# Engineering Acoustics/Sound Absorbing Structures and Materials

## Introduction

Noise can be defined as unwanted sound. There are many cases and
applications that reducing noise level is of great importance. Loss of
hearing is only one of the effects of continuous exposure to excessive
noise levels. Noise can interfere with sleep and speech, and cause
discomfort and other non-auditory effects. Moreover, high level noise
and vibration lead to structural failures as well as reduction in life
span in many industrial equipments. As an example in control valves, the
vibration caused by flow instability occasionally defects the feedback
to the control system and resulting in extreme oscillations. The
importance of noise issue could be well understood by looking at
regulations that have been passed by governments to restrict noise
production in society. Industrial machinery, air/surface transportation
and construction activities are assumed to be main contributors in noise
production or so called \"noise pollution\".

## Noise Control Mechanisms

-   **Active Noise control**

Modifying and canceling sound field by electro-acoustical approaches is
called active noise control. There are two methods for active control.
First by utilizing the actuators as an acoustic source to produce
completely out of phase signals to eliminate the disturbances. second
method is to use flexible and vibro-elastic materials to radiate a sound
field interfering with the disturbances and minimize the overall
intensity. The latter method is called active structural acoustic
control (ASAC).

-   **Passive Noise Control**

Passive noise control refers to those methods that aim to suppress the
sound by modifying the environment close to the source. Since no input
power is required in such methods, Passive noise control is often
cheaper than active control, however the performance is limited to mid
and high frequencies. active control works well for low frequencies
hence, the combination of two methods may be utilized for broadband
noise reduction.

\[\[Image:Absorbing_Mechanisms.jpg\|center\|thumb\|400px\|

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Figure 1: Noise Control Mechanisms

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## Sound Absorption

Sound waves striking an arbitrary surface are either reflected,
transmitted or absorbed; the amount of energy going into reflection,
transmission or absorption depends on acoustic properties of the
surface. The reflected sound may be almost completely redirected by
large flat surfaces or scattered by a diffused surface. When a
considerable amount of the reflected sound is spatially and temporally
scattered, this status is called a diffuse reflection, and the surface
involved is often termed a diffuser. The absorbed sound may either be
transmitted or dissipated. A simple schematic of surface-wave
interactions is shown in figure 2.
\[\[Image:Sound_treatment.jpg\|center\|thumb\|500px\|

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Figure 2: surface-sound interaction- absorption (left), reflection
(middle) and diffusing (right)

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\]\] Sound energy is dissipated by simultaneous actions of viscous and
thermal mechanisms. Sound absorbers are used to dissipate sound energy
and to minimize its reflection.[^1] The absorption coefficient $\alpha$
is a common quantity used for measuring the sound absorption of a
material and is known to be the function of the frequency of the
incident wave. It is defined as the ratio of energy absorbed by a
material to the energy incident upon its surface.

### Sound Absorbing Coefficient

The absorbing coefficient can be mathematically presented as follows:

$\alpha=1-\frac{I_R}{I_I}$

*where α, $I_R$, and $I_I$ are the sound absorption coefficient,
one-sided intensity of the reflected sound and the one-sided intensity
of the incident sound, respectively.*

from the above equation, it can be observed that the absorption
coefficient of materials varies from 0 to 1. there are several standard
methods to measure sound absorption coefficient. In one of the common
approaches,a plane wave impedance tube that is equipped with two
microphones is utilized.The experimental setup and dimensions are
according to ASTM E1050/ISO 10534-2.^2The method is done by
evaluating the transfer function, $\hat{h}(f)$ between two microphones
spaced ***s***apart, and a distance***l*** from sample to get absorption
coefficient using the following equations:

$\hat{h}=\frac{\hat{p_1}}{\hat{p_2}}$

$\hat{r}=\frac{\hat{h}-e^{-jks}}{e^{jks}+\hat{h}}$

$\alpha=1-|\hat{r}|^2$

*Where, $\hat{p_1},\hat{p_2}$ are complex pressure amplitude measured by
Mic. 1 and Mic. 2 respectively. **k** is the wave number, **s** is the
microphones spacing and **$\alpha$** is the absorption coefficent.*

According to the standard technique,[^3] frequency is limited by
microphone spacing as well as tube diameter. It is also recommended that
**$0.05\frac{c}{s} < f < 0.45\frac{c}{s}$** to guarantee the plane wave
propagation. The coefficient of commercial absorbing materials is
specified in terms of a noise reduction coefficient (NRC) which refers
to the average of absorption coefficients at **250 Hz, 500 Hz, 1,000 Hz,
and 2,000 Hz**. Average values of some acoustic insulating materials
that are used in buildings are tabulated in table 1. Based on their
construction and material structure, sound absorbers are categorized as
**non-porous** and **porous** absorbers.

\[\[Image:Imp_tube.jpg\|center\|thumb\|400px\|

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Figure 3: Two-microphone method to obtain sound absorption coefficient

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  Material                        Sound absorption coefficient
  ------------------------------- ------------------------------
  6 mm cork sheet                 0.1-0.2
  6 mm porous rubber sheet        0.1-0.2
  12 mm fiberboard on battens     0.3-0.4
  50 mm slag wool or glass silk   0.8-0.9
  Hardwood                        0.3
  100 mm mineral wool             0.65

  : **Table 1 - Sound absorbing coefficient of common absorbents** [^4]

### Non-Porous Absorbers ( Absorbing Resonators )

There are Two types of non-porous absorbers that are common in
industrial applications. **Panel (membrane) resonators** and **Helmholtz
resonators**. Panel absorbers are light, thin and non-porous sheets or
membranes that are tuned to absorb sound waves over a specific frequency
range. The structural resistance of the panel to fast shaping leads to
sound absorption. Panel absorbers are defined by their geometry and
structural vibration properties. Helmholtz Resonators or cavity
absorbers are perforated structures containing very small pores; one
example is the acoustic liners that are used inside the aircraft engine
frame to suppress the noise emission from the compression and combustion
stages. Similar structures are applied in fans and ventilators used in
ventilation and air-conditioning systems. The size of the opening, the
length of the neck, and the volume of the cavity govern the resonant
frequency of the resonator and hence the absorption performance.

### Porous Absorbers

Porous sound absorbers correspond to materials where sound propagation
takes place in a network of interconnected pores such that viscous and
thermal interaction cause acoustic energy to be dissipated and converted
to heat. Absorptive treatment such as mineral wool, glass fiber or
high-porosity foams reduces reflection sound. Porous absorbers are in
fact thermal materials and usually not effective sound barriers. The
need for significant thickness compared to operating sound wavelength
makes porous absorbers dramatically inefficient and impractical at low
frequencies.

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Figure 2: Typical variation of sound absorbing coefficient for different
absorbers

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## Physical Characteristic Properties of Porous Absorbers

The propagation of sound in a porous material is a phenomenon that
governed by physical characteristics of a porous medium, namely porosity
($\phi$) , tortuosity (*q*), flow resistivity ($\sigma$), viscous
characteristic length ($\Lambda$) and thermal characteristic
length($\Lambda'$).

-   **Porosity**

Defined as the ratio of interconnected void volume (air volume in open
pores) and the total volume. Most commercial absorbers have high
porosity (greater than 0.95). The higher porosity the easier interaction
between solid-fluid phases which leads to more sound attenuation.

$\phi = \frac{V_0}{V_T}$

$V_0$ = volume of the void space.

$V_T$ = total volume of the porous material.

-   **Tortuosity** [^5]

the physical characteristics corresponds to the "non-straightness" of
the pore network inside the porous material. It shows how well the
porous material prevents direct flow through the porous medium. The more
complex the path, the more time a wave is in contact with the absorbent
and hence the more energy dissipation and more absorbing capability. If
the Porous absorber is not conductive, one method to measure it is to
saturate the absorbent with an electrically conducting fluid and measure
the electrical resistivity of the saturated sample, $R_s$, and compare
to the resistivity of the fluid itself, $R_f$ then the tortuosity can be
expressed as follows:

$q = \phi\frac{R_s}{R_f}$

-   **Flow resistivity**

The pressure drop required to drive a unit flow through the material can
be related to the viscous losses of the propagating sound waves inside
the porous absorber and denoted as flow resistivity . For a wide range
of porous materials, the flow resistivity is the major factor for sound
absorption. The unit of flow resistance is **$N.s/m^4$ or $Rayls/m$**
and defined as the ratio of static pressure drop **$\Delta P$** to a
volume flow **(*U*)** for a small sample thickness **(*d*)**.

$\sigma = \frac{\Delta P}{Ud}$

-   **Characteristic lengths** [^6]

Two more important microstructural properties are the characteristic
viscous length $\Lambda$ and characteristic thermal length $\Lambda$'
that contribute viscous and thermal dissipation. The former is related
to smaller pores and the latter is related to the larger pores of porous
aggregate. The thermal length $\Lambda '$ is the twice ratio of volume
to surface area in connected pores. This is geometric and can be
measured directly. The viscous length, $\Lambda$, is nearly the same,
but each integral is weighted by the square of the fluid velocity
***v*** inside the pores and hence, cannot be measured directly.

$\Lambda ' = 2\frac{\int dV}{\int dS}$

$\Lambda = 2\frac{\int v^2_{fluid}dV}{\int v^2_{fluid} dS}$

# Acoustic modeling of Porous Absorbents

## Wave equation in rigid porous absorbents

the plane wave equation derived from the linearized equations of
conservation of mass and momentum should be modified to account for the
effects of porosity, tortuosity and flow resistance. The modified wave
equation[^7] that governs the sound propagation in compressible-gas
filled in rigid porous materials is given by:

$\frac{\partial^2 P}{\partial x^2} -(\frac{q\rho_0}{k_{eff} })*\frac{\partial^2 P}{\partial t^2}-(\frac{\sigma\phi}{k_{eff}})*\frac{\partial p}{\partial t} = 0$

where, p = sound pressure within the pores of material

$\rho_0$ = density of compressible gas

$k_{eff}$ = effective bulk modulus of the gas

*q* = tortuosity

$\phi$ = porosity

$\sigma$ = flow resistivity

The acoustical behavior of a absorptive porous layer can also be
investigated from its basic acoustic quantities: the complex wave number
and characteristic impedance. These quantities are obtained as a part of
solution of the modified plane wave equation and can be used to
determine the absorption coefficient and surface impedance. The most
practical and common values for the complex wave number and the surface
impedance are based on semi-empirical methods and correlated using the
regression analysis. One important correlation is suggested by Delany
and Bazely [^8]

$k' = \alpha+j\beta = \frac{\omega}{c} [1+0.0978(\frac{\rho_0 f}{\sigma})^{-0.700}-j0.189(\frac{\rho_0 f}{\sigma})^{-0.595}]$

$z' = R+jX =  \rho_0 c[1+0.0571(\frac{\rho_0 f}{\sigma})^{-0.754}-j0.087(\frac{\rho_0 f}{\sigma})^{-0.732}]$

Where, *f* = frequency *σ* = flow resistance

## Effective Density

By assuming rigid-frame pore network in the absorbent, the solid phase
would be completely motionless and the frame bulk modulus is
considerably greater than that of compressible gas, hence it can be
modeled as an effective fluid using the wave equation for a fluid with
complex effective fluid density and complex effective bulk modulus. In
this situation the dynamic density accounts for the viscous losses and
the dynamic bulk modulus for the thermal losses. The effective density
relation as a function of dynamic tortuosity was proposed by Johnson *et
al.* [^9]

$\rho_{eff}=q(1+\frac{\sigma\phi}{j\omega\rho_0 q} G(\omega))\rho_0$

$G(\omega)=\sqrt{1+j\frac{4q^2\mu\rho_0\omega}{\sigma^2\Lambda^2\phi^2}}$

where,

*μ* = gas viscosity

*ω* = 2*πf*

## Effective Bulk Module

Another factor that affects the sound propagation in the absorbent is
the thermal interaction in material due to the heat exchange between the
acoustic wave front traveling in the compressible fluid and the solid
phase. Champoux and Allard [^10]., have introduced a function
$G'(\omega)$ to evaluate effective bulk module for the gas. as it is
observed in the following formula this would be the function of thermal
characteristic length ($\Lambda '$).

$k_{eff}=\frac{\gamma p_0}{\gamma-(\gamma-1)(1-j\frac{8\mu}{\Lambda '^2 Pr^2\omega\rho_0} G'(\omega))\rho_0}$

$G'(\omega)=\sqrt{1+j\frac{\Lambda '^2 Pr^2\omega\rho_0}{16\mu}}$

where,

*γ* = gas specific heat ratio (for air \~ 1.4)

*Pr* = fluid Prantdl number

## References


[^1]: Cox, T. J. and P. D\'antonio, Acoustic Absorbers and Diffusers,
    SponPress,(2004)

[^2]: ASTM E1050 - 08 Standard Test Method for Impedance and Absorption
    of Acoustical Materials Using A Tube, Two Microphones and A Digital
    Frequency Analysis System

[^3]:

[^4]: Link common absorbing
    materials,
    absorbing coefficients.

[^5]:

[^6]:

[^7]: Fahy, F., Foundations of Engineering Acoustics, Academic Press
    London, (2001).

[^8]: Delany, M.E., and Bazley, E.N., \"Acoustical properties of fibrous
    absorbent materials\" Applied Acoustics, vol. 3, 1970, pp. 105-116.

[^9]: Johnson, D.L., Koplik, J., and Dashen, R.,\"Theory of dynamic
    permeability and tortuosity in fluid-saturated porous media,\"
    Journal of Fluid Mechanics, vol. 176, 1987, pp. 379-402

[^10]: Champoux, Y., and Allard, J.F., \"Dynamic tortuosity and bulk
    modulus in air saturated porous media,\" Journal of Applied Physics,
    vol. 70, no. 4, 1991, pp. 1975-1979.
