A comparison and review of theories of the acoustics of porous materials.

Transcription

1 A comparison and review of theories of the acoustics of porous materials. M. R. F. Kidner and C. H. Hansen Acoustics Vibration and Control Group School of Mechanical Engineering The University Of Adelaide South Australia 5005 June 24, 2008 Abstract This article reviews the research on acoustic waves in porous media. Particular emphasis is placed on the relationship between the full Biot Allard (J. Allard, Propagation of sound in porous media. Elsevier Applied Science, 1993.) model and the simpler approximations presented by Zwikker and Kosten(C. Zwikker and C. W. Kosten, Sound Absorbing Materials. Elsevier Publishing Inc, New York, 1949.), Morse and Ingard (P. M. Morse and K. U. Ingard, Theoretical Acoustics. Princeton University Press, Princeton, New Jersey, 1986.), and others. A comparison of several models used to predict the absorption characteristics of porous materials is presented. Now with Vipac Engineers & Scientists, King William St, Kent Town, SA,

2 1 Introduction Porous materials are commonly used in noise control applications because they are very easy to install and provide excellent absorption at mid to high frequencies. Although these materials are widely used, the theoretical modelling is surprisingly complex so empirical approximations 1 are often used. These approximations often prove to be valid for most practical applications; however, researchers have shown that a complete theory can shed light on unexpected behaviour. A complete theoretical description of waves in porous materials was obtained by Biot 2, 3 in The theory described the motion of the fluid in the pores and the motion of the solid pore walls, or frame. Since that time a vast quantity of work has been done to apply this theory to many diverse fields including acoustics, geo-mechanics and bio-dynamics as discussed below. S ound absorbing materials Porous materials have been used for sound and vibration absorption because of their efficacy, cost effectiveness, and simplicity to install. The focus of this paper is the modeling of the process of sound propagation within these materials. The discussion will follow the theoretical models from rigid frame assumptions as illustrated in Fig. 1 to a model that includes an elastic frame as shown in Fig. 5. Geomechanics & Ocean Acoustics Porous mechanics is applied most widely in the field of geo-mechanics. 4 The prediction of soil motion during earthquakes is a particularly important topic. The propagation of seismic waves through porous rocks can yield valuable information for mineral and fossil fuel explorations. Oceanography has also benefited from the more complete model of the ocean floor that the porous media models 5 provide. Biot s 2, 3 description of porous materials accurately predicts how acoustic energy travels through ocean sediments and rocks. Descriptions of granular materials such as those by Berryman 6 and Morse 7 are applicable to the propagation of acoustic energy through ocean sediments and rocks. Biological Systems Biological systems such as lungs and bones can also be modeled as porous materials. The prediction of the acoustic response in these materials is increasingly relevant to ultrasonic imaging as greater resolution becomes necessary. Ultrasonic propagation in bovine bones has been predicted, 89 using Biot s 2 equations. Reflection and transmission of ultrasound pulses through bone is the 2

3 focus of many bio-mechanics research papers on waves in porous media. 1.1 Paper Overview The dynamics of porous materials are complex, not only because porous materials come in different forms, but also because of the micro and macroscopic scales required to fully describe them. This results in an unsatisfying mix of crude approximations and obtuse detail, often within the same model! Attenborough 10 gave a thorough review of the detailed modelling of porous materials in which he consolidated many of the approaches and described their assumptions in great detail. This paper will attempt to summarise the most common approaches to modeling porous materials with respect to acoustic waves. The paper is laid out as follows: The simple Delany Bazley 1 approximation is reviewed, and this is probably the model with which most readers are familiar. From this point the discussion focuses on the microscopic scale and reviews descriptions of fluid flow through pores, as the majority of research from 1890 to 1950 concentrated on this mechanism. The Biot 2, 3 model, which is currently accepted as the most complete description of porous dynamics, is then discussed. In the penultimate section, the Biot equations are reviewed with reference to their application to numerical methods for solving acoustics problems such as the prediction of transmission loss through double panels with porous liners. At each stage, the sound speed predicted by the various models is shown. In the penultimate section the difference in the sound absorption predicted by each of the methods is discussed. 2 The Delany Bazley Approximation Delany and Bazley 1 developed a simple empirical model of the acoustic impedance of a porous material based on its flow resistivity, R 1. Flow resistivity is defined as the pressure required to generate a unit flow through the material per unit thickness. A material with a low porosity, φ and complicated micro structure is likely to have a higher flow resistivity than one with large simple pores. The porosity is defined as the ratio of pore volume to total volume of the material, with porosity commonly having a value between 0.95 and 0.99 for materials used for sound absorption, but generally less than 0.5 for granular materials. 3

4 Delany and Bazley proposed the following empirical fit to experimental mesurements of impedance, where positive time dependence, e jωt, is assumed in the analysis. Z = ρ 0 c 0 ( χ j0.087χ ) (1) and the following fit to calculations of the effective wavenumber (k = 2π/λ), k = 2πf ) ( χ 0.7 j0.189χ 0.595, (2) c 0 where the non-dimensional parameter χ is defined as χ = ρ 0f R 1. (3) Here f denotes the frequency in Hz, R 1 is the flow resistivity of the porous material, and ρ 0 and c 0 are, respectively, the density and speed of sound in the fluid without the presence of the porous material. The validity of Eqs. (1) and (2) extends over the range 0.01 < χ < 0.1. Bies and Hansen 30 provided a formulation to extend both the low and high frequency ranges for any value of χ. The prediction of the acoustic impedance of a material is often all that is required as once this is obtained, the absorption coefficient can be easily calculated. 11 Fig. 2 shows the variation in impedance and absorption as a function of the parameter ρ 0f R 1. It can be seen that the real and imaginary parts of the impedance, indicated by the solid and dashed line respectively in the upper figure have very similar forms but different signs. Note that the absorption shown in the lower figure can not reach unity over the range for which the model is valid. It is interesting to note the change in sound speed within the porous material, this is shown by the dashed curve in the lower part of Fig. 2. The curve shows the normalised sound speed, c/c 0 for the material, (the axis is on the right of the figure). The sound speed is less than half of that in air implying that the wavelengths in the porous materials are shorter. For many applications of porous materials for sound absorption the Delany Bazley empirical model is adequate. Bies and Hansen 12 showed that for fibrous materials, not only was the flow resistivity adequate to describe impedance, but the flow resistivity was in turn a linear function of bulk density. However, the model does not fully describe the dynamics of the materials and so the accuracy of any detailed design or optimization of the absorbing and transmission properties of the materials is limited. 4

5 2.1 The structure of porous materials Porous materials come in many forms, and yet attempts are made to apply the same mathematical models to all of them. Here we briefly review the different kind of structures that form porous solids. Rigid Frame The rigid frame assumption is the simplest and was applied by Rayleigh 13 in his initial investigations. It is assumed that the material is formed by a number of pores within a rigid solid. The geometry of the pores determines the characteristics of the material, as will be discussed in section 3. Within this class there are two further assumptions to be made about the form of the material: fibrous or granular. Fibrous In this model the frame is assumed to be formed of long thin strands of rigid material, which results in cylindrical-like pores, as in rock wool batting for example. Granular The granular assumption is often applied to materials such as sediments, rocks and soil. The material is assumed to be formed from a number of closely packed spheres. The packing geometry determines the properties of the flow in the pores and hence the attenuation of acoustic waves. Limp Frame The limp frame assumption is similar to the rigid frame assumption in that it removes the frame dynamics from the model. In this case there is some additional inertial loading on the acoustic wave which causes an expected drop in sound speed. Elastic Frame The elastic frame assumption is the most valid as it includes the frame motion. However it is more complex, as it turns out that three waves exist in a poro-elastic material. This will be discussed in section 4. In the following sections a brief history of research on acoustic wave propagation in porous media is presented. From Rayleigh s first explanations given in 1896 to the development of finite and boundary element solutions in the 1990s the field has seen continuous contributions and improvements from acoustics researchers. 5

6 3 Rigid Framed Materials Rayleigh 13 considered the reflection of sound from a haystack and in doing so set the pattern of thought for many subsequent theories. He considered a series of very small cylindrical pores of radius r in an otherwise impermeable surface, as shown in Fig. 1-a. The walls of the tubes do not move and so the honeycomb frame that they form is considered to be rigid. By calculating the acoustic impedance of a layer of air at the surface he derived a reflection coefficent in terms of the porosity, φ, the dynamic viscosity of air, η and the acoustic wavenumber, k. The determination of the correct flow resistivity was to dominate most theoretical investigations for many years. It was accepted that rigid framed porous media could be represented by a series of very small tubes. In later research an additional structural factor, ϕ was included to account for their random orientation as shown in Fig. 1c e. In 1926 Crandall 14 presented an analysis of the flow in narrow tubes. He determined that it was governed by an adiabatic process in contrast to the isothermal assumption made by Rayleigh. However as shown by Zwikker and Kosten 15 the actual process is somewhere between these two extremes; changing from isothermal (Poiseuille flow) at low frequencies to adiabatic (Helmholtz flow) at high frequencies. A detailed description of the process of energy absorption due to flow within narrow tubes depends on the fluid dynamics and a non-dimensional parameter µ is required. The parameter is defined as ωρ 0 r µ = 2, (4) η The parameter relates the mass of the fluid in the pore, ρ 0 r 2 to the viscosity η. In other words, whereas if µ is small the motion is dominated by inertial effects, if µ is large the motion is dominated by viscous processes. To describe the behaviour of acoustic waves, the effective density ρ and bulk modulus, K of the medium is required; the combination of these yields the sound speed. Zwikker and Kosten derive a complex density, shown in Eq. (5), with the assumed imaginary part being a function of the resistance coefficient, R 1. ρ = ϕρ 0 φ + R 1 jω. (5) where ϕ is the structural factor, which accounts for the misalignment between the pores and the direction of the pressure gradient. It has a value greater than 1, for example if the material is made of randomly aligned tubes ϕ = 3. 6

7 The expressions for the resistance, R 1 of the tube is a function of µ, R 1 = 8 ϕ ωρ 0 φ µ 2 µ 1 Poiseuille, (Isothermal) (6) R 1 = 2 ϕ ωρ 0 φ µ µ 1 Helmholtz, (Adiabatic), (7) The transition between the two regimes is governed by the magnitude of µ. At low frequencies or for very narrow pores µ < 1. The effective density is plotted against the non-dimensional pore size parameter µ in Fig. 3. Zwikker and Kosten then go on to present a model that is valid for all frequencies. They derived an effective density, ρ (by neglecting thermal effects) and an effective modulus, K (by neglecting viscous effects). { ρ = ρ 0 / } J 1 (ϑ), (8) ϑ J 0 (ϑ) and { K = κp 0 / } Cϑ (κ 1)J 1(Cϑ), (9) J 0 (Cϑ) where ρ 0 is the density of the fluid, P 0 is the mean pressure, κ is the bulk modulus of the fluid, C = ν = ηκ ρ 0 ν is the square root of the Prandtl number, ( 0.86 for air), λ h ρ 0 C p, λ h is the thermal conductivity of the fluid, C p is the specific heat of the fluid at constant pressure, ϑ = µ j and J n is a Bessel function of order n. The form of these equations is due to the solutions of the differential equations that describe the motion of air in a cylindrical pore. 14 The factor J 1 ()/J 0 () is the mean velocity over the pore area. By then extending Kirchoffs theory for sound propagation in pores, Zwikker and Kosten derived expressions for density and bulk modulus that are valid at high and low frequencies in which neither thermal nor viscous effects are neglected. For cylindrical pores in air the following expressions were derived: ρ = ϕ ( 4ρ ) φ 3 jµ 2 K = 1 φ P ( jµ 2 ) µ < 1, (10) ρ = ϕ ( φ ρ ) jµ ( K = 1 ) 0.92 φ κp 0 1 jµ µ > 10. (11) These expressions as well as the expression given in Eq. (9) are plotted in Fig. 4. It is important to note that although Eqs. (5) to (11) are complicated, they are only expressions for the effective density and bulk modulus for the porous materials. By inspecting Figs. 3 and 4, we can see from the solid lines that the effective 7

8 density decreases with frequency and pore size and the bulk modulus increases. The density increase for small values of µ, explains the drop in sound speed shown in Fig. 2. The bulk modulus only increases slightly with µ from the adiabatic case to the isothermal case. In summary the sound speed in porous materials is slower than in the fluid alone and the acoustic process is often closer to isothermal than adiabatic, which is contrary to the assumption made for acoustics in air. Morse and Bolt 7 presented a similar model in which they also derived an effective stiffness and density. However Beranek 16 pointed out the poor agreement with experimental data at low frequencies, stating that Morse and Bolt explain this by adjusting the porosity. 3.1 Granular Materials Morse 17 presented a theory in 1952 for the propagation of sound in granular materials in which he accounted for the viscous losses in the pores and the additional mass of the fluid due to the inertial interaction with the frame. Morse derives a complex wavenumber of the form k = ω c 0 ϕ jr 1φ ρ 0 ω. (12) where ϕ is the same structural factor as that used by Zwikker and Kosten. For the materials investigated by Morse the value of varphi was between 2 and 3.4. For cases where φr 1 ρ 0 ωϕ 1 the sound speed, c 0 and attenuation factor, β can be approximated as c = ϕc 0 (13) β = R 1φ 2ρ 0 c 0 ϕ (14) Morse and Ingard 18 present the same model, using a different notation. A complete and self consistent model for the mechanics of granular porous materials was presented by Berryman. 6, 19 He included the elastic response of the granules and showed that it was consistent with Biot s 2 model. 4 Elastic Frame In 1947 Beranek 16 presented models for rigid tiles and flexible blankets in which he considered the motion of both the frame and the fluid in the pores. His model resulted in two waves, one predominantly in the frame, the other in the fluid. 8

9 Kosten and Janssen 20 presented a model of a porous material with an elastic frame based on the work by Zwikker and Kosten. To include both frame and fluid motion an effective density and modulus for each must be defined. A coupling coefficient between the fluid and frame motion is also needed. Kosten and Jansen 20 presented a model for wave motion in terms of frame and fluid velocity and pressures as: ik 0 jωρ 1 + τ τ 0 jk τ jωρ 2 + τ jω 0 jkg jk(1 φ)k 2 0 jω jkh jkφk 2 G = p 1 p 2 v 1 v = 0 0 (15) ] [K 1 + (1 φ)2 φ (K 2 P 0 ) and H = (1 φ)(k 2 P 0 ), where P 0 is the the ambient pressure. The subscripts 1 and 2 refer to the frame and fluid respectively. The coupling coefficient is denoted τ and the moduli and effective density of the frame or fluid are indicated by K and ρ respectively with the appropriate subscripts. The frame and fluid velocities are denoted v 1 and v 2 respectively. The stress on the frame is denoted p 1 and the pressure in the fluid is indicated by p 2. The variables are illustrated in Fig. 5. The matrix shown in Eq. (15) represents the independent motion of the fluid and frame and the effect of the relative velocity between them as described by τ. The coupling coefficient is defined as a function of the structural factor, ϕ and the effective and real fluid densities: τ = jω(ϕφρ ρ 0 ). (16) As shown in Fig. 3 the effective density ρ can vary from ρ 0 by large values for small µ (low frequency / small pores). The factor K 2 in Eq. (15) is the effective bulk modulus of the fluid in the pores and is given by Eq. (9). By assuming a harmonic solution and solving for the resulting wavenumbers it can be shown that two waves exist. The normalised wave-speed as a function of ρf/r 1 is plotted with the black lines in Fig. 7. Compare this result to that shown in lower part of Fig. 2, the slower wave is similar in speed to that predicted by the Delany Bazley empirical model. In 1956 Biot 2, 3 took a fresh approach to the modeling of wave propagation in porous materials. His model could accommodate fluids and frames of similar density and it included rotational as well and longitudinal waves in the solid phase. He reduced the wave equations to a function of four parameters A, N, Q and R. 9

10 The parameter A is defined as νe/(1 + ν)(1 2ν), where ν denotes Poisson s ratio and E is the Young s modulus of the frame material. N denotes the shear modulus. R is a measure of the pressure required to force a portion of the fluid into the fluid frame aggregate while maintaining a constant aggregate volume. 2 The constant Q relates the volume changes of the fluid to that of the frame and is defined by Q/R = ɛ/e. Where ɛ is the fluid strain and e is the volumetric strain of the frame. The Poiseuille (isothermal) assumption was made for the low frequency behavior and in the second part of the article high frequency behaviour was investigated. Biot approached the problem of deriving the equation of motion for a porous material with an elastic frame from a stress-strain point of view. By deriving stress tensors for the solid and fluid phases of the material and assuming an isotropic material he arrived at the following stress strain relationships: σ n = 2Ne n + Ae + Qɛ, (17) where n = x, y, z, are the cartesian coordinates. The total volumetric strain of the frame, e is given by the divergence of the displacement vector denoted by u. The fluid volumetric strain ɛ is given by the divergence of the fluid displacement field denoted by U. Fig. 5 shows the acoustic and structural stresses and strains that occur in a poro-elastic material. The complete description of the poro-elastic dynamics involves coupled motion of the structure and the fluid. The stress, σ x, in the x-direction on the structure is due to both solid and acoustic strains. Biot then defines the fluid pressure p as: p = Rɛ Qe, (18) and finally, he also included the shear stress and strain τ nm = τ mn = Nγ nm. (19) The shear modulus is denoted N, the shear strain in the n, m plane is denoted γ nm where n, m can be any orthogonal combination of the axis x, y, z. The corresponding shear stress is indicated by τ nm. To derive equations of motion for the material, the inertial coupling between the fluid and the frame must be defined. Biot derives the following expressions to accommodate this coupling: ρ 11 = ρ 11 + b jω ; ρ 12 = ρ 12 b jω ; ρ 22 = ρ 22 + b jω ; 10 ρ 11 = ρ s + ρ a ρ 12 = ρ a ρ 22 = ρ 0 + ρ a (20)

11 Each of the effective densities ρ 11, ρ 12 and ρ 22 are respectively defined as: The mass of the fluid that couples to the motion of the frame, ρ a ; The effective moving mass of the frame, which is its actual mass plus the mass of the fluid that moves with it, (ρ 11 + ρ a ); The effective moving mass of the fluid, (ρ 22 + ρ a ). The starred quantities include the viscous damping effects modeled by a complex term b/iω. 21 The inertial coupling between the fluid and the frame, ρ 11, is a function of the structural factor ϕ, and the fluid density ρ f. ρ a = (1 ϕ)ρ 0 Note that the effective densities given in Eq. (20) are complex quantities. The imaginary part is a function of the viscous losses due to the relative motion of the fluid and the frame. These viscous losses represented by b are a function of the assumed pore geometry. There are several models for b, with the following suggested by Johnson et al. : 22 b = ρ f ϕ 2J ( 1(ϑ) 1 2J ) 1 1(ϑ) = 2J 1(ϑ) ϕρ (21) ϑ J 0 (ϑ) ϑ J 0 (ϑ) J 0 ϑ Note the similarity in form to the expressions for effective density and modulus derived by Zwikker and Kosten 15 shown in Eqs. (8) and (9). The parameter b is plotted against the non-dimensional pore size parameter µ in Fig. 6 where it can be seen that small pores have higher viscous losses. From Eqs. (17) to (19) the wavenumber for the three waves can be calculated by formulation of the wave equation in terms of the fluid and frame velocity potentials as shown in Eqs. (23) and (23) ω 2 ( ρ 11 U s + ρ 12 U f ) = P 2 U s + Q 2 U f (22) ω 2 ( ρ 22 U f + ρ 12 U s) = R 2 U f + Q 2 U s (23) where U = u and the superscipts s, f refer to structure and fluid respectively. Assuming a harmonic for the velocity potential yields the wavenumber as the roots of the resulting fourth order polynomial. The resulting wave speeds for the two in-plane waves are plotted in Fig. 7. The wave speeds were calculated using three different expressions for the effective moduli of the frame and fluid as specified by Bolton, 21 and Allard. 23 However it can be seen that they are in quite close agreement. Note that these wave speeds 11

12 calculated from the Biot equations are different than those predicted by the earlier model of Kosten and Jansen. 20 From the stress strain relationships given in Eqs. (17) and (18) the wave amplitudes for the slow, fast and shear waves can be calculated. Allard 23 derived expressions for the impedance of each wave type and hence the acoustic impedance of a porous layer. 5 Comparison of models Figs. 8 and 9 show the prediction, (by three different methods), of the acoustic impedance and absorption rigidly backed of a 30cm layer of porous material respectively. The material has a flow resistivity of Pa s m 2 and a porosity of The frame bulk modulus is Pa and the effective frame density is 130 kg/m 3. The rigid frame model of Zwikker and Kosten, 20 the Delany Bazley 1 empirical model and the Biot 2 elastic frame model were used. It can be seen that the frame resonance is only predicted by the elastic frame model. The Delany Bazley and rigid frame models predict the broadband trend well, especially given the simplicity of the Delany Bazley approach. The full Biot model is complicated to implement but does predict the resonant behaviour of foam layers, which can be important for absorbent liners made from multiple layers and narrow band applications. For example the addition of a limp mass layer can introduce resonances within the frame of the porous layers that can greatly effect performance but would not be predicted by the rigid frame model. 6 Implementation of the models It is now generally accepted that Biot s description of the dynamics of poro-elatic materials is the most accurate model. Predictions for the acoustic and structural response of a poro-elastic system can be obtained by wave based models that assume fairly simple geometries but yield exact solutions, or by numerical methods that allow more complex geometry but are computationally intensive. 6.1 Wave based solutions A matrix based approach to modeling the transmission loss through porous layers of infinite extent was proposed by Allard. 23 The method allows large numbers of 12

13 poro-elastic layers to be easily combined with structures, such a plates, in order to determine the overall transmission loss. If the wave amplitudes at the surface of the porous layer can be determined, the displacement at any point within the layer can be obtained. It is assumed that there is a matrix T such that the velocities and stresses at one point in a layer may be related to those at another. Thus, [ ] [ ] [ ] V( x l ) = T V( x 0 ) (24) where [ ] [ ] V = vy s vx s vx f σxx s σyx s σxx f Here the superscript s indicates the structural or frame variables and the superscript f indicates fluid variables. The stresses are denoted σ and the velocities are indicated by v. The transfer matrix is [ ] [ ] [ 1 T = Γ(0) Γ(l)], (25) with the components of Γ given in Allard 23 (p151 Table 7.1.). The matrix T can then be evaluated numerically. 6.2 Finite and Boundary element models Allard 23 and Bolton 21 both derived methods for predicting the transmission and absorption of sound by layers of poro-elastic material of infinite extent. However, the majority of engineering problems require analysis of more complex geometries than that of flat layers. The application of finite and boundary element techniques has therefore become necessary. Kang et al. 24 presented a finite element formulation for poro-elastic materials that could be easily coupled to existing acoustic elements. The six degrees of freedom per node for the element were the fluid and frame displacements, ( U, u). This approach leads to large frequency dependent matrices that are inefficient to solve. Göransson 25 then presented a simplified approach using the frame displacement and the fluid pressure, ( u, p) as the degrees of freedom. This simplified the coupling to the fluid and frame but ignored the elastic coupling between the fluid and frame within the material. An efficient approach to the u, U method was suggested by Panneton and Atalla. 26 The frequency dependence of the damping and stiffness matrices is approximated by very simple linear functions. 13

14 Göransson presented a symmetric finite element formulation 27 that required the lowest number of degrees of freedom per node, (only 5 instead of the 6 or 7 required by other formulations). He also presented a rigourous analysis of the coupling integrals. The symmetry was achived by using a pressure and a fluid displacement potential as parameters instead of the fluid displacement itself. Panneton and Atalla 26 express the Biot equations (see Eqs. (17) and (18)) as, σ s = D s ɛ s + D sf ɛ f (26) σ f = D f ɛ f + D sf ɛ s (27) In the above equations D s is determined from the coefficients A and N. D sf is dependent on the coupling between the fluid and the frame, Q. D f is depends on R. The vectors of stress and strain components in each axis are denoted by σ and ɛ with the subscripts s and f referring to the structure and the fluid respectively. This implementation of the finite element method has been used to predict the effect of boundary conditions on the absorption and sound transmission of poroelastic layers. 28 It has been found, consistent with Bolton s work, 21 that bonding a liner to both interior surfaces of a double panel system increases the transmission loss at low frequencies because of the additional stiffness. However a bondedunbonded configuration offers the best overall performance as the high frequency transmission loss is maintained due to the decoupling from the receiving panel. 6.3 Boundary Element Models A new boundary element formulation was presented by Tanneau et al. 29 The advantage of the boundary element method, (BEM) is the reduction in the size of the mesh required. This results in large computational savings especially in the midfrequency range. This is very relevant to the modeling of poro-elastic materials due to the frequency dependent matrices and large degrees of freedom required per node. 7 Conclusions This brief overview of the models for predicting the behaviour of porous materials has shown that it is an area of complex research. Due to the nature of the materials involved, no single model is likely to be able to capture the dynamics of limp fibrous blankets, elastic foam rubbers and packed glass beads. Delany and Baz- 14

15 leys empirical work is a valuable check for these more complex models, especially those that assume rigid frames. Biot s description is the most complete theory and with the advent of finite element representations it seems the most appropriate for detailed design work. It becomes apparent that the dynamics of porous materials are not that complicated once the effective inertial, stiffness and damping terms are acquired. The theoretical derivations of these terms are involved and there are many different models. A large number of parameters have been introduced to describe the physics of porous materials and often the parameters required for each model are different. This can make comparison of models difficult. It has been shown here that using the Biot model will result in a more complete description of the acoustics within the porous material. However, for many noise control materials the rigid frame assumption or the Delany Bazley empirical approach, extended as described in Bies and Hansen 30 is applicable. 15

16 References [1] M. E. Delaney and E. N. Bazley, Acoustical properties of fibrous materials, Applied Acoustics, vol. 3, pp , [2] M. Biot, Theory of propagation if elastic waves in a fluid saturated porous solid. 1. low-frequency range., J. Acoust. Soc. Am., vol. 28(2), pp , [3] M. Biot, Theory of propagation if elastic waves in a fluid saturated porous solid. 2.higher-frequency range., J. Acoust. Soc. Am., vol. 28(2), pp , [4] O. Zienkiewicz, A. H. C. Chan, M. Pastor, B. A. Schrefler, and T. Shiomi, Computational geomechanics with special reference to earthquake engineering. John Wiley, [5] F. B. Jensen, W. A. Kuperman, M. B. Portor, and H. Schmidt, Computational Ocean Acoustics. AIP Press, [6] J. G. Berryman, Theory of elastic properties of composite materials, Applied Physics Letters, vol. 35, no. 11, pp , [7] P. M. Morse and C. H. Bolt, Sound waves in rooms, Reviews of Modern Physics, vol. 16, pp , [8] A. Hosokawa, Simulation of ultrasound propagation though bovine cancellous bone using elastic and biot s finite-difference time-domain methods, J. Acoust. Soc. Am, vol. 118, no. 3 Pt. 1, pp , [9] K. I. Lee and S. W. Yoon, Comparison of acoustic characteristics predicted by biot s theory and the modified biot attenborough model in cancellous bone, Journal of Biomechanics, vol. 39, pp , [10] K. Attenborough, Acoustyical characteristics of porous materials, Physics Reports, vol. 82, no. 3, pp , [11] L. Kinsler, A. Frey, A. Coppens, and J. Sanders, Fundamentals of Acoustics. John Wiley & Sons, 3 rd ed., [12] D. A. Bies and C. H. Hansen, Flow resistance information for acoustical design, Applied Acoustics, vol. 13, pp , [13] J. W. S. Rayleigh, The theory of sound, vol. 2. Dover, [14] I. B. Crandall, Theory of vibrating systems and sound. London: MacMillan and Co. Ltd.,

17 [15] C. Zwikker and C. W. Kosten, Sound Absorbing Materials. Elsevier Publishing Inc, New York, [16] L. L. Beranek, Acoustical properties of homogeneous, isotropic rigid tiles and flexible blankets, J. Acoust. Soc. Am, vol. 19, no. 4, pp , [17] R. W. Morse, Acoustic propagation in granular media, J. Acoust. Soc. Am, vol. 24, no. 6, pp , [18] P. M. Morse and K. U. Ingard, Theoretical Acoustics. Princeton University Press, Princeton, New Jersey, [19] J. G. Berryman, Long wavelength propagation in composite elastic media 1. spherical inclusions, J. Acoust. Soc. Am, vol. 68, no. 6, pp , [20] C. W. Kosten and J. H. Janssen, Acoustic properties of flexible and porous materials, Acustica, vol. 7, pp , [21] J. S. Bolton, N. M. Shiau, and Y. J. Kang, Sound transmission through multipanel structures lined with elastic porous materials, J. Sound & Vibration, vol. 191(3), pp , [22] D. L. Johnson, J. Koplik, and R. Dashen, Theory of dynamics permeability and tortuosity in fluid-saturated porous media, Journal of fluid mechanics, vol. 176, pp , [23] J. Allard, Propagation of sound in porous media. Elsevier Applied Science, [24] Y. J. Kang and J. S. Bolton, Finite element modeling of isotropic elastic porous materials coupled with acoustical finite elements, The Journal of the Acoustical Society of America, vol. 98, no. 1, pp , [25] P. Göransson, Acoustic finite element formulation of a flexible porous material a correction for inertial effects, Journal of Sound and Vibration, vol. 185, no. 4, pp , [26] R. Panneton and N. Atalla, An efficient finite element scheme for solving the three dimensional poro-elasticity problem in acoustics., J. Acoust. Soc. Am., vol. 101(6), pp , [27] P. Göransson, A 3-d symmetric finite element formulation of the biot equations with application to the acoustic wave propagation through an elastic porous medium, International journal for numerical methods in engineering, vol. 41, pp ,

18 [28] F. C. Sgard, N. Atalla, and J. Nicolas, A numerical model for the low frequency diffuse field sound transmission loss of double-wall sound barriers with elastic porous linings, J. Acoust. Soc. Am, vol. 108, no. 6, pp , [29] O. Tanneau, P. Lamary, and Y. Chevalier, A boundary element method for porous media, The Journal of the Acoustical Society of America, vol. 120, no. 3, pp , [30] D.A. Bies and C.H. Hansen, Engineering Noise Control. Spon Press, London, UK, Appendix 3. 18

19 1 a) A porous material formed from narrow tubes of radius r as suggested by Rayleigh. 13 b) The velocity profile within a tube leading to the assumed form for the viscous losses (Eq. (8)). c e.) Rigid frame materials with the same porosity, φ and resistance, R 1 but different structural factor varphi, Impedance, absorption and sound speed in porous materials v.s. the non-dimensional parameter ρ 0 f/r 1 as predicted by the Delany Bazley approximation. Upper figure: impedance. Real part of normalised Imaginary part of normalised impedance. Lower figure, left axis: Sound Absorption. Right Axis : Normalised sound speed Effective density of a porous material vs the dimensionless parameter µ = ωρ0 r 2 η. Eq. (8), High frequency approximation (Eq. (11)), Low frequency approximation (Eq. (10)), V. Low frequency approximation (Eq. (6)), Effective modulus of a porous material vs the dimensionless parameter µ = ωρ0 r 2 η. Solid line: Eq. (9) Dashed line high frequency approximation Eq. (11), Dotted line Eq. (10) low frequency approximation a.) Zwikker and Kostens model of pressures and velocities in a poroelastic material. b.) Biot s model of stresses and strains in a block of poro-elastic material Normalised wave speed for the two wave types that can exist in porous materials as predicted by Kosten & Jansenn, 20 (, ) and Bolton 21 and Allard 23 (, ). Both Bolton and Allards models are based on Biots equations but they assume slightly different values for the effective bulk modulus,. The dashed line indicates the fast wave. The slow wave is shown by the solid lines Viscous loss parameter b vs non-dimensional pore size µ. Solid Line: exact expression. Dashed line: low frequency approximation. Dotted line High frequency approximation Normalised acoustic impedance of a porous material as predicted by Biot model (solid lines), Rigid Frame assumption (dotted lines), Delany Bazley approximation (Dashed lines)

20 9 Acoustic absorption of a porous material as predicted by Biot model (solid lines), Rigid Frame assumption (dotted lines), Delany Bazley approximation (Dashed lines)

21 (a) (b) 2r (c) (d) (e) Figure 1: a) A porous material formed from narrow tubes of radius r as suggested by Rayleigh. 13 b) The velocity profile within a tube leading to the assumed form for the viscous losses (Eq. (8)). c e.) Rigid frame materials with the same porosity, φ and resistance, R 1 but different structural factor varphi, 15 Z /! c !1!2! ! f / " # c / c ! f / " Figure 2: Impedance, absorption and sound speed in porous materials v.s. the nondimensional parameter ρ 0 f/r 1 as predicted by the Delany Bazley approximation. Upper figure: Real part of normalised impedance. Imaginary part of normalised impedance. Lower figure, left axis: Sound Absorption. Right Axis : Normalised sound speed 21

22 Normalised Density,! /! ! µ Figure 3: Effective density of a porous material vs the dimensionless parameter ωρ µ = 0 r 2. Eq. (8), High frequency approximation (Eq.11), Low fre- η quency approximation (Eq. (10)), V. Low frequency approximation (Eq. (6)), V. High frequency approximation (Eq. (7)) Normailsed Bulk modulus K/K ! µ Figure 4: Effective modulus of a porous material vs the dimensionless parameter µ = ωρ 0 r 2. Solid line: Eq. (9) Dashed line high frequency approximation Eq. (11), Dotted η line Eq. (10) low frequency approximation

23 a.) Fluid Frame b.) Fluid Frame Figure 5: a.) Zwikker and Kostens model of pressures and velocities in a poro-elastic material. b.) Biot s model of stresses and strains in a block of poro-elastic material. 23

24 2.5 2 b µ Figure 6: Viscous loss parameter b vs non-dimensional pore size µ. Solid Line: exact expression. Dashed line: low frequency approximation. Dotted line High frequency approximation. 24

25 Kosten& Jansenn Bolton Allard, (simple) Allard, (full) R 1 Figure 7: Normalised wave speed for the two wave types that can exist in porous materials as predicted by Kosten & Jansenn, 20 (, ) and Bolton 21 and Allard 23 (, ). Both Bolton and Allards models are based on Biots equations but they assume slightly different values for the effective bulk modulus,. The dashed line indicates the fast wave. The slow wave is shown by the solid lines. 25

26 10!(Z / " c) 5 0!5! Frequency, Hz 10 # (Z / " c) 5 0!5! Frequency, Hz Figure 8: Normalised acoustic impedance of a porous material as predicted by Biot model (solid lines), Rigid Frame assumption (dotted lines), Delany Bazley approximation (Dashed lines). 26

27 ! Frequency, Hz Figure 9: Acoustic absorption of a porous material as predicted by Biot model (solid lines), Rigid Frame assumption (dotted lines), Delany Bazley approximation (Dashed lines). 27