Engineering Acoustics/Attenuation of Sound Waves

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Introduction

When sound travels through a medium, its intensity diminishes with distance. This weakening in the energy of the wave results from two basic causes, scattering and absorption. The combined effect of scattering and absorption is called attenuation. For small distances or short times the effects of attenuation in sound waves can usually be ignored. Yet, for practical reasons it should be considered. So far in our discussions, sound has only been dissipated by the spreading of the wave, such as when we consider spherical and cylindrical waves. However this dissipation of sound in these cases is due to geometric effects associated with energy being spread over an increasing area and not actually to any loss of total energy.

Types of Attenuation

As mentioned above, attenuation is caused by both absorption and scattering. Absorption is generally caused by the media. This can be due to energy loss by both viscosity and heat conduction. Attenuation due to absorption is important when the volume of the material is large. Scattering, the second cause of attenuation, is important when the volume is small or in cases of thin ducts and porous materials.

Viscosity and Heat conduction

Whenever there is a relative motion between particles in a media, such as in wave propagation, energy conversion occurs. This is due to stress from viscous forces between particles of the medium. The energy lost is converted to heat. Because of this, the intensity of a sound wave decreases more rapidly than the inverse square of distance. Viscosity in gases is dependent upon temperature for the most part. Thus as you increase the temperature you increase the viscous forces.

Boundary Layer Losses

A special type of absorption occurs when a sound wave travels over a boundary, such as a fluid flowing over a solid surface. In such a situation, the fluid in immediate contact with the surface must be at rest. Subsequent layers of fluid will have a velocity that increases as the distance from the solid surface increases such as in the figure below.

The velocity gradient causes an internal stress associated with viscosity, that leads to a loss of momentum. This loss of momentum leads to a decrease in the amplitude of a wave close to the surface. The region over which the velocity of the fluid decreases from its nominal velocity to that of zero is called the acoustic boundary layer. The thickness of the acoustic boundary layer due to viscosity can be expressed as

$\delta _{visc}={\sqrt {\left({\frac {2*\mu }{\omega *\rho _{o}}}\right)}}$

Where $\mu \,$ is the shear viscosity number. Ideal fluids would not have a boundary layer thickness since $\mu =0\,$ .

Relaxation

Attenuation can also occur by a process called relaxation. One of the basic assumptions prior to this discussion on attenuation was that when a pressure or density of a fluid or media depended only on the instantaneous values of density and temperature and not on the rate of change in these variables. However, whenever a change occurs, equilibrium is upset and the media adjusts until a new local equilibrium is achieved. This does not occur instantaneously, and pressure and density will vary in the media. The time it takes to achieve this new equilibrium is called the relaxation time, $\theta \,$ . As a consequence the speed of sound will increase from an initial value to that of a maximum as frequency increases. Again the losses associated with relaxation are due to mechanical energy being transformed into heat.

Modeling of losses

The following is done for a plane wave. Losses can be introduced by the addition of a complex expression for the wave number

$k=\ \beta -j\alpha$

which when substituted into the time-solution yields

$\ p=Ae^{-\alpha x}e^{jwt-j\beta x}$

with a new term of $\ e^{-\alpha x}$ which resulted from the use of a complex wave number. Note the negative sign preceding $\alpha$ to denote an exponential decay in amplitude with increase values of $x$.

$\ \alpha$ is known as the absorption coefficient with units of nepers per unit distance and $\ \beta$ is related to the phase speed. The absorption coefficient is frequency dependent and is generally proportional to the square of sound frequency. However, its relationship does vary when considering the different absorption mechanisms as shown below.

The velocity of the particles can be expressed as

$\ u={\frac {k}{w*\rho _{o}}}p={\frac {1}{\rho _{o}c}}\left(1-j{\frac {\alpha }{k}}\right)p$

The impedance for this traveling wave would be given by

$\ z=\rho _{o}c{\frac {1}{1-j{\frac {\alpha }{k}}}}$

From this we can see that the rate of decrease in intensity of an attenuated wave is $\ a=8.7\alpha$

References

Wood, A. A Textbook of Sound. London: Bell, 1957.

Blackstock, David. Fundamentals of Physical Acoustics. New York: Wiley, 2000.

Attenuation considerations in Ultrasound

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