# 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

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

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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