Cellular Ceramics / p4

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4.5

Acoustic Properties

Iain D. J. Dup7re, Tian J. Lu, and Ann P. Dowling

4.5.1

Introduction

Many open-celled cellular ceramics have small cell sizes making them well suited for absorbing incident sound waves. Consequently, knowledge of the acoustic behavior of cellular ceramics is extremely important for various applications. The acoustic properties can be modeled in a number of different ways or indeed measured experimentally. A brief overview of the models and experimental techniques is given in this chapter. In particular, the acoustic behavior depends on the cell structure and is relatively independent of the material used. For example, porous metals behave in a very similar way to porous ceramics with the same cell structure. While there is not a huge amount of published data on the measured characteristics of cellular ceramics, we give an example of the acoustic behavior of a porous metal showing both a theoretical model and experimental measurements.

To consider the effect that cellular ceramics have on the propagation of acoustic waves, we begin by considering the propagation of acoustic waves in inviscid conditions. We then consider how this is modified by cellular ceramics. Readers familiar with inviscid acoustic propagation are directed to Section 4.5.6.

4.5.2

Acoustic Propagation

4.5.2.1

Linearized Equations of Motion

The amplitudes of acoustic waves are typically extremely small, enabling us to linearize the flow parameters u to give

u ¼ u þ u0

where u represents any of the usual flow parameters such as pressure p, density ,

!

temperature T, or velocity u . The no mean flow mass conservation (continuity equation) and inviscid momentum equations then become

Cellular Ceramics: Structure, Manufacturing, Properties and Applications.

Michael Scheffler, Paolo Colombo (Eds.)

Copyright 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 3-527-31320-6

These equations give us the starting point for inviscid sound propagation in the absence of mean flow. In later sections we describe how the equations are changed by the presence of a cellular material.

4.5.2.2

Wave Equation

If we take the time derivative of the continuity equation we obtain

Since this involves the time derivative of the velocity, it can be combined with the divergence of the inviscid momentum equation to give

For perfect gases, the density and pressure fluctuations are linearly related, so we choose to write

p0 ¼ c2 0

where c is a constant which will be determined later. Substituting for 0 we obtain the following second-order differential equation:

Since the thermodynamic fluctuations p0 , 0 and T0 are linearly related and since the velocity must have the same time and spatial dependence, general low-ampli- tude waves satisfy the same differential equation:

where u0 is the acoustic variable. This equation is known as the wave equation because its solutions are waves. In an inviscid pipe at low frequency, the sounds waves will be planar, that is, one-dimensional. This equation tells us that one-dimen- sional waves are described by

p0 ¼ ^p1 eixðt x=cÞ þ ^p2 eixðtþx=cÞ

4.5 Acoustic Properties 383

where ^p1 and ^p2 are the amplitudes of the forward and backward traveling waves, respectively, and x is the angular frequency. We note that the constant c, introduced earlier, is the propagation speed. These pressure waves would have associated velocity and density waves:

u0 ¼ u^1 eixðt x=cÞ þ u^2 eixðtþx=cÞ ,

0 ¼ ^1 eixðt x=cÞ þ ^2 eixðtþx=cÞ .

The above equations represent one-dimensional acoustic waves propagating in the x-direction. However, similar relations exist when the waves propagate in two dimensions or radially.

4.5.2.3

Relationships between Acoustic Parameters under Inviscid Conditions

The velocity fluctuations are related to the pressure fluctuations via the inviscid momentum equation, which for one-dimensional waves in the positive x-direction gives

p0 ¼ cu0

while for a wave propagating in the negative x-direction we have:

p0 ¼ cu0 .

Thus:

p0 ¼ cu^1 eixðt x=cÞ cu^2 eixðtþx=cÞ

The density fluctuations are related to the linearized continuity equation:

@ þ @u0 ¼ 0 , @t @x

In free space, the speed of propagation c is sufficiently high that there is insufficient time for heat flow giving isentropic flow, and thus the pressure fluctuation is related to the density fluctuation via

where c is the ratio of specific heat capacities (cp/cv) and R is the gas constant.

384 Part 4 Properties

The above relations are strictly valid only when the effects of viscosity can be neglected. In practice the effects of viscosity on the propagation of acoustic waves is usually negligible in most situations, and thus they are commonly used to describe the propagation of acoustic waves. However, cellular materials are an exception to this because the pore sizes are usually so small that viscous effects must be taken into account. This modifies the momentum equation, which should now include the effect of viscous drag, and thus the relationship between the pressure and the velocity is altered in two ways. First, the waves propagate more slowly, which changes the effective speed of sound in the material. Second, it introduces some damping which converts the energy in the acoustic wave to heat. To allow for this, acoustic properties of the cellular ceramics must be considered. This can be done either theoretically or empirically.

4.5.2.4

Acoustic Energy

Acoustic waves carry acoustic energy. In the absence of a mean flow, the acoustic intensity, Iac, that is, the rate at which this acoustic energy crosses a unit area, is given by:

Iac ¼ p0 u0

Unsurprisingly, the sign of this quantity depends upon the direction of propagation. This quantity is proportional to p02 (because u0 is itself proportional to p0 ). In the presence of a mean flow of low Mach number M, the expression for the acoustic intensity is modified [1] to give

Iac ¼ p0 u0 ð1 þ MÞ2

for waves propagating in the same direction as the mean flow, and

Iac ¼ p0 u0 ð1 MÞ2

for waves propagating in the opposite direction to the mean flow.

4.5.3

Acoustic Properties

4.5.3.1

Acoustic Impedance and Admittance

The acoustic impedance Zac of a fluid is defined as:

p0

Zac ¼ u0 ¼ Rac þ iXac .

Zac is a complex quantity and is analogous to electrical impedance with p0 analogous to electrical potential and u0 analogous to electrical current. The real part

4.5 Acoustic Properties 387

4.5.3.4

Absorption Coefficient

Although the reflection and transmission coefficients provide useful information, it should be noted that small transmission or small reflection can be obtained without damping any acoustic energy at all. It is therefore also useful to quantify the absorption of the acoustic energy. This is usually done with the absorption coefficient D which is defined as the percentage of acoustic incident acoustic energy which is absorbed. For the situation shown in Fig. 2, which effectively describes an infinite pipe, this gives a unique definition:

In practice, however, there is usually a second incident wave such as wave B in Fig. 1. In this situation, there are two possible ways to define the absorption coefficient: the absorption coefficient for the material alone and the absorption coefficient for the system as a whole.

For the material alone, there are two incident waves, I and B, and two waves propagating away from the material, R and A. The absorption coefficient is then defined as:

jRj2 þjAj2 D ¼ 1 jIj2 þjBj2 .

For the system as a whole, waves A and B are internal and so the absorption coefficient is defined as:

4.5.4

Experimental Techniques

Acoustic properties of materials are usually measured by a moving-microphone technique or the two- (or four-) microphone technique. The term “impedance tube” is used variously to describe either the moving-microphone method or both methods. To avoid this confusion, we use the term “moving microphone”. In both techniques the acoustical waves in the pipe are deduced by measuring the sound field at two points.

4.5.4.1

Moving-Microphone Technique

The moving-microphone technique is illustrated in Fig. 3. Here the material sample is mounted in a rigid tube. The tube is sealed at one end with a solid end leaving a

388 Part 4 Properties

cavity of length l behind the sample. On the other side of the sample the tube is excited over a range of frequencies by a loudspeaker. A moveable microphone is also placed in the tube on this side. It is moved along the tube and measurements of the amplitude and position (x1 in Fig. 3) are taken at the points of the first maximum and first minimum in pressure amplitude. If the sample does not absorb, then a standing wave will be set up with zero amplitude at the first minimum because the forward and backward traveling waves have the same amplitudes. When the sample absorbs, however, the forward and backward traveling waves will have different magnitudes. Measuring the ratio between the amplitude at the first minimum and that at the first maximum provides enough information to decompose the sound field into the forward and backward sound waves. By measuring the distance to the first minimum, the phase between these waves can also be determined. These waves can then be used to find the surface impedance. Since the surface impedance depends on the size of the cavity l measurements must be taken with two different cavity lengths to completely describe the acoustic properties of the material. It is important that the sound waves are plane waves, so different diameter tubes should be used for different frequency ranges.

x1

sample

Fig. 3 Moving-microphone technique.

4.5.4.2

Twoand Four-Microphone Techniques

The two-microphone technique [2] is illustrated in Fig. 4. This is similar to the mov- ing-microphone technique, except that here two microphones are used and located at fixed positions. Simultaneous measurements are then taken of the amplitude of each microphone and their relative phases. This also provides enough information to deduce the sound field and hence the surface impedance. As for the movingmicrophone method, the material impedance can be calculated if measurements are made with two cavity lengths. However, another possibility is to take measurements on either side of the sample by using four microphones. Since this enables the user to calculate the waves on either side of the sample, this negates the need to take measurements with two cavity lengths. The material impedance Zmat, which does not depend upon this length, can then be calculated directly. This is called the fourmicrophone technique by some authors, but since it essentially involves applying the two-microphone technique on either side of the sample, others refer to it as the

4.5 Acoustic Properties 389

two-microphone technique. As for the moving-microphone method, the sound waves must be planar, so different diameter tubes should be used for high-frequency measurements than for low-frequency measurements. An additional constraint for the two-microphone technique is that the microphones should be less than half a wavelength apart to prevent the matrix from being singular.

Microphones

Open

or

Closed

end

Loudspeaker

Sample

Fig. 4 Two-microphone technique.

The two-microphone technique has the disadvantage that the microphones must be calibrated relative to each other to ensure accurate measurements of the phase difference. However, it has the considerable advantage that the positions of the microphones are fixed and so can be measured more accurately.

4.5.5

Empirical Models

The characteristic impedance and the wavenumber of a material are convenient ways to characterize materials such as cellular ceramics. Delany and Bazley [3] showed empirically that these parameters are functions of the dimensionless parameter x/r, where r is the static flow resistance. The static flow resistance is often obtained from Darcy’s experimental law:

u¼ C @p.

@x

They suggest that these parameters be expressed as power-law functions derived empirically. Stinson and Champoux [4] consider the problem of rectangular and triangular pores and compare the theoretical predictions based upon Biot’s method with the empirical correlations of Delany and Bazley [3] with reasonable agreement. Design charts based on this approach are provided by Bies and Hansen [5]. Experimental results for the acoustic properties of a semi-open metal foam are given by Lu et al. [6].

390Part 4 Properties

4.5.6

Theoretical Models

In cellular materials, the effects of viscosity become important, and so the relationship between the pressure fluctuation p0 and the particle velocity fluctuation u0 is

altered. This is true both when the propagation is along the channel formed between the material elements and when it is normal to them. This alters the acoustic behavior considerably by introducing damping, changing the mean propagation speed, and changing the surface impedance. Some authors have modeled this by introducing effective physical parameters, such as a complex density, while others have gone back to the equations of motion including the viscous term, which they then solve. These techniques are reviewed in this section.

4.5.6.1

Viscous Attenuation in Channels (Rayleigh’s Model)

A simple model of a cellular ceramic, favored by many authors, is to assume that the ceramic creates small channels through which the sound waves can propagate.

The effect of viscosity on the propagation of sound was considered in detail by Lord Rayleigh [7], who considered the problem of plane sound waves propagating in free space above an infinite plane surface. By solving the one-dimensional Navier–- Stokes equation for a harmonic wave with zero tangential velocity on the plane surface, he concluded that the coefficient of decay a is given by:

8p2 l a ¼ 3k2 c

where l is the coefficient of viscosity and k is the acoustic wavelength in meters. Thus the sound must propagate over vast distances in free space above a plane for

the viscous attenuation to be nonnegligible. A similar conclusion was reached by Lighthill [8–10]. When the propagation is through very narrow tubes, such as in some porous materials, the effect of the viscosity can be much more significant. Rayleigh extended the analysis for propagation over an infinite plane to consider the case of a narrow rigid tube in which the thickness of the boundary layer was nevertheless considered small in comparison with the diameter of the tube, so that the waves could still be treated as one-dimensional. This results in a modified wave equation for a narrow circular tube of the form [7]:

where X is the local volume such that the velocity is p1r2 @@Xt ; r the radius of the circu-

lar tube, v the kinematic viscosity, and x the angular frequency. The local volume X is used here simply for convenience, since it results in a wavelike equation. From this expression it is seen that the viscosity reduces the speed of the waves to give an approximate value of: