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. Partic-

ular emphasis is placed on the relationship between the full Biot–Allard (J. Al-

lard, 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 sev-

eral models used to predict the absorption characteristics of porous materials is

presented.

∗Now with Vipac Engineers & Scientists, 17-19 King William St, Kent Town, SA, 5067

1

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. Al-

though these materials are widely used, the theoretical modelling is surprisingly

complex so empirical approximations1 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 the-

oretical description of waves in porous materials was obtained by Biot2, 3in 1956.

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 vi-

bration absorption because of their efficacy, cost effectiveness, and simplicity to

install. The focus of this paper is the modeling of the process of sound propaga-

tion 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. Oceanog-

raphy has also benefited from the more complete model of the ocean floor that the

porous media models5 provide. Biot’s2, 3 description of porous materials accurately

predicts how acoustic energy travels through ocean sediments and rocks. Descrip-

tions of granular materials such as those by Berryman6 and Morse7 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 mod-

eled as porous materials. The prediction of the acoustic response in these materials

is increasingly relevant to ultrasonic imaging as greater resolution becomes neces-

sary. Ultrasonic propagation in bovine bones has been predicted,89 using Biot’s2

equations. Reflection and transmission of ultrasound pulses through bone is the

2

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 materi-

als 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 ap-

proximations and obtuse detail, often within the same model! Attenborough10

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 mod-

eling porous materials with respect to acoustic waves. The paper is laid out as

follows: The simple Delany–Bazley1 approximation is reviewed, and this is proba-

bly 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 mecha-

nism. The Biot2, 3 model, which is currently accepted as the most complete descrip-

tion 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 Bazley1 developed a simple empirical model of the acoustic impedance

of a porous material based on its flow resistivity, R1. Flow resistivity is defined as

the pressure required to generate a unit flow through the material per unit thick-

ness. 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 poros-

ity 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

Delany and Bazley proposed the following empirical fit to experimental mesure-

ments of impedance, where positive time dependence, ejωt, is assumed in the anal-

ysis.

Z = ρ0c0(1 + 0.0571χ−0.754 − j0.087χ0.732) (1)

and the following fit to calculations of the effective wavenumber (k = 2π/λ),

k =2πfc0

(1 + 0.0978χ−0.7 − j0.189χ−0.595

), (2)

where the non-dimensional parameter χ is defined as

χ =ρ0f

R1. (3)

Here f denotes the frequency in Hz, R1 is the flow resistivity of the porous ma-

terial, and ρ0 and c0 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) ex-

tends over the range 0.01 < χ < 0.1. Bies and Hansen30 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 re-

quired 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 pa-

rameter ρ0fR1

. 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/c0 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 Hansen12 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

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

Rayleigh13 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 sed-

iments, 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 assump-

tion 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 propaga-

tion 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

3 Rigid Framed Materials

Rayleigh13 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 Crandall14 presented an analysis of the flow in narrow tubes. He deter-

mined that it was governed by an adiabatic process in contrast to the isothermal

assumption made by Rayleigh. However as shown by Zwikker and Kosten15 the

actual process is somewhere between these two extremes; changing from isother-

mal (Poiseuille flow) at low frequencies to adiabatic (Helmholtz flow) at high fre-

quencies.

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

µ =

√ωρ0r2

η, (4)

The parameter relates the mass of the fluid in the pore, ∝ ρ0r2 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, R1.

ρ =ϕρ0

φ+R1

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

The expressions for the resistance, R1 of the tube is a function of µ,

R1 = 8ϕ

φ

ωρ0

µ2µ� 1 Poiseuille, (Isothermal) (6)

R1 =√

2ϕ

φ

ωρ0

µµ� 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 frequen-

cies. They derived an effective density, ρ (by neglecting thermal effects) and an

effective modulus, K (by neglecting viscous effects).

ρ = ρ0/

{1 +

2ϑ

J1(ϑ)J0(ϑ)

}, (8)

and

K = κP0/

{1 +

2Cϑ

(κ− 1)J1(Cϑ)J0(Cϑ)

}, (9)

where ρ0 is the density of the fluid, P0 is the mean pressure, κ is the bulk modulus

of the fluid, C =√

ηκρ0ν

is the square root of the Prandtl number, (≈ 0.86 for air),

ν = λhρ0Cp

, λh is the thermal conductivity of the fluid, Cp is the specific heat of the

fluid at constant pressure, ϑ = µ√−j and Jn 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 J1()/J0() 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

φ

4ρ0

3

(1 +

6jµ2

)K =

1φP0

(1 + 0.028jµ2

) µ < 1, (10)

ρ =ϕ

φρ0

(1 +

2√jµ

)K =

1φκP0

(1−√

0.92jµ

) 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 mate-

rials. By inspecting Figs. 3 and 4, we can see from the solid lines that the effective

7

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 Bolt7 presented a similar model in which they also derived an ef-

fective stiffness and density. However Beranek16 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

Morse17 presented a theory in 1952 for the propagation of sound in granular ma-

terials 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

c0

√ϕ− jR1φ

ρ0ω. (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 φR1

ρ0ωϕ� 1 the sound speed, c0 and attenuation factor, β can be

approximated as

c =√ϕc0 (13)

β =R1φ

2ρ0c0√ϕ

(14)

Morse and Ingard18 present the same model, using a different notation. A com-

plete 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’s2 model.

4 Elastic Frame

In 1947 Beranek16 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

Kosten and Janssen20 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 Jansen20 presented a model for wave motion in terms

of frame and fluid velocity and pressures as:−ik 0 jωρ1 + τ −τ

0 −jk −τ jωρ2 + τ

jω 0 −jkG −jk(1− φ)K2

0 jω −jkH −jkφK2

p1

p2

v1

v2

=

0

0

0

0

(15)

G =[K1 + (1−φ)2

φ (K2 − P0)]

and H = (1−φ)(K2−P0), where P0 is the the am-

bient 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 v1 and v2 respectively. The stress on the

frame is denoted p1 and the pressure in the fluid is indicated by p2. 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− ρ0). (16)

As shown in Fig. 3 the effective density ρ can vary from ρ0 by large values for

small µ (low frequency / small pores). The factor K2 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/R1 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 Biot2, 3 took a fresh approach to the modeling of wave propagation in

porous materials. His model could accommodate fluids and frames of similar den-

sity 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

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 modu-

lus. 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 con-

stantQ relates the volume changes of the fluid to that of the frame and is defined by

Q/R = −ε/e. Where ε 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:

σn = 2Nen +Ae+Qε, (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 ε 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, σx, in

the x-direction on the structure is due to both solid and acoustic strains.

Biot then defines the fluid pressure p as:

p = −Rε−Qe, (18)

and finally, he also included the shear stress and strain

τnm = τmn = Nγnm. (19)

The shear modulus is denoted N , the shear strain in the n,m plane is denoted γnm

where n,m can be any orthogonal combination of the axis x, y, z. The correspond-

ing shear stress is indicated by τnm.

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 = ρ11 +b

jω; ρ11 = ρs + ρa

ρ∗12 = ρ12 −b

jω; ρ12 = −ρa

ρ∗22 = ρ22 +b

jω; ρ22 = ρ0 + ρa

(20)

10

Each of the effective densities ρ11, ρ12 and ρ22 are respectively defined as:

• The mass of the fluid that couples to the motion of the frame, −ρa ;

• The effective moving mass of the frame, which is its actual mass plus the mass

of the fluid that moves with it, (ρ11 + ρa);

• The effective moving mass of the fluid, (ρ22 + ρa).

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, ρ11, is a func-

tion of the structural factor ϕ, and the fluid density ρf .

ρa = (1− ϕ)ρ0

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 = ρfϕ2J1(ϑ)ϑJ0(ϑ)

(1− 2J1(ϑ)

ϑJ0(ϑ)

)−1

=2J1(ϑ)J0ϑ

ϕρ (21)

Note the similarity in form to the expressions for effective density and modulus

derived by Zwikker and Kosten15 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)

−ω2(ρ11Us + ρ12Uf

)= P∇2Us +Q∇2Uf (22)

−ω2(ρ22Uf + ρ12Us

)= R∇2Uf +Q∇2Us (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

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 am-

plitudes for the slow, fast and shear waves can be calculated. Allard23 derived ex-

pressions 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 re-

spectively. The material has a flow resistivity of 40000 Pa s m−2 and a porosity of

0.94. The frame bulk modulus is 22×105Pa and the effective frame density is 130

kg/m3.

The rigid frame model of Zwikker and Kosten,20 the Delany Bazley1 empirical

model and the Biot2 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 simplic-

ity 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 as-

sume 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

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(~xl)

]=[T] [

V(~x0)]

(24)

where [V]

=[vsy vsx vfx σsxx σsyx σfxx

]′Here the superscript s indicates the structural or frame variables and the super-

script f indicates fluid variables. The stresses are denoted σ and the velocities are

indicated by v.

The transfer matrix is [T]

=[Γ(0)

] [Γ(l)

]−1, (25)

with the components of Γ given in Allard23 (p151 Table 7.1.). The matrix T can

then be evaluated numerically.

6.2 Finite and Boundary element models

Allard23 and Bolton21 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 free-

dom 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.

Goransson25 then presented a simplified approach using the frame displacement

and the fluid pressure, (~u, p) as the degrees of freedom. This simplified the cou-

pling 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 ap-

proximated by very simple linear functions.

13

Goransson presented a symmetric finite element formulation27 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 Atalla26 express the Biot equations (see Eqs. (17) and (18)) as,

σs = Dsεs + Dsfεf (26)

σf = Dfεf + Dsfεs (27)

In the above equations Ds is determined from the coefficients A and N . Dsf is

dependent on the coupling between the fluid and the frame, Q. Df is depends on

R. The vectors of stress and strain components in each axis are denoted by σ and ε

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 poro-

elastic 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 bonded-

unbonded 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 ad-

vantage 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 mid-

frequency 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 materi-

als 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

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 compli-

cated once the effective inertial, stiffness and damping terms are acquired. The the-

oretical derivations of these terms are involved and there are many different mod-

els. 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 com-

plete 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 Hansen30 is applicable.

15

References

[1] M. E. Delaney and E. N. Bazley, “Acoustical properties of fibrous materials,”

Applied Acoustics, vol. 3, pp. 105–116, 1970.

[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. 168–178,

1956.

[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. 179–191,

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18

1 a) A porous material formed from narrow tubes of radius r as sug-

gested 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, R1 but different

structural factor varphi,15 . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Impedance, absorption and sound speed in porous materials v.s.

the non-dimensional parameter ρ0f/R1 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 : Nor-

malised sound speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Effective density of a porous material vs the dimensionless param-

eter µ =√

ωρ0r2

η . Eq. (8), High frequency approximation

(Eq. (11)), Low frequency approximation (Eq. (10)), � V. Low

frequency approximation (Eq. (6)), . . . . . . . . . . . . . . . . 17

4 Effective modulus of a porous material vs the dimensionless param-

eter µ =√

ωρ0r2

η . Solid line: Eq. (9) Dashed line high frequency

approximation Eq. (11), Dotted line Eq. (10) low frequency approxi-

mation.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6 Normalised wave speed for the two wave types that can exist in

porous materials as predicted by Kosten & Jansenn,20 ( ,

) and Bolton21 and Allard23 ( , ). Both Bolton and Allards

models are based on Biots equations but they assume slightly differ-

ent values for the effective bulk modulus, . The dashed line indicates

the fast wave. The slow wave is shown by the solid lines. . . . . . . . 19

7 Viscous loss parameter b vs non-dimensional pore size µ. Solid Line:

exact expression. Dashed line: low frequency approximation. Dot-

ted line High frequency approximation. . . . . . . . . . . . . . . . . . 20

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). . . . . . . . . . . . . . 20

19

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

20

(a) (b)

(c) (d) (e)

2r

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, R1 but different structural factor varphi,15

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1!3

!2

!1

0

1

2

3

! f / "

Z / !

c

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.2

0.4

0.6

0.8

1

! f / "

#

0

0.5

1

c / c

0

Figure 2: Impedance, absorption and sound speed in porous materials v.s. the non-

dimensional parameter ρ0f/R1 as predicted by the Delany–Bazley approximation. Up-

per figure: Real part of normalised impedance. Imaginary part of nor-

malised impedance. Lower figure, left axis: Sound Absorption. Right Axis :

Normalised sound speed

21

10!1 100 101 102

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

µ

Nor

mal

ised

Den

sity

, ! / ! 0

Figure 3: Effective density of a porous material vs the dimensionless parameter

µ =√

ωρ0r2

η. Eq. (8), High frequency approximation (Eq.11), Low fre-

quency approximation (Eq. (10)), � V. Low frequency approximation (Eq. (6)),

V. High frequency approximation (Eq. (7)).

10!1 100 101 1020.9

1

1.1

1.2

1.3

1.4

1.5

µ

Nor

mai

lsed

Bul

k m

odul

us K

/K0

Figure 4: Effective modulus of a porous material vs the dimensionless parameter µ =√ωρ0r2

η. Solid line: Eq. (9) Dashed line high frequency approximation Eq. (11), Dotted

line Eq. (10) low frequency approximation.15

22

Fluid

Frame

Fluid

Frame

a.)

b.)

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

0 2 4 6 8 100.5

1

1.5

2

2.5

µ

|b|

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

Kosten& Jansenn

Bolton Allard, (full)Allard, (simple)

R1

Figure 7: Normalised wave speed for the two wave types that can exist in porous ma-

terials as predicted by Kosten & Jansenn,20 ( , ) and Bolton21 and Allard23

( , ). Both Bolton and Allards models are based on Biots equations but they

assume slightly different values for the effective bulk modulus, . The dashed line indi-

cates the fast wave. The slow wave is shown by the solid lines.

25

0 500 1000 1500!10

!5

0

5

10

Frequency, Hz

!(Z

/ "

c)

0 500 1000 1500!10

!5

0

5

10

Frequency, Hz

# (Z

/ "

c)

Figure 8: Normalised acoustic impedance of a porous material as predicted by Biot

model (solid lines), Rigid Frame assumption (dotted lines), Delany–Bazley approxi-

mation (Dashed lines).

26

0 500 1000 15000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

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