Porous Materials for Sound Absorption and Transmission Control

J. Stuart Bolton

Ray W. Herrick Laboratories, 140 S. Intramural Drive, Purdue University, West Lafayette IN 47907-2031, USA

bolton@purdue.edu

Abstract This paper begins with a discussion of the general types of porous materials, i.e., rigid, limp, and elastic, and of their general physical properties. The macroscopic properties (e.g., flow resistivity, porosity, tortuosity, etc.) that control the acoustical behavior of each type of porous material are then defined and discussed, as are methods for their measurement. The acoustical characterization of porous media is considered next, followed by a discussion of modeling porous materials with particular reference to elastic porous materials such as foams. The special characteristics of elastic porous materials are illustrated through experimental and computational examples involving sound absorption and sound transmission. In particular, the importance of apparently small details of foam layer boundary conditions is emphasized. Finally, foam finite elements that are capable of predicting the behavior of finite-sized noise control treatments having realistic shapes are discussed. By using foam finite element models it is possible to perform optimal design studies: i.e., to design real treatments that yield optimal acoustical performance at a given treatment volume or weight.

1. INTRODUCTION

Porous noise control materials have two phases: the solid, usually fibrous component referred to as the frame, and the interstitial fluid contained within the pores formed by the frame. Owing to their relatively low density, porous materials are not generally useful as barriers, but they are frequently used as efficient sound absorbing materials, in which role they convert organized acoustical motion into heat. Sound absorbing materials dissipate acoustical energy largely by the interaction of their solid and fluid phases. In particular they convert acoustical energy into heat by: viscous means (associated with oscillatory shearing of the fluid in the vicinity of the fibers’ surface); thermal means (irreversible heat flow from the interstitial fluid to the fibers forming the frame); and by structural means (irreversible losses associated with flexure of the fibers comprising the frame).

Porous materials such as glass fiber, mineral wool, and open or partially open cell foams have many applications in automotive noise control in particular: e.g., as headliners, seats, under carpets, trim lining, dashmat lining, cavity interiors, parcel self lining, panel damping, etc. And of course they are widely used in appliance noise control, aircraft applications and in buildings. The objective in this article is not to discuss all possible applications of porous materials, but rather to introduce fundamental terminology and concepts related to porous materials, and especially elastic porous materials such as foams, to facilitate later discussion related to applications. In this paper, porous materials will first be categorized, and then the macroscopic physical properties that determine their acoustical performance will be discussed (as will techniques for measuring those quantities). After a brief discussion of the acoustical quantities that are used to characterize porous materials, some aspects of wave propagation in porous materials will be introduced through a discussion of approaches to theoretical modeling of these materials. A number of examples will be used to highlight the sensitivity of relatively stiff porous materials to apparently small details of their installation. Finally, recent progress in finite element modeling of foam will be described, and the use of that technology in the optimal design of noise control treatments will be illustrated.

2. CATEGORIES OF POROUS MATERIALS

Porous materials may be categorized as being rigid, limp or elastic. The types of porous materials that are simplest to characterize and model are those in which the expanded solid phase (i.e., the fibers collectively comprising the frame) may be considered either rigid or limp. The assumption of a rigid frame is appropriate when the solid phase does not move significantly compared to the motion of the fluid phase, either because individual fibers are immobilized by the stiffness of interconnecting filaments, or because they are much denser than the interstitial fluid [1]. The limp approximation is useful, on the other hand, when the fibers making up the material are sufficiently unconstrained and light that they can move freely as a result of viscous or inertial coupling with the interstitial fluid [2,3]. A porous material may generally be assumed rigid when the in vacuo bulk modulus of elasticity of its expanded solid phase is much greater than that of the interstitial fluid, usually air in noise control applications, and if its solid phase is not directly excited by attachment to a vibrating surface. Conversely, a material may be considered limp when the in vacuo bulk modulus of elasticity of its expanded solid phase is much less than that of air. When the frame bulk modulus is significantly greater or smaller than that of the interstitial fluid, a porous material can support only a single longitudinal wave type [4,5]. As a result, the material can be treated as an effective fluid characterized by frequency-dependent, complex parameters (e.g., the density and sound speed) that together account for the effects of viscosity, pore tortuosity and heat conduction (and frame motion in the case of limp materials) [5]. The acoustical properties of rigid or limp porous media may thus be described most easily by defining a complex wave number, k = β - jα, where β is the propagation factor and α is the attenuation factor, and a complex characteristic impedance, zc. On the other hand, published experimental results show that some flexible porous materials such as polyurethane foam cannot be modeled as being either “rigid” or “limp” owing to the

ability of the material’s frame to support wave motion independent of the interstitial fluid [6,7]. The in vacuo frame bulk modulus of an “elastic” porous material is of the same order as that of air, and as a result both the frame and fluid take a significant and distinct role in the wave propagation process. Elastic porous materials can support three wave types simultaneously: the so-called airborne, frame and shear waves [8]. The degree to which each of these wave types is excited depends critically on the boundary conditions at the surfaces of the porous material, particularly the manner in which those surfaces connect to vibrating solids. For this reason, apparently minor aspects of surface treatment attachment can play a very important role in determining the installed acoustical behavior of elastic porous materials such as polyurethane and polyimide foams [9-11], much more so than is the case for either rigid or limp porous media. Note that most porous materials are physically anisotropic: i.e., their physical properties are different in different directions. This is particularly true for foams that exhibit a preferred rise direction [12], but is also true for many fibrous materials that have a preferred fiber direction resulting from the process by which they are manufactured [13-15]. Some theoretical work has been directed towards the study of foam anisotropy [16]; nevertheless, it is most often assumed that noise control materials are macroscopically homogeneous (except when explicitly layered) and isotropic. It is also common experience that most porous materials are spatially inhomogeneous: i.e., their macroscopic properties vary randomly throughout the material [12]. It is not uncommon for the flow resistivities of two closely spaced samples taken from a single piece of porous material to differ by a factor of two: the same tends to be true of stiffness properties. Insofar as this effect is acknowledged, the “properties” of a porous material are usually estimated by averaging the results for a number of individual samples. Currently popular porous material theories do not explicitly account for the spatial inhomogeneity of macroscopic properties: the development of such a theory would result in a useful practical tool. However, the most important point is that the macroscopic properties of a single, small sample of porous material are unlikely to represent the average properties of the material. Porous materials also exhibit nonlinearities resulting from both nonlinear inertial effects related to pore fluid flow [17-19] and nonlinear frame elasticity [20]. Nonlinear effects, of course, become progressively more significant as the excitation levels increase. In this article, the discussion is limited to the linear regime that is normally appropriate when porous materials are used in noise control applications. As a result, some of the present discussion is not relevant to situations in which the porous material experiences large static or dynamic strains (as in seating applications, for example).

3. MACROSCOPIC PHYSICAL PROPERTIES

3.1 Introduction

Most porous material theories are formulated in terms of the macroscopically measurable physical properties of the frame and the fluid: e.g., the flow resistivity. The advantage of such an approach is that it enables the influence of the various directly measurable physical

properties of the porous material to be identified; thus a particular set of physical parameters can be specified that will result in a porous material having a specified level of acoustical performance. In addition, easy-to-use approximate models derived by neglecting the physical parameters that are judged to be least significant in a particular situation (e.g., the bulk elastic properties in the case of a nearly rigid porous material exposed to airborne incident sound) may be used in preliminary analyses. It should be noted, of course, that there is a direct link between the microscopic structure of a porous material (e.g., fiber radius, fiber shape, fiber orientation, fiber material density, number of fibers per unit material volume, etc.) and its macroscopic properties. To-date, however, there is little information available regarding the links between a porous material’s microscopic and macroscopic properties (see, however, the discussion by Bies of the relation between fiber size, bulk density and the flow resistivity of fibrous materials [21]). Although the link between material microstructure and macroscopic properties represents a promising avenue for future research, in this article the discussion will largely be limited to macroscopically measurable parameters. The most important macroscopic physical properties of a porous material are: flow resistivity, connected porosity, pore tortuosity (these first three together constituting the material’s fluid-acoustical properties), bulk density, in vacuo bulk modulus, shear modulus and loss factor (the latter four properties comprising the elastic properties of a porous material). The acoustical performance of the various types of porous media can be predicted with a knowledge of all or a sub-set of these properties: i.e., (i) when a material is “rigid”, only a knowledge of the fluid-acoustical parameters is required; (ii) when a material is “limp”, both the fluid-acoustical parameters and the bulk density must be known; and (iii) when a material is “elastic”, a knowledge of both the fluid-acoustical and elastic properties is required. In the following sub-sections each of the macroscopic properties listed above will be discussed briefly.

3.2 Porosity

The porosity of a porous material is defined as the ratio of the volume occupied by the fluid within the porous material (the volume of the voids) to the total volume of the porous material and it is thus dimensionless. By its nature, the porosity falls within the range zero to unity. Since the porosity quantifies the relative volume occupied by the solid and fluid phases within a unit volume of porous material, it is a key parameter in theories of sound propagation in porous material [1,5,8]. However, the porosity of typical noise control materials such as foam and glass fibers is normally very high: i.e., greater than 0.9, and often greater than 0.98. Since the porosity of most noise control materials is so high, and because the variation in porosity tends to fall into a narrow range, porosity is generally not very important when distinguishing amongst typical noise control materials. However, it should be remembered that much of the energy dissipation within a porous material results from the relative motion of the solid frame and the interstitial fluid, and that for this process to work there must be continuous paths through the material: i.e., the material’s pores must be connected. Therefore, materials featuring closed cells, e.g., Styrofoam, do not normally exhibit useful acoustical properties, and should be modeled as elastic solids rather than as porous materials.

The most direct way of determining a material’s porosity is to measure the volume of air contained within it. This task may be achieved, for example, by using the apparatus developed by Champoux et al. [22] that was itself based on a device developed by Beranek for the measurement of static porosity [23]. The operation of this device is based on the ideal gas law at a constant temperature (i.e., Boyle’s law). When the temperature of a rigid chamber containing a porous sample is held constant, a measurement of the change in air pressure that accompanies a known change in volume allows the volume of fluid within the sample to be determined if the change in air pressure accompanying the same volume change in a rigid, empty chamber of the same total volume is known. Note that it is assumed in this measurement that the fibers comprising the solid component of the material are themselves incompressible. The porosity may also be determined directly from a knowledge of the bulk density of the material when the density of the material comprising the solid phase is known.

3.3 Flow Resistivity

The specific (unit area) flow resistance of any layer of porous material is defined as the ratio of the pressure difference across the layer to the steady state air velocity exterior and perpendicular to the two faces of the layer [23]: it is conventionally given the units MKS Rayls. The flow resistivity is then the specific flow resistance per unit material thickness and it has SI units of MKS Rayls/m. The flow resistivities of useful noise control materials vary widely, but typically fall within the range 1 × 103 MKS Rayls/m to 1 × 106 MKS Rayls/m. The flow resistivity depends on the porosity of a material as well as its tortuosity, but for high porosity, low tortuosity fibrous materials, the flow resistivity is approximately inversely proportional to fiber radius squared at a constant bulk density. That is, a large number of small diameter fibers results in a higher flow resistivity than does a small number of large diameter fibers [21]. At a microscopic level, flow resistance results from the formation of a viscous boundary layer as fluid flows over each fiber, and the amount of shearing in those boundary layers, and hence the amount of viscous drag per unit mass of fibers exerted by the flow on the fibrous medium, increases as the mean fiber radius decreases. The flow resistivity is thus usually taken to be a measure of the viscous coupling between the fluid and solid phases of the porous material, and so is a measure of the potential for viscous dissipation of sound. Note that the steady state flow resistivity as discussed above, gives an accurate estimate of the dynamic flow resistivity experienced by oscillatory flow within the porous material only in the low frequency limit. When the flow is oscillatory, as when created by a sound field within a porous material, the steady state flow resistivity must be corrected to account for frequency-dependent boundary layer velocity profiles: that correction is an integral part of modern porous material theories [24]. Flow resistivity measurement has been standardized by the ASTM [25]. Note that the steady-state flow resistivity of porous materials is a nonlinear function of flow velocity owing to inertial effects and to turbulence at very high flow rates. Tests should normally be conducted at flow rates sufficiently low to fall within the linear range, or be extrapolated to the zero flow rate limit, since the linear flow resistivity is appropriate for acoustical situations

in which particle velocities are typically on the order of several mm/s or less. The pressure difference across the test sample can be measured by using a manometer, and airflow velocities are usually inferred from the volume flow rate through the sample.

3.4 Pore Tortuosity

The tortuosity (sometimes referred to as the structure factor [7]) is a measure of the deviation within the porous material of the pore axis from the direction of wave propagation through the porous material, and of the pore’s nonuniformity in cross-sectional dimension. Tortuosity was introduced into the theory of rigid porous materials by Zwikker and Kosten [7] to account for fluid acceleration effects associated both with the deviation of pores from straight lines in the direction of wave propagation, and with pore expansions and contractions. In the case of limp and elastic porous materials, the tortuosity is also used to quantify the inertial coupling of the solid and fluid phases of a porous material resulting from the deflection of fluid flow by the pore walls. The minimum value of pore tortuosity is unity, and fibrous materials such as glass fibers typically have tortuosities of not much more than unity: e.g., 1.5 or less. The same is true of fully open cell foams. However, when a foam is partially reticulated, i.e., when its cells are partially closed by residual membranes, it may possess a relatively high tortuosity: up to 10, say. The same may be true for relatively dense acoustical materials: e.g., acoustic plasters and porous ceramics or aluminum. There is no direct way of measuring a porous material’s tortuosity. However, Champoux and Stinson [26] have developed a procedure based on the measurement of electrical conductivity to infer the tortuosity of porous materials: their procedure is based on earlier work by Brown [27]. In this measurement, the voltage difference resulting from the passage of a high frequency alternating current through a fluid-saturated sample is measured (the saturating fluid must be electrically conductive). When the conductivity of both the fluid and of the fluid-filled sample are known by measurement, the effective path length through the material, and hence the tortuosity, may be calculated (when the sample porosity is known). Note that this technique has been most widely used to measure the tortuosity of granular media [27]. It is not clear that this procedure yields accurate results for typical noise control materials whose structural or chemical properties may be altered by immersion in a conducting fluid. More recently, it has been suggested that the tortuosity may be estimated from ultrasonic reflection measurements. This suggestion is based on the observation that the reflection coefficient from a porous material is directly related to its tortuosity [28] at very high frequencies. Note, however, that for this measurement to be successful, the probe frequency must still be such that the wavelength of sound is very large compared to the pore size and any surface irregularities: otherwise, scattering effects will render the results inaccurate.

3.5 Bulk Density

The bulk density of an expanded porous material is simply defined as the ratio of an expanded sample’s mass to its total, expanded volume. The bulk density and the porosity are simply related by the expression ρ1 = ρt(1-h), where ρ1 is the bulk density of the expanded porous material, ρt is the density of the material that comprises the solid phase of the porous material, and h is the porosity of the porous material. Note that since the porosity of noise

control materials is often greater than 0.98, the bulk density of a porous material is usually less than one-fiftieth of the density of the material from which the solid phase is formed.

3.6 Bulk Elastic Properties

The in vacuo bulk Young’s modulus of the expanded solid phase of a porous material is defined, as usual, as the ratio of axial stress to strain in the absence of lateral constraint. It is generally made complex to account for losses due to frame flexure and is of the form E(1+jη), where E is the (real) Young’s modulus, or storage modulus, and η is the loss factor or loss tangent [29-31]. The SI unit of the bulk Young’s modulus is Pa. As noted previously, porous materials can be categorized as limp, elastic or rigid depending on whether the material’s bulk modulus is less than, of the same order as, or greater than the bulk modulus of air (i.e., 1.4 × 105 Pa), respectively. For elastic porous materials in particular, the bulk Young’s modulus is an important dynamic mechanical property that significantly affects the acoustical behavior of the material since it controls (along with the Poisson’s ratio or shear modulus) the speed of the longitudinal and shear waves that propagate primarily through the frame [8]. The bulk Young’s modulus of polyurethane noise control foam of medium density, say 10 kg/m3 to 20 kg/m3, is typically on the order of that of air, but can vary widely from one foam type to another depending on the precise chemical composition of the foam and the details of its manufacture. The stiffness of fibrous materials, on the other hand, tends to be substantially less than that of air, unless the materials have been purposely stiffened by the addition of a binding agent, for example. A dynamic method should be used to measure the bulk Young’s modulus since it has been observed to be frequency-dependent. Kim, Wijesinghe and Kingsbury [29,30] determined the complex modulus of polyurethane foams by measuring the complex transmissibility of a single degree of freedom system in which a disc shaped foam specimen acted as an essentially massless stiffness element. Okuno [31] proposed a similar experimental setup that is capable of measuring the complex bulk Young’s modulus of poroelastic materials. Following Okuno, the bulk Young’s modulus of a foam sample can be estimated from measurements of the normal mechanical impedance of the sample under the assumption that the sample layer behaves like a simple spring. Note that dynamic stiffness measurements should, in principle, be conducted in a vacuum so that the damping effect of the interstitial fluid is eliminated and the frame in vacuo loss factor can be accurately identified. In vacuo measurements of porous material elastic properties have been reported by Ingard et al. [32,33]. The bulk shear modulus of an elastic porous material is also, in general, frequency-dependent and complex. The magnitudes of the bulk Young’s and the shear moduli are related by the Poisson’s ratio, ν (when it can be assumed that the material is linear, homogeneous and isotropic). The shear modulus of a foam can be estimated directly by holding two samples vertically between two fixed outer plates and a central plate that is driven vertically by a mini-shaker. The shear modulus is then estimated from the measured mechanical impedance. Finally, note that recent finite element calculations indicate that the Young’s and shear moduli as estimated from mechanical impedance measurements of the type suggested here are sometimes not accurate owing to effects related to finite sample size and sample constraint on the driven surface [34]. The latter observation would suggest that it may be

better to measure the input mechanical impedance of finite-sized blocks of material, constrained in known ways at their boundaries, and then estimate the frame stiffness properties by matching the measured results with finite element predictions [35].

4. ACOUSTICAL CHARACTERIZATION OF POROUS MATERIALS

4.1 Introduction

Acoustical porous materials are currently used in many noise control applications in the automotive, aircraft and appliance industries; they can be used to absorb airborne sound (e.g., automotive interior headliners) or to enhance the transmission loss of barrier systems (e.g., aircraft fuselage or automotive dashpanel applications). The performance of acoustical materials can be characterized in a number of ways, and when specifying performance levels it is important that they be characterized in a manner relevant to their end use.

4.2 Sound Absorption

The performance of a treatment applied to an interior surface to absorb airborne sound is most often characterized by its normal or random incidence absorption coefficients, which are measures of the fraction of the incident acoustic power that is absorbed by the treatment when sound strikes its surface at normal incidence, or with equal likelihood from all angles, respectively. The normal incidence absorption coefficient of a sample can be measured directly by using a standing wave tube. The random incidence absorption coefficient can be measured directly in a reverberant room, or it can be calculated by integrating the appropriately weighted plane wave absorption coefficient over all possible angles of incidence [36]. The plane wave absorption coefficient itself is the fraction of the incident power that is absorbed at a surface when a plane wave strikes it at a particular angle, and it is defined as α = 1 - |R|2, where R is the plane wave reflection coefficient. The plane wave reflection coefficient can, in turn, be determined from a knowledge of the normal specific impedance of a surface, and the latter quantity can be determined from theory, if the material properties and its geometry are known, or it can be measured directly. The normal incidence surface normal impedance can be measured using a standing wave tube, as can the normal incidence reflection and absorption coefficients. Since the surface normal impedance, the reflection coefficient and the absorption coefficient can all vary with angle of incidence, it is sometimes useful to measure them at oblique incidence angles. A number of free field measurement techniques have been proposed for that purpose [24,37]. All the quantities mentioned in the last paragraph should strictly be referred to as “derived” quantities since they are not inherent properties of the porous materials, but rather depend both on the material and its installation (e.g., layer depth and surface treatment). The acoustical properties that are inherent to a porous material (at least those that may be considered to be either limp or rigid, and so may be modeled as an equivalent fluid) are its characteristic impedance and wave number. If the latter two quantities are known then all other relevant acoustical quantities of a porous treatment can be derived. The characteristic impedance of a limp or rigid porous medium is the ratio of pressure to particle velocity in the direction of wave propagation in a plane wave propagating in an infinitely extended piece of material. In practice, the characteristic impedance is equal to the normal incidence surface normal impedance of a porous layer that is sufficiently deep that no

internal reflections return to the surface of the porous layer. Thus the characteristic impedance of limp or rigid porous materials may be estimated by measuring the surface normal impedance of a deep layer of porous material placed in a standing wave tube [6]. The wave number of a plane wave propagating freely within a porous material is complex and is a measure of both the speed of wave propagation (and hence the wavelength within the material) and the rate at which a propagating wave attenuates. The wave number within a porous material can be estimated directly by passing a probe tube through a deep sample exposed to normally incident sound at a single frequency and measuring the wavelength and rate of attenuation [e.g., 38,39]. In an elastic porous material, the situation is not so straightforward, however, since three wave types, each having a different speed and attenuation rate, can propagate simultaneously. Porous materials, such as glass fibers or foams, can also be categorized as being either locally reacting or non-locally reacting. Physically, a porous layer can be assumed to be locally reacting when the speed of wave propagation in the porous layer is much less than the speed of sound in the ambient medium, with the result that transmitted waves propagate normal to the material surface regardless of incidence angle. Owing to the latter effect, the surface normal impedance of a locally reacting porous material is independent of the incidence angle [36], with the important practical consequence that the reflection coefficient of a locally reacting porous material may be predicted using a simple one-dimensional model of the porous material [24,40]. In addition, simple relations between a locally reacting material’s plane wave reflection coefficient and its random incidence absorption coefficient were derived by Morse [41]. For example, Bies and Hansen [42] have combined relations developed by Morse [41] with semi-empirical formulae of Delany and Bazley [43] to obtain theoretical predictions of the random incidence absorption coefficients of unfaced, finite-depth, locally reacting fibrous layers. Thus, it is very convenient from a computational point of view if an absorbing material may be considered locally reacting. Note that non-locally reacting materials may be made locally reacting by segmenting the material into small elements by using rigid partitions: i.e., by placing the porous material within a honeycomb structure. In this case, the partitions prevent sound propagation parallel to the surface making the material effectively locally reacting. On the other hand, when a material is non-locally reacting (i.e., when the sound speed in the material is not small compared to that of air), its surface normal impedance varies with angle of incidence. As a result, the random incidence absorption coefficient of such a material must usually be found by applying numerical integration techniques [24,37]. A two-dimensional theoretical porous material model of the type described in the next section is then required to make predictions of the reflection coefficient of non-locally reacting materials [11]. Obviously, from a theoretical point of view, it would be convenient if all porous materials could be treated as surfaces of local reaction: however, in most situations the latter assumption is not strictly valid.

4.3 Sound Transmission

As noted in the Introduction, porous layers are not generally good barriers and are thus not effective when used by themselves at reducing sound transmission between two spaces.

However, they are often used to line the space between two panels: in that role they serve both to “decouple” the panels, that is to suppress the deleterious effects of the mass-air-mass resonance and inter-panel depth resonances, and to enhance the total transmission loss at high frequencies through the dissipative action of the porous material [44]. Note that a structural base panel, a layer of porous material, and a limp barrier layer (made of vinyl or rubber, for example) forms a double panel system. Dashpanel and dashmat combinations are an example of this kind of system, as are typical automotive floor treatments. Whether lined or unlined, the acoustical performance of a single or multi-panel barrier system can be quantified by measuring or predicting its power transmission coefficient (or its decibel equivalent, the transmission loss). The power transmission coefficient is defined as τ = Wt /Wi, where Wt is the sound power transmitted by a barrier and Wi is the sound power incident upon the treatment. The transmission loss is then TL = 10log(1/τ). The transmission loss is normally measured by inserting the test barrier in an aperture between two reverberation rooms [45] or between a reverberation room and an anechoic space [46]. In the latter case, the transmitted sound power can be measured directly by scanning an intensity probe over the transmission side of the barrier. The sound transmission loss of barrier treatments lined with porous materials is an important design factor in vehicles as well as in other applications, and it can be predicted when the properties of the porous lining are known. For example, Beranek and Work [47] developed systematic procedures for predicting the normal incidence transmission loss of multi-panel structures lined with porous materials (and also made measurements using a modified version of the apparatus described in Reference [48]). An excellent source of information on the fundamentals of sound transmission through single and double panel systems is the book by Fahy [49]. A modification of the two-microphone standing wave tube now makes it possible to measure the normal incidence transmission loss of acoustical materials directly [50]. Recently, Bolton, Shiau and Kang presented predictions and corresponding measurements of the random incidence transmission of foam-lined double panel structures [10,11]. The latter measurements are particularly significant since they illustrate the dramatic difference in transmission loss that can result from apparently minor changes in the way the lining material is attached to the facing panels. The theory lying behind those predictions will be presented next and illustrative transmission loss results will be given.

5. MODELING OF WAVE PROPAGATION IN ELASTIC POROUS MEDIA

5.1 Background

In this section, theoretical modeling of elastic porous materials is discussed briefly. Note that both limp and rigid porous material models may be derived from the model presented here as limiting cases [2,24]. The main difference between rigid or limp and elastic porous materials is that the latter type can support several wave types simultaneously while the former type can support only a single longitudinal wave. Thus elastic porous material theories are directed towards the prediction of wave numbers for each wave type and the definition of relations between stresses and motions in the two phases. If such information can be predicted based on a knowledge of the physical parameters of the material, then accurate theoretical predictions of the material’s acoustical behavior may be made.

Note also that almost all porous material theories are based on the concept that the real porous material can be replaced by a “smeared” medium that is homogeneous and isotropic, but that has the same macroscopic bulk properties as the real porous material. This approach limits the applicability of such models to frequencies sufficiently low that the wavelength of sound is very large compared to the typical pore size in the medium. It is also usually assumed that porous materials are homogeneous in the sense that their properties are independent of position on a scale large compared to a pore size but small compared to a wavelength. As was noted above, this assumption is rarely true in practice, but it is not yet clear how serious a problem it presents. Also note that there have been some attempts to develop theoretical models based explicitly on particular microstructures. To-date this approach has been brought to bear on both fibrous media [13-15] and foams [51]. One of the first elastic porous material models was that due to Zwikker and Kosten who showed (in earlier work summarized in reference [7]) that two types of longitudinal waves can propagate in a porous material (i.e., the frame-wave and the airborne wave). Beranek later arrived at the same conclusion using a similar theory: he also examined the rigid and limp limits. Subsequently, Biot [8] developed a very general model for elastic porous materials that allows for both longitudinal and shear modes of three-dimensional wave propagation in elastic porous media. Biot’s model has, in time, become the standard model for wave propagation in elastic porous media. In order to investigate the transmission loss of foam-lined multi-panel structures, Bolton, Shiau and Kang [10] adapted Biot’s theory to describe wave propagation in noise control materials. Since the porous layer in most foam-lined structures has a depth that is much smaller than its lateral extent, their theory was presented in two-dimensional form. This theory can be used to predict the behavior of plane, layered systems of infinite lateral extent, and thus in practice is limited to predicting the transmission loss of barriers that are large in cross-section compared to the wavelength of the incident sound. Nevertheless, all three wave types known to propagate in elastic porous materials were taken into account and their theory is capable of predicting oblique and random incidence behavior of elastic porous materials. They also developed sets of boundary conditions for several configurations of panel structures. By comparing their predictions with random incidence transmission loss measurements for various configurations of panel structures, the model was shown to be accurate [10]. Allard et al. [24,49] have also predicted the oblique incidence behavior of elastic porous materials based on Biot’s theory [8].

5.2 Theoretical Development

The development of a theory for wave propagation in elastic porous media proceeds from dynamic and stress-strain equations for the two phases of the porous material: the solid frame and the interstitial fluid. By using the sign convention shown in Fig. 1, one can write appropriate stress-strain relations and dynamic relations for the solid and fluid phases. Note that in the dynamic relations there appear inertial fluid-structural coupling terms proportional to the relative acceleration of the two phases, and viscous coupling terms proportional to the relative velocity of the two phases. The inertial coupling is directly related to the pore tortuosity, while the viscous coupling can be related to the flow resistivity. After combining the two sets of relations and performing a sequence of vector operations, one obtains two

wave equations that govern the propagation of volumetric and rotational strains within the material [10,11]: i.e.,

022

14 =+∇+∇ sss eAeAe (1)

022 =+∇ ϖϖ tk (2)

where es and ϖ are the volumetric and rotational strains, respectively, and kt is the wave number of the rotational wave. Note that a fourth order wave equation having the form of Eq. (1) indicates the existence of two longitudinal wave types; a second order wave equation having the form of Eq. (2) governs the propagation of a single transverse wave. When appropriate two-dimensional solutions of Eqs. (1) and (2) are assumed for the solid volumetric strain and rotational strain, all the field variables such as fluid and solid displacements and stresses can be expressed in terms of six wave component amplitudes (representing three wave types propagating upwards and downwards within the foam layer) and material constants. The unknown wave component amplitudes are then determined by imposing sets of boundary conditions as appropriate to represent different physical situations.

5.3 Boundary Conditions

As examples, the sets of boundary conditions for an open (i.e., unfaced) surface and a sealed surface (i.e., the elastic porous material is assumed to be bonded to a facing panel) are shown below. In the latter case, the panel has been modeled as an Euler-Bernoulli panel: i.e., one possessing flexural stiffness. At the open surface of the elastic porous layer (illustrated schematically in Fig. 2(a)) there are four boundary conditions to be satisfied: two normal stress continuity conditions, a normal volume velocity continuity condition and one shear stress-free condition. These conditions, are, respectively [10,11]:

shP =− (3a)

yPh σ=−− )1( (3b)

yyy hUjuhjv ωω +−= )1( (3c)

0=xyτ (3d)

where h is the porosity, P is the pressure in the exterior acoustic field at the interface, vy is the normal component of the acoustic particle velocity in the exterior medium at the interface, s is the fluid stress within the porous material, σy is the solid stress within the porous material, τxy is the shear stress acting on the solid phase, and uy and Uy are, the solid and fluid displacements, respectively. At an open surface of this type there is no constraint placed on the individual velocities of the solid and fluid phases of the material: only the volume velocity must be continuous at the interface. As a result, the partial impedances of the fluid and solid phases add in parallel, and the motion in the external fluid region (i.e., the external sound field) primarily drives the phase of the porous material that has the lowest impedance, which would normally be the fluid phase of the porous material. Thus, at an open surface, airborne incident sound couples

most directly to the airborne wave within the material [10]. This is usually advantageous since the airborne wave is usually much more heavily damped than is the frame wave [11]. When the elastic porous layer is bonded directly to an elastic panel (illustrated schematically in Fig. 2(b)) there are six boundary conditions to be satisfied: i.e., normal velocity continuity, normal displacement continuity, tangential displacement, in-plane acceleration, and force (i.e., the panel equation of motion). These conditions are, respectively [10,11]:

ty Wjv ω= (4a)

ty Wu = (4b)

ty WU = (4c)

dx

dWhWu tp

px 2)/( +−= (4d)

( ) psxpxy WmkD 22)/( ωτ −=−+ (4e)

( ) tsxxyp

xp WmDkh

jkqP 242

)/()/( ωτ −=−+−−+ (4f)

where hp is the panel thickness, D is the panel flexural stiffness per unit width, Dp is the panel longitudinal stiffness per unit width, ms is the panel mass per unit area, qp is the normal force per unit panel area exerted on the panel by the elastic porous material, Wt is the transverse panel displacement, Wp is the panel in-plane displacement, and the first sign is chosen when the foam is attached to the panel surface facing in the positive coordinate direction. In this case, since the panel is directly attached to the solid phase of the porous material, the velocities of the solid and fluid phases are constrained to be equal to each other (and to the normal velocity of the panel and the external acoustical field). As a result of this velocity constraint, the impedances of the fluid and solid phases add in series and, since the impedance of the solid component is usually higher than that of the fluid component, the former largely determines the response of the system. In practice this means that it is the relatively lightly damped frame wave that is primarily excited in this configuration. The shear wave may also be excited in this case [10]. The open and bonded boundary conditions have been discussed in some detail here to emphasize that the degree to which each of the wave types that can propagate within an elastic porous material is actually excited depends on the boundary conditions at the material’s surface. The acoustical performance of a foam layer, for example, may appear to be completely different depending on whether it is directly attached to a facing panel, or whether it is separated from it by a small air gap. The difference in performance may be traced to the dominant role played by the frame wave in one case and the airborne wave in the other. Thus, when using elastic porous materials in noise control applications it is very important to understand the effect of boundary conditions on the installed performance of a treatment so that the optimum arrangement can be chosen in each particular instance. When the boundary conditions appropriate to a particular panel configuration have been defined, the various assumed field solutions can then be substituted into them. The result is a matrix equation for the wave amplitudes in the elastic porous material, facing panels, air gaps

(if any), as well as for the reflected and transmitted wave amplitudes that are usually the objective of the calculation. Sample results will be discussed in the next section.

6. EXPERIMENTAL AND COMPUTATIONAL EXAMPLES

In this section, theoretical predictions of the transmission loss of foam-lined double panel structures are compared with experimental results [10,11] with the intention of emphasizing the importance of the boundary conditions at the foam surface. The foam parameters used in the predictions were: bulk density of the solid phase ρ1 = 30 kg/m3, in vacuo bulk Young’s modulus Em = 8 × 105 Pa, in vacuo loss factor η = 0.265, bulk Poisson’s ratio ν = 0.4, flow resistivity σ = 25 × 103 MKS Rayls/m, tortuosity ε' = 7.8, and porosity h = 0.9. The thickness of the aluminum panels on the incident and transmitted sides were, respectively, 0.05 in and 0.03 in, and the panels were separated by distances ranging from 21 mm to 41 mm. The measurements were performed by placing the double panel system in a 1.14 m × 1.14 m aperture between a reverberant source room and an anechoic receiving space. The transmission loss was measured using the intensity technique described earlier. Figures 3(a) and (b) show the predicted and measured transmission losses for the BB (foam bonded directly to both panels) and UU (foam separated from the panels by small air gaps) configurations. It may be seen that the theoretical predictions follow the measured results reasonably closely. While the BB configuration gives the higher transmission loss below 400 Hz, the UU configuration transmission loss displays an approximately 18 dB/octave slope above the double panel resonance near 250 Hz, with the result that the transmission loss of the UU configuration is substantially higher than that of the BB configuration (by approximately 15 dB) at frequencies above 400 Hz. The double panel resonance occurs when the panel masses vibrate against the lining stiffness. Note that the BB double panel resonance frequency is elevated compared to that of the UU configuration due to the relatively high stiffness of the directly attached foam lining in the former case. The relatively poor performance of the BB configuration results from the direct coupling of the facing panels and the solid phase of the porous material, which in turn results in strong excitation of the frame wave. By contrast, when the foam is separated from the facing panels by even the smallest of air spaces, it is the highly damped airborne wave that is primarily excited in the foam, thus causing the transmission loss in this configuration to be relatively high. It is important to realize that the masses per unit area of both the UU and BB configurations shown here were exactly the same (as, for practical purposes, were their thicknesses). The dramatic difference in their transmission losses originates solely in the difference in the foam/panel boundary conditions in the two cases.

7. FINITE ELEMENT MODELING OF FOAMS

Although analytical models [8,10,24,52] can account for all three wave-types propagating within elastic porous materials and are useful and easy to implement, they can in practice only be used to model laterally infinite planar treatments. Since realistic noise control elements feature foam pieces that are of finite size, and usually, irregular shape, there is clearly a need for numerical tools to allow the performance of such treatments to be predicted accurately. Thus, foam finite elements, based on Biot’s elastic porous material theory, that can be coupled with existing acoustical and/or structural finite elements have recently been developed [53-58]. Here, examples of the use of foam finite elements in noise control design

procedures will be given: the detailed mathematical formulation of foam finite elements and the procedures required to couple them to adjacent media are described elsewhere [53-58]. The first example involves shape optimization of a foam treatment. In particular, the shape of a foam wedge terminating a waveguide was optimized to offer the maximum absorption over a specified frequency range: 500 Hz to 2 kHz. The model considered here is illustrated in Fig. 4: a foam wedge was placed in a two-dimensional hardwalled waveguide (27 cm long and 5.4 cm wide) and a rigid piston was considered to generate a plane sound field that was incident on the front surface of the wedge. When the volume of the wedge and the duct width were held constant, the geometry of the wedge could be defined by a single design parameter, the wedge tip angle, θ. The same foam properties as those used in the previous section were used in the calculations presented here. By calculating the normal incidence absorption offered by wedges having various tip angles, the optimal wedge angle was found to be 36o for the foam material considered here. In Fig. 5(a), the absorption offered by the optimized wedge is compared with the absorption of non-optimal wedges having tip angles of θ = 132o and 180o (i.e., a flat foam layer). Generally, it was found that wedge absorption increased at high frequencies at the expense of a reduction of absorption at low frequencies as the wedge was made sharper and longer (causing the constrained edges to become shorter). In Fig. 5(b), the absorption coefficient averaged over the frequency range 500 Hz to 2 kHz has been plotted for various wedge angles. An extensive discussion of wedge shape optimization (including inverse and hybrid wedges) can be found in reference [55]. The main point to be taken from this example, though, is that the numerical tools are now in place to use the techniques of optimal design to identify the best possible foam configuration for a particular application. As another example, foam finite elements were applied to investigate normal incidence sound transmission characteristics through a foam wedge placed in a hardwalled duct as shown in Fig. 6(a). As may be seen from Fig. 7(a), it was found that in some frequency bands the transmission loss of the wedge was significantly higher than that of a plane foam layer of the same volume (especially when the wedge tip-angle was smaller than 48o). Further studies identified the increase in transmission loss as resulting from the conversion, within the foam, of the incident plane wave into higher order symmetric waveguide modes that could not radiate efficiently from the rear surface of the foam wedge. Although this characteristic may be useful when good high frequency transmission loss is desired, the installation thickness of a wedge makes this approach of only limited utility in most noise control applications. However, it was found that a similar sound transmission loss increase could be achieved by grading the tortuosity of a plane foam layer across the width (not the depth) of the channel. For the configuration shown in Fig. 6(b), the tortuosity was decreased in piecewise-constant steps from 7.8 at the center, to a minimum value in the foam adjacent to the hard walls: the variation was symmetrical with respect to the duct center-line. The dimensions of the foam layer were 5.4 cm × 5.4 cm: i.e., the foam block had the same volume as the wedges. As may be seen from Fig. 7(b), the high frequency transmission loss increased as the spatial variation of tortuosity was made larger. Unlike the wedge case, there was not a decrease in transmission loss at low frequencies in these cases. As was the case for the wedge, the axial fluid displacement in the central and edge regions was found to be out-of-phase across the width of the foam layer, thus accounting for the higher transmission. Therefore, it was

concluded that the phase variations across the width of the foam layer resulting from the spatial variation of tortuosity caused the net axial volume velocity at the rear surface of the plane foam layer to be minimized in certain frequency ranges, thus enhancing the layer’s transmission loss. Extensive investigation of this case and further discussion can be found in reference [56]. A finite element approach has also been used to guide the design of sound absorption treatments having inhomogeneous porosity distributions [59]. As illustrated here, foam finite element techniques provide a powerful design tool in noise control. With the availability of these new numerical tools, the full power of optimal design procedures can be applied to the design of noise control treatments featuring foam elements.

8. CONCLUSIONS

It has been shown in this article that porous materials may be grouped into three categories, i.e., rigid, limp and elastic, and that rigid and limp porous materials differ from elastic porous materials in the number of wave types that can propagate within them. In addition, it has been shown that the important acoustical properties of a porous material can be predicted when its macroscopic properties such as porosity, flow resistivity, tortuosity, bulk density, bulk Young’s shear and modulus of elasticity, are known. Acoustically, porous materials can be characterized by reflection and absorption coefficients, surface impedance and transmission coefficients, etc. It has also been shown that wave propagation in elastic porous materials can be modeled accurately based on the Biot’s general elastic porous material theory: in particular, theoretical and experimental results show good agreement in the case of random incidence sound transmission loss through foam-lined double panel structures. Finally, it has been demonstrated that foam finite element models can successfully be applied to the optimal design of noise control treatments. Although much progress has been made in this area, many issues that relate to porous noise control materials remain to be resolved. For example, it would be desirable to pursue the following topics: (i) development of more accurate and efficient techniques for measuring the tortuosity and the bulk elastic properties of foams (including the loss factor); (ii) development of theories that can connect a foam’s microstructure to its macroscopic properties; (iii) systematic optimization of foam microstructure to yield specified acoustical properties; (iv) development of SEA-compatible models for efficient prediction of the high frequency behavior of foam elements; (v) use of porous materials in damping applications; (vi) development of nonlinear models in the context of the Biot theory; (vii) development of specialized models to allow increased calculation efficiency under special circumstances, e.g., for limp materials; (viii) development of models to account for the anisotropy inherent in most noise control materials; and (ix) development of models that explicitly account for the inhomogeneity of typical noise control materials.

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σ x

τ xy

τ yx σ y

s τ xy

σ y τ yx

s

s

s

y

xsolid phase fluid phase

x

y

vy

τ yx

s σ y jωUy jωuy

(a) P

jωuxx

y

vy

w p

s σ y

jωUy jωuy

jωwy

τ yx

(b)P

foam

hard wall

air

c dL

constrained edges

xy

rigid piston

uo ejωt θ a

Figure 3: Measured (solid line) and predicted (dashed line) transmission loss for foam- lined double panel systems: (a) Bonded- Bonded (BB) configuration and (b) Unbonded-Unbonded (UU) configuration.

Figure 4: Configuration for shape optimization of foam wedge.

Figure 1: Sign convention for stresses acting on the solid and fluid phases of foam.

Figure 2: Boundary conditions for: (a) open surface and (b) bonded Euler-Bernoulli plate.

0.0 0.

0.2 0.2

0.4 0.4

0.6 0.

0.8 0.

1.0 1.α

0 500 1000 1500 2000

θ = 36o (optimal wedge)

Frequency (Hz)

θ = 132o

θ = 180o

(a)

0.0 0

0.2 0

0.4 0

0.6 0

0.8 0

1.0

α

16 28 36 41 48 59 74 97 132 180wedge tip angle (θ)

(b)

0 0

10 10

20 20

30 30

40 40

102

Tran

smis

sion

Los

s (d

B)

103

180o132o97o74o59o

Frequency (Hz)

48o41o36o28o

(a)

Figure 6: System configurations for calculation of sound transmission through (a) a wedge having uniform properties, and (b) a plane foam layer (L = 5.4 cm, d = 5.4 cm) having a spatial variation of tortuosity.

Figure 5: (a) Absorption coefficient vs. frequency and (b) frequency-averaged absorption coefficient vs. wedge tip angle.

0 0

10 10

20 20

30 30

40 40

10 2

Tran

smis

sion

Los

s (d

B)

10 3

case0 (7.8-7.8)case1 (7.8-7.0)case2 (7.8-6.0)case3 (7.8-5.0)case4 (7.8-4.0)

Frequency (Hz)

case5 (7.8-3.0)case6 (7.8-2.0)case7 (7.8-1.0)

(b)

Figure 7: Sound transmission through (a) a wedge having uniform properties, and (b) a plane foam layer having a spatial variation of tortuosity.

θ

foam

lubricated edges

air

d

air

xy

hard wall

lubricated edges

air

d

air

xy

(a)

(b)

z n=

ρ0c

z n=

ρ 0c

hard wall

foam

u0 e jωt

u0 e jωt