2

Acoustic impedance at normalincidence of fluids. Substitutionof a fluid layer for a porous layer

2.1 Introduction

The concept of acoustic impedance is very useful in the field of sound absorption. In thischapter, the impedance at normal incidence of one or several layers of fluid is calculated.The laws of Delany and Bazley (1970) are presented and used to replace a layer of porousmaterial by a layer of equivalent fluid. The surface impedance at normal incidence for alayer of porous material backed by a rigid wall with and without an air gap is calculated.The main properties of both the reflection coefficient and the absorption coefficient arealso discussed in this chapter.

2.2 Plane waves in unbounded fluids

2.2.1 Travelling waves

As indicated in the previous chapter, a simple displacement potential solution of thelinear wave equation (1.56) in a compressible lossless fluid is

ϕ(x, t) = A

ρω2exp[j (ωt − kx)] (2.1)

In this equation, ω is the angular frequency and k the wave number, given by

k = ω(ρ/K)1/2 (2.2)

Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials, Second Edition J. F. Allard and N. Atalla© 2009 John Wiley & Sons, Ltd. ISBN: 978-0-470-74661-5

16 ACOUSTIC IMPEDANCE AT NORMAL INCIDENCE OF FLUIDS

K and ρ are the bulk modulus and the density of the fluid, respectively. The quantity A

is the amplitude of the acoustic pressure. From Equations (1.63) and (1.64), it follows thatthe acoustic pressure p and the components of the displacement vector u are respectively

p(x, t) = A exp[jω(t − kx)] (2.3)

and

uy = uz = 0, ux(x, t) = −jAk

ρω2exp[jω(t − kx)] (2.4)

Only the x component υx of the velocity vector does not vanish:

υx(x, t) = kA

ρωexp[jω(t − x/c)] (2.5)

Equations (2.3) and (2.5) describe a travelling harmonic plane wave propagating alongthe x direction. Pressure and velocity are related by

υx(x, t) = 1

Zc

p(x, t) (2.6)

with

Zc = (ρK)1/2 (2.7)

The quantity Zc is the characteristic impedance of the fluid.

2.2.2 Example

As an example, for air at the normal conditions of temperature T and pressure p (18 ◦Cand 1 033 × 105 Pa), the density ρ0, the adiabatic bulk modulus K0, the characteristicimpedance Z0, and the speed of sound c0 are as follows (Gray 1957):

ρ0 = 1 · 213 kg m−3

K0 = 1 · 42 × 105 Pa

Z0 = 415 · 1 Pa m−1 s

c0 = 342 m s−1

2.2.3 Attenuation

In a free field in air at acoustical frequencies, the damping can be neglected to a firstapproximation when the order of magnitude of the propagation length is 10 m or less.In the previous example, the effects of viscosity, heat conduction, and other dissipativeprocesses have been neglected. The phenomena of viscosity and thermal conductionin fluids are a consequence of their molecular constitution. The description of soundpropagation in viscothermal fluids can be found in the literature (Pierce 1981, Morse andIngard 1986). The effects of viscosity and heat conduction on sound propagation in tubes

MAIN PROPERTIES OF IMPEDANCE AT NORMAL INCIDENCE 17

are described in Chapter 4. Viscosity and heat conduction in tubes lead to dissipativeprocesses, and in a macroscopic description of sound propagation, the density ρ andthe bulk modulus K must be replaced by complex quantities. The wave number k andthe characteristic impedance Zc given by Equations (2.2) and (2.7) respectively, becomecomplex:

k = Re(k) + j Im(k)

Zc = Re(Zc) + j Im(Zc)(2.8)

If the amplitude of the waves decreases in the direction of propagation, the quantityIm(k)/Re(k) must be negative if the time dependence is chosen as exp(jωt). In thealternative convention, exp(−jωt), Im(k)/Re(k) must be positive (see Section 2.7).

2.2.4 Superposition of two waves propagating in opposite directions

The subscript x is removed for clarity. The pressure and the velocity, for a wave propa-gating toward the negative abcissa are, respectively,

p′(x, t) = A′ exp[j (kx + ωt)] (2.9)

υ ′(x, t) = −A′

Zc

exp[j (kx + ωt)] (2.10)

If the acoustic field is a superposition of the two waves described by Equations (2.3)and (2.5) and by Equations (2.9), (2.10), the total pressure pT and the total velocity vT

are

pT (x, t) = A exp[j (−kx + ωt)] + A′ exp[j (kx + ωt)] (2.11)

υT (x, t) = A

Zc

exp[j (−kx + ωt)] − A′

Zc

exp[j (kx + ωt)] (2.12)

A superposition of several waves of the same ω and k propagating in a given directionis equivalent to one resulting wave propagating in the same direction. The acoustic fielddescribed by Equations (2.11) and (2.12) is the most general unidimensional monochro-matic field. The ratio pT (x, t)/vT (x, t) is called the impedance at x. The main propertiesof the impedance are studied in the following sections.

2.3 Main properties of impedance at normal incidence

2.3.1 Impedance variation along a direction of propagation

In Figure 2.1, two waves propagate in opposite directions parallel to the x axis. Theimpedance Z(M1) at M1 is known. By employing Equations (2.11) and (2.12) for thepressure and the velocity, the impedance Z(M1) can be written

Z(M1) = pT (M1)

vT (M1)= Zc

A exp[−jkx(M1)] + A′ exp[jkx(M1)]

A exp[−jkx(M1)] − A′ exp[jkx(M1)](2.13)

18 ACOUSTIC IMPEDANCE AT NORMAL INCIDENCE OF FLUIDS

M1M2

d

x

Z(M1)Z(M2)

p

p′

Figure 2.1 Plane waves propagate both in the x direction and in the opposite direction.The impedance at M1 is Z(M1).

At M2, the impedance Z(M2) is given by

Z(M2) = Zc

A exp[−jkx(M2)] + A′ exp[jkx(M2)]

A exp[−jkx(M2)] − A′ exp[jkx(M2)](2.14)

From Equation (2.13) it follows that

A′

A= Z(M1) − Zc

Z(M1) + Zc

exp[−2jkx(M1)] (2.15)

By the use of Equations (2.14) and (2.15) we finally obtain

Z(M2) = Zc

−jZ(M1) cotg kd + Zc

Z(M1) − jZc cotg kd(2.16)

where d is equal to x(M1) − x(M2). Equation (2.16) is known as the impedance trans-lation theorem.

2.3.2 Impedance at normal incidence of a layer of fluid backedby an impervious rigid wall

A layer of fluid 1 backed by a rigid impervious plane of infinite impedance at x = 0 isrepresented in Figure 2.2. Two points M2 and M3 are shown at the boundary of fluids 1and 2, M3 being in fluid 2 and M2 in fluid 1. The impedance at M2 at the surface of the

M2 M1

x=−d

xp

p′M3

Fluid 1

x=0

Fluid 2

Figure 2.2 A layer of fluid of finite thickness in contact with another fluid on its frontface and backed by a rigid impervious wall on its rear face.

REFLECTION COEFFICIENT AND ABSORPTION COEFFICIENT 19

M1M6

x

Π

M2

d1

M3M4M5

d2 d3

Figure 2.3 Three layers of fluid backed by an impedance plane II.

layer of fluid 1 is obtained from Equation (2.16) with Z(M1) infinite:

Z(M2) = −jZc cotg kd (2.17)

where Zc is the characteristic impedance and k the wave number in fluid 1.The pressure and the velocity are continuous at the boundary. The impedance at M3

is equal to the impedance at M2, the velocities and pressures being the same on eitherside of the boundary:

Z(M3) = Z(M2) (2.18)

2.3.3 Impedance at normal incidence of a multilayered fluid

A multilayered fluid is represented in Figure 2.3. If the impedance Z(M1) is known, theimpedance Z(M2) inside fluid 1 can be obtained from Equation (2.16). The impedanceZ(M3) is equal to the impedance at M2. The impedance at M4, M5 and M6 can beobtained successively in the same way.

2.4 Reflection coefficient and absorption coefficientat normal incidence

2.4.1 Reflection coefficient

The reflection coefficient R at the surface of a layer is the ratio of the pressures p′ and p

created by the outgoing and the ingoing waves at the surface of the layer. For instance,at M3, in Figure 2.2, the reflection coefficient R(M3) is equal to

R(M3) = p′(M3, t)/p(M3, t) (2.19)

This coefficient does not depend on t because the numerator and the denominatorhave the same dependence on t . Using Equation (2.15), the reflection coefficient at M3

can be written as

R(M3) = (Z(M3)) − Z′c)/(Z(M3) + Z′

c) (2.20)

where Z′c is the characteristic impedance in fluid 2. The ingoing and outgoing waves at

M3 have the same amplitude if

|R(M3)| = 1 (2.21)

20 ACOUSTIC IMPEDANCE AT NORMAL INCIDENCE OF FLUIDS

This occurs if |Z(M3)| is infinite or equal to zero. If |Z(M3)| is finite, a moregeneral condition is Z∗(M3)Z

′c + Z(M3)Z

′∗c = 0. If Z′

c is real, this occurs if Z(M3) isimaginary. If |Z(M3)| is greater than 1, the amplitude of the outgoing wave is larger thanthe amplitude of the ingoing wave. If Z′

c is real, this occurs if the real part of Z(M3) isnegative. More generally, the coefficient R can be defined everywhere in a fluid where aningoing and an outgoing wave propagate in opposite directions. For instance, it has beenshown previously that the ratio p/v for a travelling wave propagating in the positive x

direction is Z′c. As indicated by Equation (2.20) there exists only an ingoing wave at

M inside a fluid if the impedance at M is the characteristic impedance. The behaviourof the reflection coefficient as a function of x is much simpler than the behaviour ofthe impedance. Returning to Figure 2.1, Equations (2.3) and (2.9) provide the followingrelation between R(M2) and R(M1):

R(M2) = R(M1) exp(−2jkd) (2.22)

where d = x(M1) − x(M2). Hence, the reflection coefficient describes a circle in thecomplex plane if k is real. If k is complex, the reflection coefficient describes a spiral.

It should be noted that the propagation of electromagnetic plane waves in a waveguidecan be described with impedances and reflection coefficients in a similar way to the useof those concepts in describing sound propagation. The Smith chart, which provides agraphical representation for the propagation of electromagnetic waves, can also be usedto describe the acoustic plane-wave propagation.

2.4.2 Absorption coefficient

The absorption coefficient α(M) is related to the reflection coefficient R(M) as follows

α(M) = 1 − |R(M)|2 (2.23)

The phase of R(M) is removed, and the absorption coefficient does not carry as muchinformation as the impedance or the reflection coefficient. The absorption coefficient isoften used in architectural acoustics, where this simplification can be advantageous. Itcan be rewritten as

α(M) = 1 − E′(M)

E(M)(2.24)

where E(M) and E′(M) are the average energy flux through the plane x = x(M) of theincident and the reflected waves, respectively.

2.5 Fluids equivalent to porous materials: the lawsof Delany and Bazley

2.5.1 Porosity and flow resistivity in porous materials

Porosity

Materials such as fibreglass and plastic foam with open bubbles consist of an elasticframe which is surrounded by air. The porosity φ is the ratio of the air volume Va to the

FLUIDS EQUIVALENT TO POROUS MATERIALS 21

p2

p1

V

h

Figure 2.4 A slice of porous material is placed in a pipe. A differential pressure p2 − p1

induces a steady flow V of air per unit area of material.

total volume of porous material VT . Thus,

φ = Va/VT (2.25)

Let Vb be the volume occupied by the frame in VT . The quantities Va , Vb and VT arethen related by

Va + Vb = VT (2.26)

Only the volume of air which is not locked within the frame must be considered in Va

and thus in the calculation of the porosity. The latter is also known as the open porosityor the connected porosity. For instance, a closed bubble in a plastic foam is consideredlocked within the frame, and its volume therefore belongs to Vb. For most of the fibrousmaterials and plastic foams, the porosity lies very close to 1. Methods for measuringporosity are given in Zwikker and Kosten (1949) and Champoux et al. (1991).

Flow resistivity

One of the important parameters governing the absorption of a porous material is itsflow resistance. It is defined by the ratio of the pressure differential across a sample ofthe material to the normal flow velocity through the material. The flow resistivity σ isthe specific (unit area) flow resistance per unit thickness. A sketch of the set-up for themeasurement of the flow resistivity σ is shown in Figure 2.4.

The material is placed in a pipe, and a differential pressure induces a steady flow ofair. The flow resistivity σ is given by

σ = (p2 − p1)/V h (2.27)

In this equation, the quantities V and h are the mean flow of air per unit area ofmaterial and the thickness of the material, respectively. In MKSA units, σ is expressedin N m−4 s. More information about the measurement of flow resistivity can be foundin standards ASTM C-522, ISO 9053 (1991), Bies and Hansen (1980), and Stinson andDaigle (1988).

It should be pointed out that fibrous materials are generally anisotropic (Attenborough1971, Burke 1983, Nicolas and Berry 1984, Allard et al. 1987, Tarnow 2005). A panelof fibreglass is represented in Figure 2.5. Fibres in the material generally lie in planesparallel to the surface of the material. The flow resistivity in the normal direction isdifferent from that in the planar directions. In the former case, air flows perpendicularly

22 ACOUSTIC IMPEDANCE AT NORMAL INCIDENCE OF FLUIDS

Planar direction

Normal direction

Figure 2.5 A panel of fibrous material. The normal direction is perpendicular to thesurface of the panel, and the planar directions lie in planes parallel to the surface.

to the surface of the panel while in the latter case it flows parallel to the surface ofthe layer. The normal flow resistivity σN is larger than the planar flow resistivity σP .The flow resistivity of fibreglass and open-bubble foam generally lies between 1000 and100 000 N m−4 s.

2.5.2 Microscopic and macroscopic description of sound propagationin porous media

The quantities that are involved in sound propagation can be defined locally, on a micro-scopic scale, for instance in a porous material with cylindrical pores having a circularcross-section, as functions of the distance to the axis of the pores. On a microscopic scale,sound propagation in porous materials is generally difficult to study because of the com-plicated geometries of the frames. Only the mean values of the quantities involved are ofpractical interest. The averaging must be performed on a macroscopic scale, on a homog-enization volume with dimensions sufficiently large for the average to be significant. Atthe same time, these dimensions must be much smaller than the acoustic wavelength.The description of sound propagation in porous material can be complicated by the factthat sound also excites and moves the frame of the material. If the frame is motionless,in a first step, the air inside the porous medium can be replaced on the macroscopic scaleby an equivalent free fluid. This equivalent fluid has a complex effective density ρ anda complex bulk modulus K . The wave number k and the characteristic impedance Zc ofthe equivalent fluid are also complex. In a second step, as shown in Chapter 5, Section5.7, the porous layer can be replaced by a fluid layer of density ρ/φ and of bulk modulusK /φ.

2.5.3 The Laws of Delany and Bazley and flow resistivity

The complex wave number k and the characteristic impedance Zc have been measuredby Delany and Bazley (1970) for a large range of frequencies in many fibrous materialswith porosity close to 1. According to these measurements, the quantities k and Zc

depend mainly on the angular frequency ω and on the flow resistivity σ of the material.A good fit of the measured values of k and Zc has been obtained with the followingexpressions:

Zc = ρoco[1 + 0.057X−0.754 − j0.087X−0.732] (2.28)

k = ω

co

[1 + 0.0978X−0.700 − j0.189X−0.595] (2.29)

EXAMPLES 23

where ρo and co are the density of air and the speed of sound in air (see Section 2.2.2),and X is a dimensionless parameter equal to

X = ρof /σ (2.30)

f being the frequency related to ω by ω = 2πf .Delany and Bazley suggest the following boundary for the validity of their laws in

terms of boundaries of X, as follows:

0.01 < X < 1.0 (2.31)

It may not be expected that single relations provide a perfect prediction of acousticbehaviour of all the porous materials in the frequency range defined by Equation (2.31).More elaborate models will be studied in Chapter 5. Nevertheless, the laws of Delanyand Bazley are widely used and can provide reasonable orders of magnitude for Zc

and k. With fibrous materials which are anisotropic, as indicated previously, the flowresistivity must be measured in the direction of propagation for waves travelling in eitherthe normal or the planar direction. The case of oblique incidence is more complicatedand is considered in Chapter 3. It should be pointed out that after the work by Delanyand Bazley, several authors suggested slightly different empirical expressions of k andZc for specific frequency ranges and for different materials (Mechel 1976, Dunn andDavern 1986, Miki 1990).

2.6 Examples

As a first example, the impedance at the surface of a layer of fibrous material of thicknessd equal to 10 cm, and of normal flow resistivity equal to 10 000 Nm−4 s, fixed on a rigidimpervious wall (Figure 2.6), has been calculated with the use of Equations (2.17), (2.28)and (2.29).

The real and the imaginary parts of the impedance Z are shown in Figure 2.7.As a second example, the impedance of the same layer of fibrous material with an

air gap of thickness d ′ equal to 10 cm (Figure 2.8) has been calculated.The general method of calculating Z(M) is given in Section 2.3.3. In the example

considered, Equation (2.16) can be used, Z(M1) being the impedance of the air gap.The values of Zo and co are used to calculate Z(M1) with Equation (2.17). Expressions(2.28) and (2.29) for Zc and k for the fibrous material have been used. The impedance is

d

M

Figure 2.6 A layer of porous material fixed on a rigid impervious wall.

24 ACOUSTIC IMPEDANCE AT NORMAL INCIDENCE OF FLUIDS

0 0.5 1 1.5 2 2.5 3 3.5 4−1000

−800

−600

−400

−200

0

200

400

600

800

1000

Frequency (kHz)

Z(P

a m

−1s)

Re Z

Im Z

Figure 2.7 The impedance Z at normal incidence of a layer of fibrous material ofthickness d = 10 cm, of normal flow resistivity σ = 10 000 N m−4 s, calculated accordingto the laws of Delany and Bazley.

d′

M M2 M1

airFibrous

layer

Figure 2.8 A layer of fibrous material with an air gap between the material and therigid wall.

represented in Figure 2.9, and the absorption coefficient for the material with and withoutan air gap is shown in Figure 2.10.

The interesting effect of the air gap appears clearly in Figure 2.10. The air gapincreases significantly the absorption at low frequencies. This is explained by the factthat sound absorption is mainly due to the viscous dissipation, related to the velocity ofair in the porous medium. When the material is bonded onto a hard wall, the particlevelocity at the wall is zero, and thus the absorption deteriorates rapidly at low frequen-cies. When backed by an air gap, the particle velocity at the rear face of the materialoscillates and reaches a maximum at the quarter-wavelength of the lowest frequency ofinterest, thus increasing the absorption. This is an alternative to an increase of the materialthickness.

EXAMPLES 25

0 0.5 1 1.5 2 2.5 3 3.5 4−1000

−800

−600

−400

−200

0

200

400

600

800

1000

Frequency (kHz)

Z(P

a m

−1s)

Re Z

Im Z

Figure 2.9 The impedance Z at normal incidence of a layer of fibrous material of thick-ness d = 10 cm, and flow resistivity σ = 10 000 N m−4 s, with an air gap of thicknessd ′ = 10 cm.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

Frequency (kHz)

Abs

orpt

ion

coef

ficie

nt

no gapgap

Figure 2.10 The absorption coefficient for the two previous configurations.

26 ACOUSTIC IMPEDANCE AT NORMAL INCIDENCE OF FLUIDS

2.7 The complex exponential representation

In the complex representation, the function cos(ωt − kx) is replaced by exp[j (ωt − kx)].The function exp[−j (ωt − kx)] is removed from the cosine which is given by

cos(ωt − kx) = exp[j (ωt − kx)] + exp[−j (ωt − kx)]

2(2.32)

and the remaining function is multiplied by two. The result, in signal processing, is calledan analytical signal (Papoulis 1984). It is simpler to handle this quantity rather than theinitial function, as the negative frequencies have been removed. Many authors use thetime dependence exp(−jωt), where the positive frequencies are removed. The complexrepresentation of a wave travelling in the direction of positive x will be exp[j (k∗x −ωt)], because decreasing the amplitude in the direction of propagation implies that thenew wave number is the complex conjugate of k. When adding the positive frequencycomponents of a real signal to its negative frequency components, the initial real signalmust be obtained. This will be possible simultaneously for both pressure and velocity, thecharacteristic impedances in both representations being complex conjugate. For instance,the real damped wave characterized by

p(x, t) = A exp(x Im k) cos(ωt − x Re k) (2.33)

υ(x, t) = A/|Zc| exp(x Im k) cos(ωt − x Re k − ArgZc) (2.34)

has the following two representations:

p+(x, t) = A exp[j (ωt − (Re k + j Im k)x)] (2.35)

υ+(x, t) = (A/Zc) exp[j (ωt − (Re k + j Im k)x)] (2.36)

p−(x, t) = A exp[j (−ωt + (Re k − j Im k)x)] (2.37)

υ−(x, t) = (A/Z∗c) exp[j (−ωt + (Re k − j Im k)x)] (2.38)

The quantities p− and p+ are related by

(p+ + p−)/2 = A exp(x Im k) cos(ωt − x Re k) (2.39)

In the same way

(υ+ + υ−)/2 = υ(x, t) (2.40)

The characteristic impedance Zc becomes Z∗c for the time dependence exp(−j ωt).

The impedances present the same property.Similar arguments about the reconstruction of a real signal can be used to demonstrate

that the bulk moduli K in both representations are related by complex conjugation; thesame is true for the density ρ.

ReferencesAllard, J. F., Bourdier, R. and L’Esperance, A. (1987) Anisotropy effect in glass wool on normal

impedance at oblique incidence. J. Sound Vib., 114, 233–8.

REFERENCES 27

Attenborough, K. (1971) The prediction of oblique-incidence behaviour of fibrous absorbents.J. Sound Vib., 14, 183–91.

Bies, D. A. and Hansen, C.H. (1980) Flow resistance information for acoustical design. AppliedAcoustics , 13, 357–91.

Burke, S. (1983) The absorption of sound by anisotropic porous layers. Paper presented at 106thMeeting of the ASA, San Diego, CA.

Champoux, Y, Stinson, M. R. and Daigle, G. A. (1991) Air-based system for the measurement ofporosity, J. Acoust. Soc. Amer ., 89, 910–6.

Delany, M. E. and Bazley, E. N. (1970) Acoustical properties of fibrous materials. Applied Acous-tics , 3, 105–16.

Dunn, I. P. and Davern, W. A. (1986) Calculation of acoustic impedance of multilayer absorbers.Applied Acoustics , 19, 321–34.

Gray, D. E., ed., (1957) American Institute of Physics Handbook . McGraw-Hill, New York.ISO 9053: (1991) Acoustics-Materials for acoustical applications-Determination of airflow resis-

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Acustica , 35, 210–13.Miki, Y. (1990) Acoustical properties of porous materials – Modifications of Delany–Bazley mod-

els. J. Acoust. Soc. Japan , 11, 19–24.Morse, M. K., and Ingard K. U. (1986) Theoretical Acoustics . Princeton University Press, Prince-

ton.Nicolas, J. and Berry, J. L. (1984) Propagation du son et effet de sol. Revue d’Acoustique, 71,

191–200.Papoulis, A. (1984) Signal Analysis . McGraw-Hill, Singapore.Pierce, A. D. (1981) Acoustics: An Introduction to its Physical Principles and Applications .

McGraw-Hill, New York.Stinson, M. R. and Daigle, G. A. (1988) Electronic system for the measurement of flow resistance.

J. Acoust. Soc. Amer . 83, 2422–2428Tarnow, V. (2005) Dynamic measurement of the elastic constants of glass wool. J. Acoust. Soc.

Amer . 118, 3672–3678.Zwikker, C. and Kosten, C. W. (1949) Sound Absorbing Materials . Elsevier, New York.