s is the total scattering cross section.[5] Similarly, Waterman and Truell also analyzed the multiple scattering effects due to a random array of obstacles, [10] employing a configurational averaging" procedure. From the condition that the multiple scattering field caused by the insertion of a bubble should be much smaller than the field exciting that bubble, the following criterion is derived: nσ s k 1 (10) An equivalent expression can be found in Frisch [11]: 4πn f b 2 k 1, which was explored in a recent article by Raveau and Feuillade.[12] Henceforward, for simplicity, we will generally refer to Clay and Medwin (eq.(8)), Commander and Prosperetti (eq.(9)) and Waterman and Truell (eq.(10)) criteria, using the abbreviations C-M, C-P, and W-T, respectively. Let us note the similarity between C-M and C-P criteria: if only radiative losses are considered (i.e., σ e = σ s ) and d is replaced by n 1/3 in eq.(8), this expression yields to n 1/3 σ s < 1 which corresponds to the square root of eq.(9). Therefore, even when both C-P and C-M were satisfied, C-M would be a stronger criterion than C-P. On the order hand, W-T has certain similarities with C-P, but is less strict in higher frequencies. When viscous losses are incorporated, the relation between C-P and W-T remains the same, but the extinction greatly increases, and C-M becomes a much stricter criterion. A comparison among these criteria is presented later in Section

5 2.4 How to consider the whole ensemble? In eq.(1), as was originally developed by Clay and Medwin, σ e is the extinction cross section for a single bubble, not for a bubble cloud. But the optical theory allow us to estimate the extinction for a cloud σ es, by calculating the scattering amplitude for the whole ensemble. This can be done by using a mathematical formalism based upon the harmonic solution of sets of coupled differential equations (CDE),[13] which includes all multiple scattering interactions between the bubbles, and calculates the aggregate scattering field by coherent summation. Therefore, eq.(1) may be used for bubble ensembles with σ es instead of σ e, where n would be the density of clouds per cubic meter. This approach was used in a recent work by Raveau and Feuillade [12], to model the attenuation data published by Diachok in 1999.[14] A brief description of the experiment is given in Section 3.1. However, the optic theory assumes that the receiver is located at a sufficiently large distance from the scattering object, so we can approximate the scatter function inside the total scattering integral, with its value in the forward direction. Outside of this region, the scattering function would fluctuate too rapidly to contribute to the extinction. This assumption may not be true for controlled experiments, when the distances among the source, bubbles and receiver is usually in the order of a few meters.[15, 16] In such cases, an alternative approach is to calculate the insertion loss, by computing the ratio of the frequency responses before and after the scatterers are incorporated in the experiment: IL = 20log 10 H ref ( f ) H bubs ( f ) (11) where H ref ( f ) is the frequency response without bubbles in the water and H bubs ( f ) is the frequency response when added bubbles. This method was recently used by Dolder to model sound attenuation by encapsulated balloons.[16] See section 3.2 for a brief description of the experiment. 3 Available data A number of experimental studies on sound attenuation in liquid environments can be found in the literature, but only some of them contain sufficient detail to compare with the theory. We will briefly describe some of these studies, but focus on the laboratory experiment performed by Davies. [15] 3.1 Diachok (1999) At low frequencies, scattering from swim bladder fish is dominated by the monopole resonance of the swim bladder, an internal gas-filled organ that allows the fish to control its buoyancy. Acoustically, a swim bladder behaves like a highly damped gas bubble, and therefore the sound propagation models for bubbly water has been also applied for schools of fish. In the 1999 article by Diachok,[14] an analysis of acoustic transmission data obtained in the Mediterranean Sea in September 1995, is presented. The purpose of this analysis was to study absorption 5

6 due to fish in shallow water ( 80 m depth), by isolating their sound attenuation effect from the overall propagation losses measured in the experiment. In his analysis, Diachok presented absorption coefficient curves α b, obtained by matching the measurements with a computational model of transmission loss. The main purpose of Diachok s analysis was to estimate the abundance of sardines in a school, which is related to the peak frequency of the collective resonance of the school. A new interpretation of these data was recently published by Raveau and Feuillade,[17] which estimates the number of fish in fish schools by matching the peak frequency of the absorption data with computed values of σ es for the school, obtained using CDE method. In a later article by Raveau and Feuillade,[12] the frequency variation of σ es was computed and compared with Diachok s absorption data via the relationship between the extinction cross section and the absorption coefficient given in eq.(1). In this case, n is the number of fish schools per m 3, which are assumed to be approximately the same size. 3.2 Encapsulated bubbles In a recent work by Dolder, [16] a series of experiments were performed in Lake Travis Test Station (ARL, The University of Texas at Austin), to measure the attenuation of sound through an artificial bubble cloud made of 14 fixed air-filled latex balloons of 4.68 cm surface radius. Linear chirps from 30 Hz to 2 khz were produced by an approximately omnidirectional electromagnetic loudspeaker (J-13), and recorded by nine hydrophones, located from 2 meters to 18 meters depth at a horizontal distance of 11.7 meters from the center of the bubble cloud (the details of the equipment and the positions of the bubbles can be found in Ref.[16]). Transfer functions were measured with and without the bubble cloud, subtracting both quantities to determine the impact of adding bubbles to the system. [18] The same approach was followed to estimate a theoretical attenuation, by calculating the Insertion Loss of the system (eq.(11)). The transfer function for the bubble cloud H bubs was calculated using a full-scattering solution, which involves solving the coupled system of differential equations for the scattered field of the bubbles. This result was also compared with the attenuation per meter predicted by the Church model and multiplying it by the total traveled distance, and a discussion of the results can be found in Section 4.3 of Ref.[16]. For our purposes, let us note that the full scattering model, even when is the most computationally intensive and requires knowledge of the bubble geometry, best captures the IL profile. 3.3 Davies (1973) The 1973 article by Davies describes a laboratory experiment using live marine fish as an attempt to collect attenuation data by fish schools and confront it against theoretical values. [15] The experiment was performed in 1972 at the TRANSDEC calibration pool of the Naval Undersea Center. A 132 cm diameter plexi-glass sphere was constructed to enclose the fish (Northern anchovy of 10.6 cm mean length). The sphere was placed in a direct path between the source and and the hydrophone, all three component at a depth of 6 m. The distances from the source to the sphere, and from the sphere to the hydrophone were 4 m and 2 m, 6

7 respectively. Several acoustic runs were made on the water filled sphere without the fish to establish a reference. The fish were then transferred into the sphere, and six sweeps were made over the frequency spectrum from 1 khz to 20 khz. Attenuation values were normalized to the no-fish reference levels previously measured. The mean curve was computed over the six runs, and subsequently converted to db. From underwater observations, it was estimated that the live fish (650) were in a tight group in the center of the sphere, occupying approximately 1 m 3 of the available volume Insertion Loss CDE Data (min. value) Attenuation, [db] Frequency, [Hz] Figure 1: Theoretical attenuation coefficient using eq.(11) and data extracted from Ref.[15] The first peak attenuation, near 3 khz, drew Davies s attention (see Fig.(1)). Since the resonance frequency of a 10.6 cm anchovy should be around 1.2 khz, the 3 khz peak is not likely to be due to resonance. He also compared the results with a theoretical model, by calculating the extinction coefficient for a distribution of swim bladder volumes,[19] and the absorption coefficient with an expression similar to eq.(1), but applicable for distributions (see eq.(6.4.2) from Ref.[5]). The attenuation that he predicted with this model shows a similar frequency distribution than the ones calculated using eqs.(3) and (5) (see Fig.2(b)). As he noted, there is a considerable difference between these two curves and the data, both in frequency response and amplitude. While Davies provided no answer to this discrepancy between model and data, it is known that eq.(1) does not consider multiple scattering effects, which is relevant for modeling the scattering response of dense schools near the resonance frequency. Given a density of 650 fish/m 3, Fig.2(a) shows C-M and C-P criteria introduced in Section 2.3. Since C-M 1 near resonance, multiple scattering effect should not be neglected on this case, so the attenuation expressions proposed by Clay and Medwin, and by Commander and Prosperetti should not be used in this case. It can be noticed that due to the effect of viscous damping, [20] the 7

8 extinction makes C-M criterion very hard to fulfill. Figure (1) also shows a good comparison between the data and the theoretical Insertion Loss (eq.(11)), which was calculated by using the CDE approach, and therefore includes multiple scattering among the swim bladders (a) CM Eq.(8) CP Eq.(9) Attenuation, [db/m] (b) Attenuation Eq.(3) Attenuation Eq.(5) Frequency, [Hz] Frequency, [Hz] Figure 2: (a) Frequency variation of C-M and C-P criteria, as defined in eq.(8) and eq.(9), respectively. The inputs to run the simulations were extracted from Ref.[15]. The W-T curve (eq.(10), not shown here) is similar in amplitude to C-P. (b) Theoretical attenuation computed by eq.(3) and eq.(5). 4 Conclusions Multiple scattering in dense bubble clouds near the resonance frequency is a limitation for current attenuation models, as in the expression proposed by Clay and Medwin, and in the one based on Foldy s effective wave number. Even when it is possible to use full scattering models (which include multiple scattering) to calculate the pressure scattered by bubble clouds, it is unclear how to calculate the extinction when the source, bubbles and receivers are close to each other, which is the case in many controlled experiments and noise mitigation systems. For long range applications, it is also unclear how to relate the extinction to a attenuation coefficient. Also, the optical theory may not be useful when the source is in the middle of the bubble cloud, since it is not possible to define a forward direction. Use of insertion loss to estimate the attenuation is still new, and further investigation is required to explore its limitations. 8

9 Acknowledgements This work was supported by the Fondo Nacional de Desarrollo Cientifico y Tecnologico" N (FONDECYT, Chile). References [1] Richardson, W.J.; Greene Jr, C.R.; Malme, C.I.; Thomson, D.H. Marine mammals and noise. Academic press, [2] Loye, D.P.; Arndt, W.F. A sheet of air bubbles as an acoustic screen for underwater noise. The Journal of the Acoustical Society of America, 20 (2), 1948, [3] Ross, W.S.;Lee P.J.;Heiney, S.E.; Young, J.V.; Drake, E.N.; Tenghamn, R.; Stenzel, A. Mitigating seismic noise with an acoustic blanket - the promise and the challenge. The Leading Edge, 24 (3), 2005, [4] Domenico, S.N. Acoustic wave propagation in air-bubble curtains in water-part i: History and theory. Geophysics, 47 (3), 1982, [5] Clay, C.S.; Medwin, H. Acoustical Oceanography, Wiley, New York, [6] Ishimaru, A. Wave propagation and scattering in random media, IEEE Press, New York (US), [7] Minnaert, M. On musical air-bubbles and the sounds of running water, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Series 7, 16:104, 1933, [8] Foldy, L.L. The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers. Physical Review, 67 (3 4), 1945, [9] Commander K.W.; Prosperetti, A. Linear pressure waves in bubbly liquids: Comparison between theory and experiments. The Journal of the Acoustical Society of America, 85 (2), 1989, [10] Waterman, P.C.; Truell, R. Multiple Scattering of Waves. Journal of Mathematical Physics, 2 (4), 1961, [11] Frisch, U. Wave propagation in random media. Probabilistic Methods in Applied Mathematics, Academic Press, San Diego (US), [12] Raveau, M.P.; Feuillade, C. Resonance scattering by fish schools: A comparison of two models. The Journal of the Acoustical Society of America, 139 (1), 2016,

10 [13] Feuillade, C.; Nero, R.W.; Love, R.H. A low-frequency acoustic scattering model for small schools of fish. The Journal of the Acoustical Society of America, 99 (1), 1996, [14] Diachok, O. Effects of absorptivity due to fish on transmission loss in shallow water. The Journal of the Acoustical Society of America, 105 (4), 1999, [15] Davies, I.E. Attenuation of sound by schooled anchovies, The Journal of the Acoustical Society of America, 54 (1), 1973, [16] Dolder, C. Direct Measurement of Effective Medium Properties of Model Fish School. The University of Texas at Austin, (Doctoral Dissertation), [17] Raveau, M.P.; Feuillade, C. Sound extinction by fish schools: Forward scattering theory and data analysis. The Journal of the Acoustical Society of America, 137 (2), 2015, [18] Leighton, T.G.; White, P.R.; Morfey, C.L.; Clarke, J.W.L.; Heald, G.J.; Dumbrell H.A.; Holland, K.R. The effect of reverberation on the damping of bubbles. The Journal of the Acoustical Society of America, 112 (4), 2002, [19] Holliday, D.V. Resonance structure in echoes from schooled pelagic fish. The Journal of the Acoustical Society of America, 51 (4), 1972, [20] Love, R.H. Resonant acoustic scattering by swim bladder-bearing fish. The Journal of the Acoustical Society of America, 64(2), 1978,