# Engineering Acoustics/Outdoor Sound Propagation

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## Introduction

Outdoor sound propagation or atmospheric sound propagation is of special
interest in environmental acoustics which is concerned with the control
of sound and vibrations in an outdoor environment. Outdoor sound
propagation is affected by spreading, absorption, ground configuration,
terrain profile, obstacles, pressure, wind, turbulence, temperature,
humidity, etc. The subjects covered in this page are speed of sound in
air, decibel scales, spreading losses, attenuation by atmospheric
absorption, attenuation over the ground, refraction, diffraction and
sound reduction examples.

## Speed of sound in air

Sound speed in air varies with pressure, density, temperature, humidity,
wind speed, etc. The expression for the speed of sound $c$ in a fluid is
given in terms of its thermodynamic properties. The equation is given by

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$c^2 =\ \left( \frac{ \partial P}{ \partial \rho} \right) _{adiabat}$

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where $\rho$ is the fluid density and $P$ is the fluid pressure.

This equation can be simplified for an ideal gas leading to

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$c^2 = \gamma \frac{P_0}{\rho_0}$

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where $\gamma$ is the ratio of heat capacities

For air at 0 °C and 1 atm, the speed of sound is

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```
$c_0 = 331.5 \ \mbox{m/s}$

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For air at 20 °C and 1 atm, the speed of sound is

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$c_0 = 343 \ \mbox{m/s}$

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An equivalent expression for the speed of sound in terms of the
temperature in Kelvin $T_K$ is

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$c^2 = \ \gamma r T_{K}$

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where $r$ is the specific gas constant.

## Decibel scale

The decibel dB or dB SPL (Sound pressure level) in acoustics is used to
quantify sound pressure levels and intensities relative to a reference
on a logarithmic scale.

The intensity level $IL$ of a sound intensity $I$ is defined by

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```
$IL = 10 \log_{10} \bigg(\frac{I}{I_{ref}}\bigg) \mbox{ dB} \,$

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```
where $I_{ref}$ is a reference intensity.

Since the intensity carried by a traveling wave is proportional to the
square of the pressure amplitude, the intensity level can be expressed
as the sound pressure level

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```
$SPL = 10 \log_{10} \bigg(\frac{{P_e}^2}{{P_{ref}}^2}\bigg) = 20 \log_{10} \bigg(\frac{P_e}{P_{ref}}\bigg) \mbox{ dB} \,$

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```
where $P_e$ is the measured effective pressure amplitude of the sound
wave and $P_{ref}$ is the reference effective pressure amplitude. The
effective sound pressure is the root mean square of the instantaneous
sound pressure over a given interval of time. $SPL$ is also called sound
level $L_p$.

For air, the pressure reference is taken to be

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```
$P_{ref} = 20 \cdot 10^{-6} \mbox{ Pa} \$,

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```
the lowest sound pressure a human can hear, also called the threshold of
hearing.

![](Plot_of_decibel_and_inverse.png "Plot_of_decibel_and_inverse.png")

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**Figure 1 - Plot of decibel and inverse**

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## Sound attenuation

Study of outdoor sound activity requires the definition of a source and
a receiver in order to explain the different phenomenon involved in the
process. The sound attenuation due to its propagation in the atmosphere
can be described in terms of its total attenuation $A_{T}$ in dB between
the source and the receiver. The total attenuation $A_{T}$ can be
expressed as

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```
$A_{T} = L_{ps} - L_{pr} = 10 \log_{10} \bigg(\frac{I_{ps}}{I_{pr}}\bigg) = 20 \log_{10} \bigg( \frac{P_{s}}{P_{r}} \bigg) \mbox{ dB} \,$

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where $L_{ps}$ is the sound pressure level of the root-mean-square (rms)
sound pressure $P_s$ at a distance $s$ near the source and $L_{pr}$ is
the corresponding sound pressure level with an rms sound pressure $P_r$
measured at a distance r from the source.

The total attenuation is defined as the sum of the attenuation due to
geometric spreading $A_s$, the attenuation due to atmospheric absorption
$A_a$, and the excess attenuation due to all other effects $A_e$, namely

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```
$A_{T} = A_s + A_a + A_e \mbox{ dB} \,$

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The excess attenuation $A_e$ can include attenuation from the ground in
a homogeneous atmosphere $A_g$, refraction by a non-homogeneous
atmosphere, attenuation by diffraction and reflection by a barrier or
obstacle, and scattering or diffraction effects due to turbulence. The
values of attenuation are normally positive.

## Spreading losses

The geometric spreading loss $A_s$, in dB, between two points at a
distance $r_1$ and $r_2$ from a source can be expressed as

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```
$A_{s} = 20 g \log_{10} \bigg( \frac{r_{2}}{r_{1}} \bigg) \quad \mbox{dB}$

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where $g$ is a constant given by the geometry of the problem. $g = 0$
for plane wave propagation (uniform pipe), $g = 0.5$ for cylindrical
propagation from a line source, and $g = 1$ for spherical wave
propagation from a point source. It is noticed that for spherical wave
propagation from a point source, doubling the distance from the source
($r_2 = 2 r_1$) corresponds to a loss of $6 \mbox{dB}$.

## Attenuation by atmospheric absorption

Absorption of sound through the atmosphere is due to shear viscosity,
thermal conductivity or heat dissipation, and molecular relaxation due
to oxygen, nitrogen, and water vapor vibrational, rotational, and
translational energy. The attenuation $A_a$, in dB due to atmospheric
absorption can be expressed as

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```
$A_{a} = -20 \log_{10} \left[ \frac{P(r)}{P(0)} \right] = -20 \log_{10} [ \exp (- \alpha r) ] = a r \quad \mbox{dB}$

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where $r$, in meters, is the path length of the traveling wave, $P(r)$
is the sound pressure after traveling the distance $r$, $P(0)$ is the
initial sound pressure at $r = 0$, $\alpha$ is the attenuation
coefficient in Nepers per meter, and $a$ is the attenuation coefficient
in dB per meter.

Spreading losses are dependent on the pressure, relative humidity and
frequency for air in still atmosphere. The attenuation coefficient $a$
for pure tone frequencies can be expressed as

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```
$\frac{a}{p_s} = \frac{20}{\ln 10}\ \frac{F^2}{p_{s0}} \left \{ 1.84 \times 10^{-11}   \left( \frac{T}{T_0} \right)^{1/2} + \left( \frac{T}{T_0} \right)^{-5/2} \left[ 0.01275 \frac{e^{-2239.1/T}}{F_{r,O}+F^2/F_{r,O}} + 0.1068\frac{e^{-3352/T}}{F_{r,N}+F^2/F_{r,N}}  \right]  \right \} \quad \frac{\mathrm{dB}}{\mathrm{m} \cdot \mathrm{atm}} \,$

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with $F = f/p_s$, $F_{r,O} = f_{r,O}/p_s$, and $F_{r,N} = f_{r,N}/p_s$,
and where $f$ is the acoustic frequency in Hz, $p_s$ is the atmospheric
pressure, $p_{s0}$ is the reference atmospheric pressure (1 atm), $T$ is
the atmospheric temperature in K, $T_0$ is the reference temperature
(293.15 K), $f_{r,O}$ is the relaxation frequency of molecular oxygen
and $f_{r,N}$ is the relaxation frequency of molecular nitrogen. Scaled
relaxation frequencies for oxygen and nitrogen formulas from
experimental measurements are given by

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$F_{r,O} = \frac{1}{p_{s0}} \left( 24 + 4.04 \times 10^4 h \frac{0.02 + h}{0.391 + h} \right) \quad \frac{\mathrm{Hz}}{\mathrm{atm}}$

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and

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$F_{r,N} = \frac{1}{p_{s0}} \left(  \frac{T_0}{T} \right)^{1/2}  \left( 9 + 280 h \times \exp \left\{  -4.17 \left[ \left(  \frac{T_0}{T} \right)^{1/3} - 1 \right] \right\}  \right) \quad \frac{\mathrm{Hz}}{\mathrm{atm}}$

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where $h$ is the molar concentration of water vapor (absolute humidity)
in percent. $h$ is calculated from the relative humidity $h_r$ as
follows

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```
$h = p_{s0}  \left( \frac{h_r}{p_s} \right) \left( \frac{p_{sat}}{p_{s0}} \right) \quad \%$

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where the saturated vapor pressure $p_{sat}$ is given by

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$p_{sat} = p_{s0} \times 10^{ -6.8346 (T_{01}/T)^{1.261} + 4.6151  }  \quad \mbox{atm}$

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with $T_{01} = 273.16 \mbox{K}$.

The formulas, are valid for a pressure under 2 atm, a temperature under
330 K (57 C, or 134 F) and up to an altitude of 3 km. One can see from
the graph and formulas that the absorption coefficient is higher for a
higher frequency and/or a higher pressure.

The attenuation coefficient $\alpha$ for pure tone frequencies is shown
in Figure 2 for air at 20 °C as a function of frequency per atmosphere
and relative humidity per atmosphere. The matlab script used to produce
the graph is shown in Appendix A.

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```
![](Atmospheric_sound_absorption_coefficient_2.svg "Atmospheric_sound_absorption_coefficient_2.svg"){width="650"}

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```
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```
**Figure 2 - Attenuation coefficient for atmospheric absorption per
atmosphere as a function of frequency and relative humidity, for air at
20 °C.
**

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```
Values of attenuation can also be obtained from Table 1 for different
temperatures, relative humidities and pure tone frequencies at 1
atmosphere.

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**Table 1 - Atmospheric attenuation coefficient $a$ (dB/km) at selected
frequencies at 1 atm**

  Temperature   Relative humidity (%)   62.5 Hz   125 Hz   250 Hz   500 Hz   1000 Hz   2000 Hz   4000 Hz   8000 Hz
  ------------- ----------------------- --------- -------- -------- -------- --------- --------- --------- ---------
  30 °C         10                      0.362     0.958    1.82     3.40     8.67      28.5      96.0      260
  20            0.212                   0.725     1.87     3.41     6.00     14.5      47.1      165
  30            0.147                   0.543     1.68     3.67     6.15     11.8      32.7      113
  50            0.091                   0.351     1.25     3.57     7.03     11.7      24.5      73.1
  70            0.065                   0.256     0.963    3.14     7.41     12.7      23.1      59.3
  90            0.051                   0.202     0.775    2.71     7.32     13.8      23.5      53.5
  20 °C         10                      0.370     0.775    1.58     4.25     14.1      45.3      109       175
  20            0.260                   0.712     1.39     2.60     6.53     21.5      74.1      215
  30            0.192                   0.615     1.42     2.52     5.01     14.1      48.5      166
  50            0.123                   0.445     1.32     2.73     4.66     9.86      29.4      104
  70            0.090                   0.339     1.13     2.80     4.98     9.02      22.9      76.6
  90            0.071                   0.272     0.966    2.71     5.30     9.06      20.2      62.6
  10 °C         10                      0.342     0.788    2.29     7.52     21.6      42.3      57.3      69.4
  20            0.271                   0.579     1.20     3.27     11.0     36.2      91.5      154
  30            0.225                   0.551     1.05     2.28     6.77     23.5      76.6      187
  50            0.160                   0.486     1.05     1.90     4.26     13.2      46.7      155
  70            0.122                   0.411     1.04     1.93     3.66     9.66      32.8      117
  90            0.097                   0.348     0.996    2.00     3.54     8.14      25.7      92.4
  0 °C          10                      0.424     1.30     4.00     9.25     14.0      16.6      19.0      26.4
  20            0.256                   0.614     1.85     6.16     17.7     34.6      47.0      58.1
  30            0.219                   0.469     1.17     3.73     12.7     36.0      69.0      95.2
  50            0.181                   0.411     0.821    2.08     6.83     23.8      71.0      147
  70            0.151                   0.390     0.763    1.61     4.64     16.1      55.5      153
  90            0.127                   0.367     0.760    1.45     3.66     12.1      43.2      138

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The effective atmospheric attenuation of a constant-percentage band of a
broadband noise is normally less than for pure-tone sound due to the
finite bandwidth and slope of the filter
skirts. Some atmospheric
attenuation also occurs in fog and
precipitation,
in dust in
air,
and at frequencies below 10 Hz due to electromagnetic radiation of moist
air
molecules.

## Reflection from the surface of a solid

When a wavefront comes in to contact with a solid surface, it is
reflected away from that surface. The angle of reflection of the sound
wave is equal to the angle of incidence of the wave. Reflected waves can
interfere with incident waves causing constructive and destructive
interference. This can cause a standing wave pattern and resonance
because the incident and reflected waves travel in opposite directions.
Near the surface of the solid, sound pressure intensity is enhanced
because the pressure of the reflected wave adds up to the pressure of
the incident wave.

![](Sound_Angle_of_Incidence.png "Sound_Angle_of_Incidence.png")

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**Figure 3 - Angle of incidence for a reflected sound wave**

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## Attenuation over the ground

Sound propagation near the ground is affected by absorption and
reflection of the sound waves by the ground. Sound can either leave a
source and follow a straight path to a receiver or be reflected and/or
absorbed by the ground. How the sound wave reacts with the ground is
influenced by the ground impedance which relates pressure and speed.

## Refraction

Refraction of sound is normally defined as the deviation of a sound wave
leaving a fluid and entering another one with different speed of sound.
For outdoor sound propagation refraction of sound waves causes the waves
to bend due to a change in the speed of sound. This change in the speed
of sound is caused by strong wind speeds and temperature gradients. An
upward refraction reduces sound levels near the ground, while a downward
refraction helps sound travel over obstacle such as noise barriers. A
downward refraction will occur if air above the earth is warmer than the
air at the surface. The warmer air above earth will have a faster speed
of sound causing the wave to bend back towards the earth. The same
phenomena will occur for hot air over a cold ground.

![](Outdoor_Sound_Refraction.png "Outdoor_Sound_Refraction.png")

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**Figure 4 - Outdoor sound refraction caused by temperature gradients**

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## Diffraction

Diffraction is the mechanism by which sound can propagate and spread out
beyond an opening and around obstacles. The traveling waves tend to bend
around obstacles. Diffraction is related to the wavelength of the sound
produced by the source. It is stronger for lower frequencies. High
frequencies propagate in a more directional manner. This is why low
frequencies can be heard better from behind obstacles and in shadows
zones. If a wavefront travels towards a small opening, diffraction will
cause the waves to spread out past the opening in a spherical manner.
When a wavefront that passes through a small obstacle, compared to the
wavelength, diffraction will cause the sound waves to bend and the
wavefront will reconstruct past the obstacle. This means that one could
not identify the presence of that small obstacle from sound measurement
far from the obstacle and the source.

![](Sound_diffraction_paths.png "Sound_diffraction_paths.png")

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**Figure 5 - Different paths of sound propagation behind a barrier**

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![](Sound_Diffraction_from_a_Hole.png "Sound_Diffraction_from_a_Hole.png")

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**Figure 6 - Outdoor sound diffraction behind a small opening**

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## Sound reduction

Several mechanisms can be used in order to reduce noise in outdoor
environment. Different noise barriers used to reduce highway noise are
shown in Figure 7. Another way of reducing noise is to use vegetation.
Figure 8 shows pictures of vertical vegetation walls.

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```
  ![](Geluidswal_123.jpg "Geluidswal_123.jpg"){width="300"}   ![](TullamarineFwy.jpg "TullamarineFwy.jpg"){width="300"}   ![](Slunečný_vršek,_K_Horkám_-_Bratislavská.jpg "Slunečný_vršek,_K_Horkám_-_Bratislavská.jpg"){width="300"}
  ----------------------------------------------------------- ----------------------------------------------------------- -------------------------------------------------------------------------------------------------------------

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**Figure 7 - Different types of noise barriers**

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```
  ![](Jardin_vertical_de_Plaza_del_Pericón,_Málaga..JPG "Jardin_vertical_de_Plaza_del_Pericón,_Málaga..JPG"){width="300"}   ![](Mur_vegetal_avignon_nuit1.jpg "Mur_vegetal_avignon_nuit1.jpg"){width="300"}   ![](CaixaForumMadridyJardinVertical.jpg "CaixaForumMadridyJardinVertical.jpg"){width="300"}
  ------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------

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**Figure 8 - Vertical vegetation walls**

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```
## Useful Websites

-       HyperPhysics,
    Section on the basics of sound propagation

-   Acoustical Porous Material Recipes

## References and further reading

References are presented in order of date of publication.

**General**

1.  Piercy, Embleton, Sutherland, *Review of noise propagation in the
    atmosphere*, J. Acoust. Soc.
    Am. Volume 61, Issue 6, pp. 1403--1418, June 1977
2.  Delany,

    *Sound propagation in the atmosphere - A historical
    review*,
    Acustica. Vol. 38, pp. 201--223, October 1977
3.  Piercy, Embleton, *Review of sound propagation in the atmosphere
    (A)*, J. Acoust. Soc. Am.
    Volume 69, Issue S1, pp. S99-S99, May 1981
4.  Crocker,  *Handbook
    of
    Acoustics*,
    John Wiley & Sons, February 1998
5.  Kinsler, Frey, Coppens, Sanders,
     *Fundamentals of
    Acoustics*,
    4th Ed., John Wiley & Sons, New York, 2000

**Speed of sound**

1.  Wong, *Speed of sound in standard
    air*, J. Acoust. Soc. Am.
    Volume 79, Issue 5, pp. 1359--1366, May 1986

**Absorption of sound in the atmosphere**

1.  Calvert, Coffman, Querfeld,
    *Radiative
    Absorption of Sound by Water Vapor in the
    Atmosphere*, J. Acoust. Soc.
    Am. Volume 39, Issue 3, pp. 532--536, March 1966
2.  Henley and Hoidale,

    *Attenuation and dispersion of acoustic energy by atmospheric
    dust*, J. Acoust. Soc. Am.
    Volume 54, Issue 2, pp. 437--445, August 1973
3.  Sutherland, Piercy, Bass, Evans, *Method for calculating the
    absorption of sound by the
    atmosphere*, J. Acoust. Soc.
    Am. Volume 56, Issue S1, pp. S1-S1, November 1974
4.  Bass, Sutherland, Piercy, Evans,
    \[<http://adsabs.harvard.edu/abs/1984papm>\...17..145B *Absorption
    of sound by the atmosphere*\], Physical acoustics: Principles and
    methods. Volume 17 (A85-28596 12-71). Orlando, FL, Academic Press,
    Inc., p. 145-232, 1984
5.  Bass, Sutherland, Zuckerwar,

    *Atmospheric absorption of sound:
    Update*, J. Acoust. Soc. Am.
    Volume 88, Issue 4, pp. 2019--2021, October 1990
6.  Bass, Sutherland, Zuckerwar, Blackstock, Hester,

    *Atmospheric absorption of sound: Further
    developments*, J. Acoust. Soc.
    Am. Volume 97, Issue 1, pp. 680--683, January 1995

**Attenuation over the ground**

1.  Embleton, Piercy, and Olson, *Outdoor sound propagation over ground
    of finite impedance*, J.
    Acoust. Soc. Am. Volume 59, Issue 2, pp. 267--277, February 1976
2.  Bolen and Bass, *Effects of ground cover on the propagation of
    sound through the
    atmosphere*, J. Acoust. Soc.
    Am. Volume 69, Issue 4, pp. 950--954, April 1981
3.  Embleton, Piercy, and Daigle, *Effective flow resistivity of ground
    surfaces determined by acoustical
    measurements*, J. Acoust. Soc.
    Am. Volume 74, Issue 4, pp. 1239--1244, October 1983
4.  Rasmussen, *On the effect of terrain profile on sound propagation
    outdoors*,
    Journal of Sound and Vibration, Volume 98, Issue 1, Pages 35--44,
    January 1985
5.  Attenborougha, *Review of ground effects on outdoor sound
    propagation from continuous broadband
    sources*,
    Applied Acoustics, Volume 24, Issue 4, Pages 289-319, 1988
6.  Hess, Attenborough, and Heap, *Ground characterization by
    short-range propagation
    measurements*, J. Acoust. Soc.
    Am. Volume 87, Issue 5, pp. 1975--1986, May 1990
7.  Attenborough, Taherzadeh, Bass, Di, and others *Benchmark cases for
    outdoor sound propagation
    models*, J. Acoust. Soc. Am.
    Volume 97, Issue 1, pp. 173--191, January 1995

**Vegetation**

1.  Aylor, *Noise Reduction by Vegetation and
    Ground*, J. Acoust. Soc. Am.
    Volume 51, Issue 1B, pp. 197--205, January 1972
2.  Bullen, Fricke, *Sound propagation through
    vegetation*,
    Journal of Sound and Vibration, Volume 80, Issue 1, 8 January 1982,
    Pages 11--23, May 1981
3.  Price, Attenborough, Heap, *Sound attenuation through trees:
    Measurements and models*, J.
    Acoust. Soc. Am. Volume 84, Issue 5, pp. 1836--1844, November 1988

**Barriers and screens**

1.  Maekawa, *Noise reduction by
    screens*,
    Applied Acoustics, Volume 1, Issue 3, Pages 157-173, July 1968
2.  Jonasson, *Sound reduction by barriers on the
    ground*,
    Journal of Sound and Vibration, Volume 22, Issue 1, Pages 113-126,
    May 1972
3.  Kurze, *Noise reduction by
    barriers*, J. Acoust. Soc. Am.
    Volume 55, Issue 3, pp. 504--518, March 1974
4.  Isei, Embleton, Piercy, *Noise reduction by barriers on finite
    impedance ground*, J. Acoust.
    Soc. Am. Volume 67, Issue 1, pp. 46--58, January 1980
5.  Li, Law, Kwok, *Absorbent parallel noise barriers in urban
    environments*,
    Journal of Sound and Vibration Volume 315, Issues 1-2, Pages
    239-257, August 2008

**Absorbent materials**

1.  Delany and Bazley, *Acoustical properties of fibrous absorbent
    materials*,
    Applied Acoustics, Volume 3, Issue 2, April 1970, Pages 105-116
2.  Attenborough, *Acoustical characteristics of porous
    materials*,
    Physics Reports Volume 82, Issue 3, Pages 179-227, February 1982
3.  Lauriksa, Copsa, Verhaegena, *Acoustical properties of elastic
    porous
    materials*,
    Journal of Sound and Vibration Volume 131, Issue 1, Pages 143-156,
    22 May 1989

## Appendices

### Appendix A - Matlab program for the plot of the attenuation coefficient

``` matlab
clear all;
clc ;
close all;

T_0 = 293.15;
T_01 = 273.16 ;
T = 20 + 273.15;
p_s0 = 1;
F = logspace(1,6);
ler=length(F);
hrar=[0 10 20 40 60 80 100];
a_ps_ar=zeros(7,ler);
for k=1:7
    hr=hrar(k);
    psat = p_s0*10^(-6.8346*(T_01/T)^1.261 + 4.6151);
    h = p_s0*(hr)*(psat/p_s0);
    F_rO = 1/p_s0*(24 + 4.04*10^4*h*(0.02+h)/(0.391+h));
    F_rN = 1/p_s0*(T_0/T)^(1/2)*( 9 + 280*h*exp(-4.17*((T_0/T)^(1/3)-1)) );
    alpha_ps= 100*F.^2./p_s0.*( 1.84*10^(-11)*(T/T_0)^(1/2)...
    + (T/T_0)^(-5/2)*(0.01275*exp(-2239.1/T)./(F_rO + F.^2/F_rO)...
    + 0.1068*exp(-3352/T)./(F_rN + F.^2/F_rN) ) );
    a_ps_ar(k,:) = alpha_ps*20/log(10);

end
psvg = figure (1);
loglog(F,a_ps_ar(1,:), F,a_ps_ar(2,:), F,a_ps_ar(3,:), F,a_ps_ar(4,:),...
    F,a_ps_ar(5,:), F,a_ps_ar(6,:), F,a_ps_ar(7,:),'LineWidth',1);
xlabel({'f / p_s [Hz/atm]';'Frequency/pressure'},'FontSize',10,...
    'FontWeight','normal','FontName','Times New Roman');
ylabel('Absorption coefficient/pressure a / p_s [dB/100 m atm]',...
    'FontSize',10,'FontWeight','normal','FontName','Times New Roman');
title({'Sound absorption coefficient per atmosphere for air at 20°C ';...
    'according to relative humidity per atmosphere'},...
    'FontSize',10,'FontWeight','bold','FontName','Times New Roman')
hleg = legend('   0','   10',...
    '   20','   40','   60',...
    '   80','   100');
v = get(hleg,'title');
set(v,'string',{'h_r / p_s [%/atm]'},'FontName','Times New Roman','FontSize',10,...
    'BackgroundColor', 'white','EdgeColor','white','HorizontalAlignment','center');
set(hleg,'Location','SouthEast','EdgeColor','black')
axis([1e1 1e6 1e-3 1e4]);
grid on;
set(gca,'gridlinestyle','-');
set(gca,'MinorGridLineStyle','-')
%plot2svg('Absorption_coefficient.svg',psvg);
```

### Appendix B - Python program for the plot of the attenuation coefficient

``` python
#!/usr/bin/python3

import math
import numpy as np
import matplotlib.pyplot as plt

## 1 atm in Pa
ps0 = 1.01325e5

def absorption(f, t=20, rh=60, ps=ps0):
    """ In dB/m

        f: frequency in Hz
        t: temperature in °C
        rh: relative humidity in %
        ps: atmospheric pressure in Pa

        From http://en.wikibooks.org/wiki/Engineering_Acoustics/Outdoor_Sound_Propagation
        See __main__ for actual curves.
    """
    T = t + 273.15
    T0 = 293.15
    T01 = 273.16

    Csat = -6.8346 * math.pow(T01 / T, 1.261) + 4.6151
    rhosat = math.pow(10, Csat)
    H = rhosat * rh * ps0 / ps

    frn = (ps / ps0) * math.pow(T0 / T, 0.5) * (
            9 + 280 * H * math.exp(-4.17 * (math.pow(T0 / T, 1/3.) - 1)))

    fro = (ps / ps0) * (24.0 + 4.04e4 * H * (0.02 + H) / (0.391 + H))

    alpha = f * f * (
        1.84e-11 / ( math.pow(T0 / T, 0.5) * ps / ps0 )
        + math.pow(T / T0, -2.5)
        * (
            0.10680 * math.exp(-3352 / T) * frn / (f * f + frn * frn)
            + 0.01278 * math.exp(-2239.1 / T) * fro / (f * f + fro * fro)
            )
        )

    return 20 * alpha / math.log(10)

def plot():
    ## Figure in http://en.wikibooks.org/wiki/Engineering_Acoustics/Outdoor_Sound_Propagation
    ax = plt.subplot(111)
    fs = np.logspace(1, 6, num=100, endpoint=True, base=10)
    ys = np.zeros(fs.shape)
    rh = (0, 10, 20, 40, 60, 80, 100)
    for r in rh:
        for i in np.arange(fs.shape[0]):
            ys[i] = absorption(fs[i], rh=r)
        ax.loglog(fs, 100 * ys, label='rh:%d'%r)
    ax.grid(True)
    ax.set_xlabel('Frequency/pressure [Hz/atm]')
    ax.set_ylabel('Absorption coefficient/pressure [dB/100m.atm]')
    ax.legend(loc='lower right')
    plt.show()

def table():
    p = ps0
    for t in [30, 20, 10, 0]:
        for rh in [10, 20, 30, 50, 70, 90]:
            print("T=%2d RH=%2d " % (t, rh), end='')
            for f in [62.5, 125, 250, 500, 1000, 2000, 4000, 8000]:
                a = absorption(f, t, rh, p)
                print("%7.3f " % (a*1000), end='')
            print()

if __name__ == '__main__':
    table()
    plot()
```

Back to main page


# Engineering Acoustics/Anechoic Tiling

## Introduction

! HMS Triumph (S93))
anechoic
tiling. anechoic tiling."){width="325"}

One important application of acoustics research is the testing, design,
and construction of anechoic tiles for underwater acoustic stealth.
Anechoic tiling, first used in the Second World War by German U-boats
(codenamed "Alberich" [^1]), are used to reduce the acoustic signature
of naval vessels. The tiles reduce both reflected active SONAR off the
pressure hull, and reduce internal noise transmitted through the
pressure hull that can be picked up by passive SONAR. In modern times,
anechoic tiles are present on nearly all submarines. The frequency range
of SONAR, and consequently the range of interest of anechoic tiling to
minimize detection, is around 1-30 kHz [^2].

The main sound attenuation mechanism in the tiles comes from resonant
scattering the sound waves due to air cavities in the rubber. The use of
air bubbles to attenuate sound was first published by German
acousticians Erwin Meyer and Eugen Skudrzyk in a report written in the
British occupied zone of Germany in 1946, translated to English in 1950
for an unclassified release by the U.S. Navy [^3]. The air bubbles work
to attenuate sound by acting as resonant oscillators and dissipate
acoustic energy through thermal losses, frictional losses, and other
processes[^4].

Tank measurements of acoustic parameters with the panel surround on both
sides by water, or "free-field", are theoretically simple measurements
that can be conducted in an interior laboratory setting [^5], and will
be described in later sections. Audoly [^6] presents a method to
transfer the results of free-field acoustic properties from tank
acoustic measurements to the acoustic properties of panels with
arbitrary backing, such as the rigid backing of a submarine hull.

## Planar Waves

The transmission and reflection coefficients $\hat{T}$ and $\hat{R}$ are
presented as ratios of the incident, reflected, and transmitted acoustic
pressure magnitudes:

{\\hat{P}\_{incident}},\\quad \\quad \\hat{R} =
\\frac{\\hat{P}\_{reflected}}{\\hat{P}\_{incident}}`</math>`{=html}\|}}

The conservation of acoustic power from transmitted waves through a
panel is as follows: $|\hat{T}|^2+|\hat{R}|^2+|\hat{A}|^2=1$

Where $\hat{A}$ is the acoustic absorption coefficient. From careful
observation of the conservation of energy equation, it is evident that
to minimize $|\hat{T}|$ and $|\hat{R}|$, acoustic energy dissipated
$|\hat{A}|$ should be increased. In materials such as metals in the
frequency ranges pertinent to this study, $|\hat{A}| \approx 0$. In
rubber materials, and especially those with air voids [^7] [^8],
$|\hat{A}|$ can no longer be assumed to be negligible [^9].

The terms \"insertion loss\" ($IL$) or \"echo reduction\" ($ER$) are
used. The insertion loss is the reduction (in decibels) of the acoustic
power of the insertion of a panel, related to the transmission
coefficient: $IL = -10\log|\hat{T}|^2$, and the echo reduction is the
reduction (in decibels) of the acoustic power after a reflection:
$ER = -10\log|\hat{R}|^2$.

### Three-layered media: No panel absorption

For three-layered medium with two infinite fluid layers, each with
impedance $r_{1/3}=\rho_{1/3}c_{1/3}$ on either side of a sample of
thickness $L$ and impedance impedance $r_{2}=\rho_{2}c_{2}$, and
$k=\omega/c_2$, at normal incidence the following equation for the
reflection coefficient $\hat{R}$ may be used [^10]:

For the symmetrical case of the first fluid and third fluid identical
$r_3=r_1$, equation () can be reduced
into the following simplified formulations [^11]:

Where $m=\frac{r_{1}}{r_2}$. By inspection of equations
(), resonances of minima of $\hat{R}$
and maxima of $\hat{T}$ occur at $kL = n\pi$. These lossless $IL$ and
$ER$ are plotted over $kL$ in the following figures in the next section
(for aluminum panels suspended in water) as the black lines.

### Three-layered media: With panel absorption

For panel media with acoustic attenuation, the following formulas in
() can be used to describe the insertion
loss and echo reduction, where $\alpha$ is the attenuation constant in
$dB/m$, and $r=\frac{\alpha}{\omega c_2}$[^12]. For the case of no
acoustic attenuation $\alpha=0$, equations
() are recovered.

A sketch of the effect of absorption in panel materials on $IL$ and $ER$
is shown in the following figures:

For a general formulation of n-layer solid panels with absorption,
methods are described in references [^13] [^14].

### Determination of *α(ω)* from experimental data

A 2nd order approximation for $\alpha$ is used (Equation
()) as it is not constant over frequency
[^15], as shown for the case of aluminum and nitrile rubber in the
figure below.

It is possible to experimentally determine $\alpha (\omega)$ from tank
acoustic tests. First the magnitude of the absorption coefficient is
determined from conservation of energy and measured values of
$|\hat{T}|$ and $|\hat{R}|$ [^16]:

The coefficient $a$ in equation () is
estimated using the following formula, then $b$ is then fit to data
[^17].

## Experimental Investigations

To characterize $|\hat{T}|$ and$|\hat{R}|$ of panel materials, free
field acoustic measurements are performed: In a water-filled tank, a
parametric array source $(a)$ produces an highly directional discrete
acoustic wave [^18] with the far-field directionality function
$D(\theta)$ [^19]. For demonstration, the far field directionality of an
underwater array source is included in the insertion loss sample
measurement figure below. The shape of the discrete wave is shown in
[^20].

### Insertion Loss

Using a hydrophone $(c)$, one recording is made with the sample $(b)$ in
place to record transmitted pressure $P_t$ and one measurement without
the sample in place to record incident pressure $P_i$. The measurement
configurations are shown in the following figures.

### Echo Reduction

For reflection experiments, the reflected pressure $P_r$ off the sample
is measured, and for the incident pressure $P_i$ measurement a foam
reflector is used. A foam reflector has a high acoustic impedance
mismatch with the water and reflects sound efficiently.

### Pressure Measurements

Shown in the following figures, the measured pressure signals over time
are are recorded and processed with a Fourier transform to determine
pressure over frequency. The coefficients $|\hat{R}|$ and $|\hat{T}|$
over frequency are simply the ratio of the resultant pressure spectra.

### Results

Insertion loss and echo reduction plots for, aluminum test samples, such
as those measured in [^21], are shown in the following figures. Aluminum
samples are conducted as it is a material with well known acoustic
properties, and negligibly low absorption over the frequencies studied.
The experimental setup described shows good agreement with theory [^22]
[^23].

### Other Considerations

For applications of anechoic tiling on submarines, the conditions
surrounding the submarine change drastically with submerged depth.
Varying pressure, salinity, temperature all affect acoustic properties
of the rubber, the surrounding water, and the anechoic tile in general.
Environmental tank such as the ones described in reference [^24] can be
used to simulate ocean conditions. The wavelength of the sound produced
by the parametric array is limited by the physical size of the tank.
Achieving lower frequency measurements necessitates the use of larger
tanks for experiments, or semi-anechoic siding on the tank walls [^25].

## References

Back to main page

------------------------------------------------------------------------

By: Geoffrey Chase

[^1]: James L Lastinger and Gerald A Sabin. Underwater sound absorbers:
    A review of published research with an annotated
    bibliography .
    Technical report, NAVAL RESEARCH LAB ORLANDO FL UNDERWATER SOUND
    REFERENCE DIV, 1970.

[^2]: N. Friedman and United States Naval Institute. The Naval
    Institute guide to world naval weapons
    systems. The Naval
    Institute Guide To\... Series. Naval Institute Press, 1989.

[^3]: Walter Kuhl. Sound absorption and sound absorbers in water
    (Dynamic properties of rubber and rubberlike substances in the
    acoustic frequency
    region). Dept. of
    the Navy, Bureau of Ships, 1950.

[^4]:

[^5]: V.F. Humphrey. The measurement of acoustic properties of limited
    size panels by use of a parametric
    source90403-1). Journal of
    Sound and Vibration, 98(1):67 -- 81, 1985.

[^6]: Christian Audoly. Determination of efficiency of anechoic or
    decoupling hull coatings using water tank acoustic
    measurements. In
    Societe Francaise d'Acoustique, editor, Acoustics 2012, Nantes,
    France, April 2012.

[^7]:

[^8]:

[^9]: E. Eugene Mikeska and John A. Behrens. Evaluation of transducer
    window materials. The Journal of
    the Acoustical Society of America, 59(6):1294--1298, 1976.

[^10]: Lawrence E Kinsler, Austin R Frey, Alan B Coppens, and James V
    Sanders. Fundamentals of
    Acoustics. 4th Edition, pp. 560.
    ISBN 0-471-84789-5. Wiley-VCH, December 1999., page 560, 1999.

[^11]: Robert J Bobber. Underwater electroacoustic
    measurements.
    Technical report, NAVAL RESEARCH LAB ORLANDO FL UNDERWATER SOUND
    REFERENCE DIV, 1970.

[^12]:

[^13]: A. K. Mal, C.-C. Yin, and Y. Bar-Cohen. The Influence of
    Material Dissipation and Imperfect Bonding on Acoustic Wave
    Reflection from Layered
    Solids,
    pages 927--934. Springer US, Boston, MA, 1988.

[^14]: Bernard Hosten and Michel Castaings. Transfer matrix of
    multilayered absorbing and anisotropic media. Measurements and
    simulations of ultrasonic wave propagation through composite
    materials. The Journal of the
    Acoustical Society of America, 94(3):1488--1495, 1993.

[^15]:

[^16]:

[^17]:

[^18]:

[^19]: P. D. Thorne. A broad-band acoustic source for underwater
    laboratory applications.
    IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency
    Control, 34(5):515--523, Sept 1987.

[^20]:

[^21]:

[^22]:

[^23]: Victor F Humphrey, Stephen P Robinson, John D Smith, Michael J
    Martin, Graham A Beamiss, Gary Hayman, and Nicholas L Carroll.
    Acoustic characterization of panel materials under simulated ocean
    conditions using a parametric array
    source. The Journal of the
    Acoustical Society of America, 124(2):803--814, 2008.

[^24]:

[^25]:


# Engineering Acoustics/Phonograph Sound Reproduction

## Phonograph Sound Reproduction

The content of this article is intended as an electro-mechanical
analysis of phonograph sound reproduction. For a general history and
overview of phonograph technology refer to Wikipedia entries on
Phonograph and Magnetic
Cartridges.

## A Simplified Phono Model for Phono (Magnetic) Cartridges

The basic principle of phonograph sound reproduction stems from a small
diameter diamond needle follows a groove cut into the surface of a
record. The resulting needle velocity is mechanically coupled to one
element of an electrical coil transducer to produce an electrical
current.

Two main variants of cartage design exist. Moving Magnet (MM) designs
couple a permanent magnet to the needle, causing the magnet to move near
an electrical coil solenoid. Moving Coil (MC) cartridges couple an
electrical coil to the needle, causing the coil to move in a fixed
permanent magnetic field. In both cartridge designs the relative motion
of the magnetic flux field induces current flow in the electrical coil.

Figure 1 demonstrates this process with a simplified MM cartridge
schematic. In this configuration the position of the magnet alters the
magnetic domains of the surrounding
ferromagnetic transducer
core. Similarly, the velocity of the magnet induces a change in the
magnetic flux of the
transducer core, and according to the principle of electromagnetic
induction, a
current in the electrical coil is produced.

\
!Figure 1: Top view of simplified (MM) phono cartridge
schematic. phono cartridge schematic."){width="600"}

## Electro-Mechanical Analogy of a Phono Cartridge

Figure 2 gives an electrical analogue model for the simplified MM
cartridge show in Figure 1. This circuit representation of the system
was obtained according to the Mobility Analogue for
Mechanical-Acoustical
Systems.
The following assumptions are included in this model:

-   Motion is limited to the horizontal plane.
-   Angular velocities are proportional to linear velocities according
    to the small angle assumption.
-   The stylus cantilever and tonearm are perfectly rigid acting only as
    mechanical transformers.
-   All compliant and damping elements are represented by ideal
    linearized elements.
-   The MM transducer element is represented by an ideal transformer
    with an aggregate coefficient *μBl*.

\
!Figure 2: Mechanical mobility circuit analogy for MM phonograph
system.{width="600"}

As an estimate of the phonograph system frequency response can be
obtained by calculating the complex input impedance, $Z_o$. An
analytical expression for $Z_o$ is more easily obtained by neglecting
the stylus mass *M~s~* and electrical system influence. These
assumptions are consistent with a low frequency approximaiton of the
system, shown in Figure 2. The resulting system input impedance is given
by the equation for *Z~o~*.

\
!Figure 3: Simplified low frequency circuit analogy for MM phonograph
system.{width="400"}\

```{=html}
<center>
```
$\begin{matrix}
\hat{Z}_o = &

            \left[\frac{
                  R_pL_t^2R_t + \omega^2R_p(M_c+L_t^2M_t)^2 + \frac{R_t}{w^2C_p^2}
                 }{
                 \left( R_p L_t^2 R_t + \frac{M_c+L_t^2M_t}{C_p} \right)^2 +
                 \left(\omega (M_c+L_t^2M_t) R_p - \frac{L_t^2 R_t}{\omega C_p} \right)^2
                 }\right]
            \\
            &
            \\
            &
            +j \left[\frac{
                 \omega(M_c+L_t^2M_t)\left(\frac{M_c+L_t^2M_t}{C_p} - R_p^2\right)
                 - \frac{1}{\omega C_p}\left(\frac{M_c+L_t^2M_t}{C_p} - L_t^4 R_t^2\right)
                 }{
                 \left( R_p L_t^2 R_t + \frac{M_c+L_t^2M_t}{C_p} \right)^2 +
                 \left(\omega (M_c+L_t^2M_t) R_p - \frac{L_t^2 R_t}{\omega C_p} \right)^2
                 }\right]
\end{matrix}$

```{=html}
</center>
```
## References

The technique of applying a lumped element system analysis was a
standard method used in the development of phonograph cartridges. In
addition to the low frequency analysis shown it is also possible to
conduct a simplified high frequency analysis for which the properties of
the stylus mass and vinyl surface compliance dominate the response.
Interestingly, poor performance in the low frequency extreme of a
phonograph cartridge response can have substantial and detrimental
effects on the high frequency response capability. For further reading
on this topic a list of relevant reference material is given below.

1.  Hunt, F. V. (1962). \"The Rational Design of Phonograph
    Pickups.\" J. Audio Eng. Soc 10(4): 274-289.
2.  Bauer, B. B. (1963). \"On the Damping of Phonograph Arms.\" J. Audio
    Eng. Soc 11(3): 207-211.
3.  Walton, J. (1963). \"Stylus Mass and Reproduction Distortion.\" J.
    Audio Eng. Soc 11(2): 104-109.
4.  Bauer, B. B. (1964). \"On the Damping of Phonograph Styli.\" J.
    Audio Eng. Soc 12(3): 210-213.
5.  Anderson, C. R. K., J. H.; Samson, R. S., (1965). Optimizing the
    Dynamic Characteristics of a Phonograph Pickup. Audio Engineering
    Society Convention 17.
6.  Anderson, C. R. K., James H.; Samson, Robert S., (1966).
    \"Optimizing the Dynamic Characteristics of a Phonograph
    Pickup.\" J. Audio Eng. Soc 14(2): 145-152.
7.  White, J. V. (1972). \"Mechanical Playback Losses and the Design of
    Wideband Phonograph Pickups.\" J. Audio Eng. Soc 20(4): 265-270.
8.  Nakai, G. T. (1973). \"Dynamic Damping of Stylus Compliance/Tone-Arm
    Resonance.\" J. Audio Eng. Soc 21(7): 555-562.
9.  Kates, J. M. (1976). \"Low-Frequency Tracking Behavior of Pickup
    Arm-Cartridge Systems.\" J. Audio Eng. Soc 24(4): 258-262.
10. Bauer, B. B. (1977). \"The High-Fidelity Phonograph Transducer.\" J.
    Audio Eng. Soc 25(10/11): 729-748.
11. Kogen, J. H. (1977). \"Record Changers, Turntables, and Tone Arms-A
    Brief Technical History.\" J. Audio Eng. Soc 25(10/11): 749-758.
12. Barlow Donald A.; Garside, G. R. (1978). \"Groove Deformation and
    Distortion in Recordings.\" J. Audio Eng. Soc 26(7/8): 498-510.
13. Lipshitz, S. P. (1978). \"Impulse Response of the Pickup
    Arm-Cartridge System.\" J. Audio Eng. Soc 26(1/2): 20-35.
14. Takahashi, S. T., Sadao; Kaneko, Nobuyuki; Fujimoto, Yasuhiro,
    (1979). \"The Optimum Pivot Position on a Tone Arm.\" J. Audio Eng.
    Soc 27(9): 648-656.
15. Happ, L. R. (1979). \"Dynamic Modeling and Analysis of a Phonograph
    Stylus.\" J. Audio Eng. Soc 27(1/2): 3-12.
16. Pardee, R. P. (1981). \"Determination of Sliding Friction Between
    Stylus and Record Groove.\" J. Audio Eng. Soc 29(12): 890-894.

## References

Back to Main page


# Engineering Acoustics/Bass Reflex Enclosure Design

## Introduction

```{=html}
<div style="float:right;margin:0 0 1em 1em;">
```
![](Bassreflex-Gehäuse_(enclosure).png "Bassreflex-Gehäuse_(enclosure).png")

```{=html}
</div>
```
Bass-reflex enclosures improve the low-frequency response of loudspeaker
systems. Bass-reflex enclosures are also called \"vented-box design\" or
\"ported-cabinet design\". A bass-reflex enclosure includes a vent or
port between the cabinet and the ambient environment. This type of
design, as one may observe by looking at contemporary loudspeaker
products, is still widely used today. Although the construction of
bass-reflex enclosures is fairly simple, their design is not simple, and
requires proper tuning. This reference focuses on the technical details
of bass-reflex design. General loudspeaker information can be found
here.

## Effects of the Port on the Enclosure Response

Before discussing the bass-reflex enclosure, it is important to be
familiar with the simpler sealed enclosure system performance. As the
name suggests, the sealed enclosure system attaches the loudspeaker to a
sealed enclosure (except for a small air leak included to equalize the
ambient pressure inside). Ideally, the enclosure would act as an
acoustical compliance element, as the air inside the enclosure is
compressed and rarified. Often, however, an acoustic material is added
inside the box to reduce standing waves, dissipate heat, and other
reasons. This adds a resistive element to the acoustical lumped-element
model. A non-ideal model of the effect of the enclosure actually adds an
acoustical mass element to complete a series lumped-element circuit
given in Figure 1. For more on sealed enclosure design, see the Sealed
Box Subwoofer
Design
page.

```{=html}
<center>
```
*Figure 1. Sealed enclosure acoustic circuit.*

```{=html}
</center>
```
In the case of a bass-reflex enclosure, a port is added to the
construction. Typically, the port is cylindrical and is flanged on the
end pointing outside the enclosure. In a bass-reflex enclosure, the
amount of acoustic material used is usually much less than in the sealed
enclosure case, often none at all. This allows air to flow freely
through the port. Instead, the larger losses come from the air leakage
in the enclosure. With this setup, a lumped-element acoustical circuit
has the following form.

```{=html}
<center>
```
![](Vented_box_ckt.gif "Vented_box_ckt.gif")\
*Figure 2. Bass-reflex enclosure acoustic circuit.*

```{=html}
</center>
```
In this figure, $Z_{RAD}$ represents the radiation impedance of the
outside environment on the loudspeaker diaphragm. The loading on the
rear of the diaphragm has changed when compared to the sealed enclosure
case. If one visualizes the movement of air within the enclosure, some
of the air is compressed and rarified by the compliance of the
enclosure, some leaks out of the enclosure, and some flows out of the
port. This explains the parallel combination of $M_{AP}$, $C_{AB}$, and
$R_{AL}$. A truly realistic model would incorporate a radiation
impedance of the port in series with $M_{AP}$, but for now it is
ignored. Finally, $M_{AB}$, the acoustical mass of the enclosure, is
included as discussed in the sealed enclosure case. The formulas which
calculate the enclosure parameters are listed in Appendix
B.

It is important to note the parallel combination of $M_{AP}$ and
$C_{AB}$. This forms a Helmholtz resonator (click here for more
information).
Physically, the port functions as the "neck" of the resonator and the
enclosure functions as the "cavity." In this case, the resonator is
driven from the piston directly on the cavity instead of the typical
Helmholtz case where it is driven at the "neck." However, the same
resonant behavior still occurs at the enclosure resonance frequency,
$f_{B}$. At this frequency, the impedance seen by the loudspeaker
diaphragm is large (see Figure 3 below). Thus, the load on the
loudspeaker reduces the velocity flowing through its mechanical
parameters, causing an anti-resonance condition where the displacement
of the diaphragm is a minimum. Instead, the majority of the volume
velocity is actually emitted by the port itself instead of the
loudspeaker. When this impedance is reflected to the electrical circuit,
it is proportional to $1/Z$, thus a minimum in the impedance seen by the
voice coil is small. Figure 3 shows a plot of the impedance seen at the
terminals of the loudspeaker. In this example, $f_B$ was found to be
about 40 Hz, which corresponds to the null in the voice-coil impedance.

```{=html}
<center>
```
![](_Za0_Zvc_plots.gif "_Za0_Zvc_plots.gif")\
*Figure 3. Impedances seen by the loudspeaker diaphragm and voice coil.*

```{=html}
</center>
```
## Quantitative Analysis of Port on Enclosure

The performance of the loudspeaker is first measured by its velocity
response, which can be found directly from the equivalent circuit of the
system. As the goal of most loudspeaker designs is to improve the bass
response (leaving high-frequency production to a tweeter), low frequency
approximations will be made as much as possible to simplify the
analysis. First, the inductance of the voice coil, $\it{L_E}$, can be
ignored as long as $\omega \ll R_E/L_E$. In a typical loudspeaker,
$\it{L_E}$ is of the order of 1 mH, while $\it{R_E}$ is typically
8$\Omega$, thus an upper frequency limit is approximately 1 kHz for this
approximation, which is certainly high enough for the frequency range of
interest.

Another approximation involves the radiation impedance, $\it{Z_{RAD}}$.
It can be shown \[1\] that this value is given by the following equation
(in acoustical ohms):

```{=html}
<center>
```
$Z_{RAD} = \frac{\rho_0c}{\pi a^2}\left[\left(1 - \frac{J_1(2ka)}{ka}\right) + j\frac{H_1(2ka)}{ka}\right]$

```{=html}
</center>
```
Where $J_1(x)$ and $H_1(x)$ are types of Bessel functions. For small
values of *ka*,

```{=html}
<table align=center width=50% cellpadding=10>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$J_1(2ka) \approx ka$

```{=html}
</td>
```
```{=html}
<td>
```
and

```{=html}
</td>
```
```{=html}
<td>
```
$H_1(2ka) \approx \frac{8(ka)^2}{3\pi}$

```{=html}
</td>
```
```{=html}
<td>
```
$\Rightarrow Z_{RAD} \approx j\frac{8\rho_0\omega}{3\pi^2a} = jM_{A1}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
Hence, the low-frequency impedance on the loudspeaker is represented
with an acoustic mass $M_{A1}$ \[1\]. For a simple analysis, $R_E$,
$M_{MD}$, $C_{MS}$, and $R_{MS}$ (the transducer parameters, or
*Thiele-Small* parameters) are converted to their acoustical
equivalents. All conversions for all parameters are given in Appendix
A.
Then, the series masses, $M_{AD}$, $M_{A1}$, and $M_{AB}$, are lumped
together to create $M_{AC}$. This new circuit is shown below.

```{=html}
<center>
```
![](VB_LF_ckt.gif "VB_LF_ckt.gif")\
*Figure 4. Low-Frequency Equivalent Acoustic Circuit*

```{=html}
</center>
```
Unlike sealed enclosure analysis, there are multiple sources of volume
velocity that radiate to the outside environment. Hence, the diaphragm
volume velocity, $U_D$, is not analyzed but rather
$U_0 = U_D + U_P + U_L$. This essentially draws a "bubble" around the
enclosure and treats the system as a source with volume velocity $U_0$.
This "lumped" approach will only be valid for low frequencies, but
previous approximations have already limited the analysis to such
frequencies anyway. It can be seen from the circuit that the volume
velocity flowing *into* the enclosure, $U_B = -U_0$, compresses the air
inside the enclosure. Thus, the circuit model of Figure 3 is valid and
the relationship relating input voltage, $V_{IN}$ to $U_0$ may be
computed.

In order to make the equations easier to understand, several parameters
are combined to form other parameter names. First, $\omega_B$ and
$\omega_S$, the enclosure and loudspeaker resonance frequencies,
respectively, are:

```{=html}
<table align=center width=40%>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$\omega_B = \frac{1}{\sqrt{M_{AP}C_{AB}}}$

```{=html}
</td>
```
```{=html}
<td>
```
$\omega_S = \frac{1}{\sqrt{M_{AC}C_{AS}}}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
Based on the nature of the derivation, it is convenient to define the
parameters $\omega_0$ and *h*, the Helmholtz tuning ratio:

```{=html}
<table align=center width=25%>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$\omega_0 = \sqrt{\omega_B\omega_S}$

```{=html}
</td>
```
```{=html}
<td>
```
$h = \frac{\omega_B}{\omega_S}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
A parameter known as the *compliance ratio* or *volume ratio*, $\alpha$,
is given by:

```{=html}
<center>
```
$\alpha = \frac{C_{AS}}{C_{AB}} = \frac{V_{AS}}{V_{AB}}$

```{=html}
</center>
```
Other parameters are combined to form what are known as *quality
factors*:

```{=html}
<table align=center width=45%>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$Q_L = R_{AL}\sqrt{\frac{C_{AB}}{M_{AP}}}$

```{=html}
</td>
```
```{=html}
<td>
```
$Q_{TS} = \frac{1}{R_{AE}+R_{AS}}\sqrt{\frac{M_{AC}}{C_{AS}}}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
This notation allows for a simpler expression for the resulting transfer
function \[1\]:

```{=html}
<center>
```
$\frac{U_0}{V_{IN}} = G(s) = \frac{(s^3/\omega_0^4)}{(s/\omega_0)^4+a_3(s/\omega_0)^3+a_2(s/\omega_0)^2+a_1(s/\omega_0)+1}$

```{=html}
</center>
```
where

```{=html}
<table align=center width=70%>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$a_1 = \frac{1}{Q_L\sqrt{h}}+\frac{\sqrt{h}}{Q_{TS}}$

```{=html}
</td>
```
```{=html}
<td>
```
$a_2 = \frac{\alpha+1}{h}+h+\frac{1}{Q_L Q_{TS}}$

```{=html}
</td>
```
```{=html}
<td>
```
$a_3 = \frac{1}{Q_{TS}\sqrt{h}}+\frac{\sqrt{h}}{Q_L}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
## Development of Low-Frequency Pressure Response

It can be shown \[2\] that for $ka < 1/2$, a loudspeaker behaves as a
spherical source. Here, *a* represents the radius of the loudspeaker.
For a 15" diameter loudspeaker in air, this low frequency limit is about
150 Hz. For smaller loudspeakers, this limit increases. This limit
dominates the limit which ignores $L_E$, and is consistent with the
limit that models $Z_{RAD}$ by $M_{A1}$.

Within this limit, the loudspeaker emits a volume velocity $U_0$, as
determined in the previous section. For a simple spherical source with
volume velocity $U_0$, the far-field pressure is given by \[1\]:

```{=html}
<center>
```
$p(r) \simeq j\omega\rho_0 U_0 \frac{e^{-jkr}}{4\pi r}$

```{=html}
</center>
```
It is possible to simply let $r = 1$ for this analysis without loss of
generality because distance is only a function of the surroundings, not
the loudspeaker. Also, because the transfer function magnitude is of
primary interest, the exponential term, which has a unity magnitude, is
omitted. Hence, the pressure response of the system is given by \[1\]:

```{=html}
<center>
```
$\frac{p}{V_{IN}} = \frac{\rho_0s}{4\pi}\frac{U_0}{V_{IN}} = \frac{\rho_0Bl}{4\pi S_DR_EM_AS}H(s)$

```{=html}
</center>
```
Where $H(s) = sG(s)$. In the following sections, design methods will
focus on $|H(s)|^2$ rather than $H(s)$, which is given by:

```{=html}
<table align=center cellpadding=15>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$|H(s)|^2 = \frac{\Omega^8}{\Omega^8 + \left(a^2_3 - 2a_2\right)\Omega^6 + \left(a^2_2 + 2 - 2a_1a_3\right)\Omega^4 + \left(a^2_1 - 2a_2\right)\Omega^2 + 1}$

```{=html}
</td>
```
```{=html}
<td>
```
$\Omega = \frac{\omega}{\omega_0}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
This also implicitly ignores the constants in front of $|H(s)|$ since
they simply scale the response and do not affect the shape of the
frequency response curve.

## Alignments

A popular way to determine the ideal parameters has been through the use
of alignments. The concept of alignments is based upon well investigated
electrical filter theory. Filter development is a method of selecting
the poles (and possibly zeros) of a transfer function to meet a
particular design criterion. The criteria are the desired properties of
a magnitude-squared transfer function, which in this case is $|H(s)|^2$.
From any of the design criteria, the poles (and possibly zeros) of
$|H(s)|^2$ are found, which can then be used to calculate the numerator
and denominator. This is the "optimal" transfer function, which has
coefficients that are matched to the parameters of $|H(s)|^2$ to compute
the appropriate values that will yield a design that meets the criteria.

There are many different types of filter designs, each which have
trade-offs associated with them. However, this design approach is
limited because of the structure of $|H(s)|^2$. In particular, it has
the structure of a fourth-order high-pass filter with all zeros at *s* =
0. Therefore, only those filter design methods which produce a low-pass
filter with only poles will be acceptable methods to use. From the
traditional set of algorithms, only Butterworth and Chebyshev low-pass
filters have only poles. In addition, another type of filter called a
quasi-Butterworth filter can also be used, which has similar properties
to a Butterworth filter. These three algorithms are fairly simple, thus
they are the most popular. When these low-pass filters are converted to
high-pass filters, the $s \rightarrow 1/s$ transformation produces $s^8$
in the numerator.

More details regarding filter theory and these relationships can be
found in numerous resources, including \[5\].

## Butterworth Alignment

The Butterworth algorithm is designed to have a *maximally flat* pass
band. Since the slope of a function corresponds to its derivatives, a
flat function will have derivatives equal to zero. Since as flat of a
pass band as possible is optimal, the ideal function will have as many
derivatives equal to zero as possible at *s* = 0. Of course, if all
derivatives were equal to zero, then the function would be a constant,
which performs no filtering.

Often, it is better to examine what is called the *loss function*. Loss
is the reciprocal of gain, thus

```{=html}
<center>
```
$|\hat{H}(s)|^2 = \frac{1}{|H(s)|^2}$

```{=html}
</center>
```
The loss function can be used to achieve the desired properties, then
the desired gain function is recovered from the loss function.

Now, applying the desired Butterworth property of maximal pass-band
flatness, the loss function is simply a polynomial with derivatives
equal to zero at *s* = 0. At the same time, the original polynomial must
be of degree eight (yielding a fourth-order function). However,
derivatives one through seven can be equal to zero if \[3\]

```{=html}
<center>
```
$|\hat{H}(\Omega)|^2 = 1 + \Omega^8 \Rightarrow |H(\Omega)|^2 = \frac{1}{1 + \Omega^8}$

```{=html}
</center>
```
With the high-pass transformation $\Omega \rightarrow 1/\Omega$,

```{=html}
<center>
```
$|H(\Omega)|^2 = \frac{\Omega^8}{\Omega^8 + 1}$

```{=html}
</center>
```
It is convenient to define $\Omega = \omega/\omega_{3dB}$, since
$\Omega = 1 \Rightarrow |H(s)|^2 = 0.5$ or -3 dB. This definition allows
the matching of coefficients for the $|H(s)|^2$ describing the
loudspeaker response when $\omega_{3dB} = \omega_0$. From this matching,
the following design equations are obtained \[1\]:

```{=html}
<table align=center cellspacing=20>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$a_1 = a_3 = \sqrt{4+2\sqrt{2}}$

```{=html}
</td>
```
```{=html}
<td>
```
$a_2 = 2+\sqrt{2}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
## Quasi-Butterworth Alignment

The quasi-Butterworth alignments do not have as well-defined of an
algorithm when compared to the Butterworth alignment. The name
"quasi-Butterworth" comes from the fact that the transfer functions for
these responses appear similar to the Butterworth ones, with (in
general) the addition of terms in the denominator. This will be
illustrated below. While there are many types of quasi-Butterworth
alignments, the simplest and most popular is the 3rd order alignment
(QB3). The comparison of the QB3 magnitude-squared response against the
4th order Butterworth is shown below.

```{=html}
<table align=center cellpadding=15>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$\left|H_{QB3}(\omega)\right|^2 = \frac{(\omega/\omega_{3dB})^8}{(\omega/\omega_{3dB})^8 + B^2(\omega/\omega_{3dB})^2 + 1}$

```{=html}
</td>
```
```{=html}
<td>
```
$\left|H_{B4}(\omega)\right|^2 = \frac{(\omega/\omega_{3dB})^8}{(\omega/\omega_{3dB})^8 + 1}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
Notice that the case $B = 0$ is the Butterworth alignment. The reason
that this QB alignment is called 3rd order is due to the fact that as
*B* increases, the slope approaches 3 dec/dec instead of 4 dec/dec, as
in 4th order Butterworth. This phenomenon can be seen in Figure 5.

```{=html}
<center>
```
![](QB3_gradient.GIF "QB3_gradient.GIF")\
*Figure 5: 3rd-Order Quasi-Butterworth Response for*$0.1 \leq B \leq 3$

```{=html}
</center>
```
Equating the system response $|H(s)|^2$ with $|H_{QB3}(s)|^2$, the
equations guiding the design can be found \[1\]:

```{=html}
<table align=center cellpadding=15>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$B^2 = a^2_1 - 2a_2$

```{=html}
</td>
```
```{=html}
<td>
```
$a_2^2 + 2 = 2a_1a_3$

```{=html}
</td>
```
```{=html}
<td>
```
$a_3 = \sqrt{2a_2}$

```{=html}
</td>
```
```{=html}
<td>
```
$a_2 > 2 + \sqrt{2}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
## Chebyshev Alignment

The Chebyshev algorithm is an alternative to the Butterworth algorithm.
For the Chebyshev response, the maximally-flat passband restriction is
abandoned. Now, a *ripple*, or fluctuation is allowed in the pass band.
This allows a steeper transition or roll-off to occur. In this type of
application, the low-frequency response of the loudspeaker can be
extended beyond what can be achieved by Butterworth-type filters. An
example plot of a Chebyshev high-pass response with 0.5 dB of ripple
against a Butterworth high-pass response for the same $\omega_{3dB}$ is
shown below.

```{=html}
<center>
```
![](Butt_vs_Cheb_HP.gif "Butt_vs_Cheb_HP.gif")\
*Figure 6: Chebyshev vs. Butterworth High-Pass Response.*

```{=html}
</center>
```
The Chebyshev response is defined by \[4\]:

```{=html}
<center>
```
$|\hat{H}(j\Omega)|^2 = 1 + \epsilon^2C^2_n(\Omega)$\

```{=html}
</center>
```
$C_n(\Omega)$ is called the *Chebyshev polynomial* and is defined by
\[4\]:

```{=html}
<table align=center>
```
```{=html}
<tr>
```
```{=html}
<td valign=center rowspan=2>
```
$C_n(\Omega) = \big\lbrace$

```{=html}
</td>
```
```{=html}
<td>
```
$\rm{cos}[\it{n}\rm{cos}^{-1}(\Omega)]$

```{=html}
</td>
```
```{=html}
<td>
```
$|\Omega| < 1$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$\rm{cosh}[\it{n}\rm{cosh}^{-1}(\Omega)]$

```{=html}
</td>
```
```{=html}
<td>
```
$|\Omega| > 1$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
Fortunately, Chebyshev polynomials satisfy a simple recursion formula
\[4\]:

```{=html}
<table align=center cellpadding=15>
```
```{=html}
<td>
```
$C_0(x) = 1$

```{=html}
</td>
```
```{=html}
<td>
```
$C_1(x) = x$

```{=html}
</td>
```
```{=html}
<td>
```
$C_n(x) = 2xC_{n-1} - C_{n-2}$

```{=html}
</td>
```
```{=html}
</table>
```
For more information on Chebyshev polynomials, see the Wolfram
Mathworld: Chebyshev
Polynomials
page.

When applying the high-pass transformation to the 4th order form of
$|\hat{H}(j\Omega)|^2$, the desired response has the form \[1\]:

```{=html}
<center>
```
$|H(j\Omega)|^2 = \frac{1+\epsilon^2}{1+\epsilon^2C^2_4(1/\Omega)}$

```{=html}
</center>
```
The parameter $\epsilon$ determines the ripple. In particular, the
magnitude of the ripple is $10\rm{log}[1+\epsilon^2]$ dB and can be
chosen by the designer, similar to *B* in the quasi-Butterworth case.
Using the recursion formula for $C_n(x)$,

```{=html}
<center>
```
$C_4\left(\frac{1}{\Omega}\right) = 8\left(\frac{1}{\Omega}\right)^4 - 8\left(\frac{1}{\Omega}\right)^2 + 1$\

```{=html}
</center>
```
Applying this equation to $|H(j\Omega)|^2$ \[1\],

```{=html}
<table align=center cellpadding=15>
```
```{=html}
<tr>
```
```{=html}
<td colspan=2>
```
$\Rightarrow |H(\Omega)|^2 = \frac{\frac{1 + \epsilon^2}{64\epsilon^2}\Omega^8}{\frac{1 + \epsilon^2}{64\epsilon^2}\Omega^8 + \frac{1}{4}\Omega^6 + \frac{5}{4}\Omega^4 - 2\Omega^2 + 1}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$\Omega = \frac{\omega}{\omega_n}$

```{=html}
</td>
```
```{=html}
<td>
```
$\omega_n = \frac{\omega_{3dB}}{2}\sqrt{2 + \sqrt{2 + 2\sqrt{2+\frac{1}{\epsilon^2}}}}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
Thus, the design equations become \[1\]:

```{=html}
<table align=center cellpadding=15>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$\omega_0 = \omega_n\sqrt[8]{\frac{64\epsilon^2}{1+\epsilon^2}}$

```{=html}
</td>
```
```{=html}
<td>
```
$k = \rm{tanh}\left[\frac{1}{4}\rm{sinh}^{-1}\left(\frac{1}{\epsilon}\right)\right]$

```{=html}
<td>
```
$D = \frac{k^4 + 6k^2 + 1}{8}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$a_1 = \frac{k\sqrt{4 + 2\sqrt{2}}}{\sqrt[4]{D}},$

```{=html}
</td>
```
```{=html}
<td>
```
$a_2 = \frac{1 + k^2(1+\sqrt{2})}{\sqrt{D}}$

```{=html}
</td>
```
```{=html}
<td>
```
$a_3 = \frac{a_1}{\sqrt{D}}\left[1 - \frac{1 - k^2}{2\sqrt{2}}\right]$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
## Choosing the Correct Alignment

With all the equations that have already been presented, the question
naturally arises, "Which one should I choose?" Notice that the
coefficients $a_1$, $a_2$, and $a_3$ are not simply related to the
parameters of the system response. Certain combinations of parameters
may indeed invalidate one or more of the alignments because they cannot
realize the necessary coefficients. With this in mind, general
guidelines have been developed to guide the selection of the appropriate
alignment. This is very useful if one is designing an enclosure to suit
a particular transducer that cannot be changed.

The general guideline for the Butterworth alignment focuses on $Q_L$ and
$Q_{TS}$. Since the three coefficients $a_1$, $a_2$, and $a_3$ are a
function of $Q_L$, $Q_{TS}$, *h*, and $\alpha$, fixing one of these
parameters yields three equations that uniquely determine the other
three. In the case where a particular transducer is already given,
$Q_{TS}$ is essentially fixed. If the desired parameters of the
enclosure are already known, then $Q_L$ is a better starting point.

In the case that the rigid requirements of the Butterworth alignment
cannot be satisfied, the quasi-Butterworth alignment is often applied
when $Q_{TS}$ is not large enough.. The addition of another parameter,
*B*, allows more flexibility in the design.

For $Q_{TS}$ values that are too large for the Butterworth alignment,
the Chebyshev alignment is typically chosen. However, the steep
transition of the Chebyshev alignment may also be utilized to attempt to
extend the bass response of the loudspeaker in the case where the
transducer properties can be changed.

In addition to these three popular alignments, research continues in the
area of developing new algorithms that can manipulate the low-frequency
response of the bass-reflex enclosure. For example, a 5th order
quasi-Butterworth alignment has been developed \[6\]; its advantages
include improved low frequency extension, and much reduced driver
excursion at low frequencies and typically bi-amping or tri-amping,
while its disadvatages include somewhat difficult mathematics and
electronic complication (electronic crossovers are typically required).
Another example \[7\] applies root-locus techniques to achieve results.
In the modern age of high-powered computing, other researchers have
focused their efforts in creating computerized optimization algorithms
that can be modified to achieve a flatter response with sharp roll-off
or introduce quasi-ripples which provide a boost in sub-bass frequencies
\[8\].

Back to Engineering Acoustics

## References

\[1\] Leach, W. Marshall, Jr. *Introduction to Electroacoustics and
Audio Amplifier Design*. 2nd ed. Kendall/Hunt, Dubuque, IA. 2001.

\[2\] Beranek, L. L. *Acoustics*. 2nd ed. Acoustical Society of America,
Woodbridge, NY. 1993.

\[3\] DeCarlo, Raymond A. "The Butterworth Approximation." Notes from
ECE 445. Purdue University. 2004.

\[4\] DeCarlo, Raymond A. "The Chebyshev Approximation." Notes from ECE
445. Purdue University. 2004.

\[5\] VanValkenburg, M. E. *Analog Filter Design*. Holt, Rinehart and
Winston, Inc. Chicago, IL. 1982.

\[6\] Kreutz, Joseph and Panzer, Joerg. \"Derivation of the
Quasi-Butterworth 5 Alignments.\" *Journal of the Audio Engineering
Society*. Vol. 42, No. 5, May 1994.

\[7\] Rutt, Thomas E. \"Root-Locus Technique for Vented-Box Loudspeaker
Design.\" *Journal of the Audio Engineering Society*. Vol. 33, No. 9,
September 1985.

\[8\] Simeonov, Lubomir B. and Shopova-Simeonova, Elena.
\"Passive-Radiator Loudspeaker System Design Software Including
Optimization Algorithm.\" *Journal of the Audio Engineering Society*.
Vol. 47, No. 4, April 1999.

## Appendix A: Equivalent Circuit Parameters

```{=html}
<table align=center border=2>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Name

```{=html}
</th>
```
```{=html}
<th>
```
Electrical Equivalent

```{=html}
</th>
```
```{=html}
<th>
```
Mechanical Equivalent

```{=html}
</th>
```
```{=html}
<th>
```
Acoustical Equivalent

```{=html}
</th>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Voice-Coil Resistance

```{=html}
</th>
```
```{=html}
<td>
```
$R_E$

```{=html}
</td>
```
```{=html}
<td>
```
$R_{ME} = \frac{(Bl)^2}{R_E}$

```{=html}
</td>
```
```{=html}
<td>
```
$R_{AE} = \frac{(Bl)^2}{R_ES^2_D}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Driver (Speaker) Mass

```{=html}
</th>
```
```{=html}
<td>
```
See $C_{MEC}$

```{=html}
</td>
```
```{=html}
<td>
```
$M_{MD}$

```{=html}
</td>
```
```{=html}
<td>
```
$M_{AD} = \frac{M_{MD}}{S^2_D}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Driver (Speaker) Suspension Compliance

```{=html}
</th>
```
```{=html}
<td>
```
$L_{CES} = (Bl)^2C_{MS}$

```{=html}
</td>
```
```{=html}
<td>
```
$C_{MS}$

```{=html}
</td>
```
```{=html}
<td>
```
$C_{AS} = S^2_DC_{MS}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Driver (Speaker) Suspension Resistance

```{=html}
</th>
```
```{=html}
<td>
```
$R_{ES} = \frac{(Bl)^2}{R_{MS}}$

```{=html}
</td>
```
```{=html}
<td>
```
$R_{MS}$

```{=html}
</td>
```
```{=html}
<td>
```
$R_{AS} = \frac{R_{MS}}{S^2_D}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Enclosure Compliance

```{=html}
</th>
```
```{=html}
<td>
```
$L_{CEB} = \frac{(Bl)^2C_{AB}}{S^2_D}$

```{=html}
</td>
```
```{=html}
<td>
```
$C_{MB} = \frac{C_{AB}}{S^2_D}$

```{=html}
</td>
```
```{=html}
<td>
```
$C_{AB}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Enclosure Air-Leak Losses

```{=html}
</th>
```
```{=html}
<td>
```
$R_{EL} = \frac{(Bl)^2}{S^2_DR_{AL}}$

```{=html}
</td>
```
```{=html}
<td>
```
$R_{ML} = S^2_DR_{AL}$

```{=html}
</td>
```
```{=html}
<td>
```
$R_{AL}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Acoustic Mass of Port

```{=html}
</th>
```
```{=html}
<td>
```
$C_{MEP} = \frac{S^2_DM_{AP}}{(Bl)^2}$

```{=html}
</td>
```
```{=html}
<td>
```
$M_{MP} = S^2_DM_{AP}$

```{=html}
</td>
```
```{=html}
<td>
```
$M_{AP}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Enclosure Mass Load

```{=html}
</th>
```
```{=html}
<td>
```
See $C_{MEC}$

```{=html}
</td>
```
```{=html}
<td>
```
See $M_{MC}$

```{=html}
</td>
```
```{=html}
<td>
```
$M_{AB}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Low-Frequency Radiation Mass Load

```{=html}
</th>
```
```{=html}
<td>
```
See $C_{MEC}$

```{=html}
</td>
```
```{=html}
<td>
```
See $M_{MC}$

```{=html}
</td>
```
```{=html}
<td>
```
$M_{A1}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Combination Mass Load

```{=html}
</th>
```
```{=html}
<td>
```
$C_{MEC} = \frac{S^2_DM_{AC}}{(Bl)^2}$\
$= \frac{S^2_D(M_{AB} + M_{A1}) + M_{MD}}{(Bl)^2}$

```{=html}
</td>
```
```{=html}
<td>
```
$M_{MC} = S^2_D(M_{AB} + M_{A1}) + M_{MD}$

```{=html}
</td>
```
```{=html}
<td>
```
$M_{AC} = M_{AD} + M_{AB} + M_{A1}$\
$= \frac{M_{MD}}{S^2_D} + M_{AB} + M_{A1}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
## Appendix B: Enclosure Parameter Formulas

```{=html}
<center>
```
![](_Vented_enclosure.gif "_Vented_enclosure.gif")\
*Figure 7: Important dimensions of bass-reflex enclosure.*

```{=html}
</center>
```
Based on these dimensions \[1\],

```{=html}
<table align=center cellpadding=5>
```
```{=html}
<tr align=center>
```
```{=html}
<td>
```
$C_{AB} = \frac{V_{AB}}{\rho_0c^2_0}$

```{=html}
</td>
```
```{=html}
<td>
```
$M_{AB} = \frac{B\rho_{eff}}{\pi a}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td>
```
$B = \frac{d}{3}\left(\frac{S_D}{S_B}\right)^2\sqrt{\frac{\pi}{S_D}} + \frac{8}{3\pi}\left[1 - \frac{S_D}{S_B}\right]$

```{=html}
</td>
```
```{=html}
<td>
```
$\rho_0 \leq \rho_{eff} \leq \rho_0\left(1 - \frac{V_{fill}}{V_B}\right) + \rho_{fill}\frac{V_{fill}}{V_B}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td colspan=2>
```
$V_{AB} = V_B\left[1-\frac{V_{fill}}{V_B}\right]\left[1 + \frac{\gamma - 1}{1 + \gamma\left(\frac{V_B}{V_{fill}} - 1\right)\frac{\rho_0c_{air}}{\rho_{fill}c_{fill}}}\right]$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td>
```
$V_B= hwd$ (inside enclosure gross volume)

```{=html}
</td>
```
```{=html}
<td>
```
$S_B = wh$ (baffle area of the side the speaker is mounted on)

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td>
```
$c_{air} =$specific heat of air at constant isovolumetric process (about
$0.718 \frac{\rm kJ}{\rm kg.K}$ at 300 K)

```{=html}
</td>
```
```{=html}
<td>
```
$c_{fill} =$specific heat of filling at constant volume ($V_{filling}$)

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td>
```
$\rho_0 =$mean density of air (about $1.3 \frac{\rm kg}{\rm m^3}$ at 300
K)

```{=html}
<td>
```
$\rho_{fill} =$ density of filling

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td>
```
$\gamma =$ ratio of specific heats (Isobaric/Isovolumetric processes)
for air (about 1.4 at 300 K)

```{=html}
</td>
```
```{=html}
<td>
```
$c_0 =$ speed of sound in air (about 344 m/s)

```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td colspan=2>
```
$\rho_{eff}$ = effective density of enclosure. If little or no filling
(acceptable assumption in a bass-reflex system but not for sealed
enclosures), $\rho_{eff} \approx \rho_0$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```


# Engineering Acoustics/source-filter theory

The source-filter theory (Fant 1960) hypothesizes that an acoustic
speech signal can be seen as a source signal, filtered with the
resonances in the cavities of the vocal tract downstream from the
glottis or the constriction. This simple model for speech synthesis is
based on an assumption that the dynamics of the system is linear and
separable into three main blocks: a glottal energy (soure), the vocal
tract (filter) and the effect of modeling radiation sound that are
independent(As shown in the figure on the right).

The glottal source roughly matches the subglottal systems, while vocal
tract (VT) corresponds to the supra-glottal system. The radiation block
can be considered as a converter, which converts volume velocity in to
acoustic pressure.In general, the radiation characteristic R(f) and the
spectrum envelop of the source function S(f) for the glottal source are
smooth and monotonic functions of frequency. The transfer function T(f),
however, is usually characterized by several peaks corresponding to
resonances of the acoustic cavities that form the vocal tract.
Manipulating the shape of these cavities results in the positions and
amplitudes of the peaks. Figure on the left qualitatively shows the
configuration of vocal tract corresponding to a vowel. The forms of the
source spectrum S(f), the transfer function T(f), the radiation
characteristic R(f), and the sound pressure pr(f) is shown in each case.

The transfer function T(f) is determined by applying the theory of sound
propagation in tubes of arbitrary shape. For frequencies up to 5000 Hz,
the cross dimensions of vocal tract are less than a wavelength of the
sound. Therefore, the sound propagation can be considered as plane waves
parallel to the axis of the tube and the vocal tract can be viewed as an
acoustic tube of varying diameter.

## vocal tract transfer function

The vocal tract is approximated as an acoustic tube of a given length
composed of a number of sections with different section areas. This is
equivalent to the modelling of the sampled vocal tract transfer function
(H(s)) as a superposition of a given number of spectral poles and zeros,
which in spectral domain can be represented by

$H(s)= k\begin{matrix} \prod_{i=1}^N \frac{s-s_{ai}}{s-s_i}\end{matrix}$

Where K is constant, s~a1~,s~a2~,..are the zeros of H(s), s~1~,s~2~,\...
are the poles. for this equation, the poles and zeros mostly occur in
complex conjugate pairs, and the real parts of these complex frequencies
are much less than the imaginary parts which means that the peak energy
lost in one cycle is much less than the energy stored in one cycle of
the oscillation. Therfeore, the poles of H(s) can be expressed as below:

$\frac{1}{k_n}\begin{matrix} \prod_{i=1}^N \frac{s_i{s_i}^*}{(s-s_i)(s-{s_i}^*)}\end{matrix}$

Where Kp is a constant and the stars indicate complex conjugates.
Natural frequencies of vocal tract are represented by poles and the
imaginary parts indicate formant frequencies which refers to frequencies
at which oscillations happen in the absence of excitation and the real
parts give the rates of decay of theses oscillations. In the other
words, depending on the shape of the acoustic tube (mainly influenced by
tongue position), a sound wave travelling through it will be reflected
in a certain way so that interferences will generate resonances at
certain frequencies. These resonances are called
formants . Their location
largely determines the speech sound that is heard.

## Acoustic interpretation of transfer function

!Vocal tract as tubes with varying cross
section\|300px

According to acoustics of tubes, Pressure and volume velocity of a tube
at the end of tube (x=L) can be related to the variables at beginning of
the tube (x=0). The following transfer matrix expresses the acoustical
relationship between two sides of a tube in frequency domain:

$$\begin{bmatrix}
P_0 \\
U_0\\
\end{bmatrix}
=T(\omega)* \begin{bmatrix}
P_L \\
U_L \\
\end{bmatrix},   T(\omega)=
\begin{bmatrix}
 a_{11} & a_{22} \\
 a_{21} & a_{22}\\
 \end{bmatrix}$$

$$a_{11}=cos(KL), a_{21}=jsin(KL)/(\rho*c), a_{21}=\rho*c*jsin(KL), a_{22}=cos(KL)$$

where K is the wave number, and L is the length of the tube.The aboved
mentioned relation can be used to calculate the state of the wave field
at one location, having on hand te state of the field at another
location.

Since the vocal tract can be considered as n tubes with different cross
section (see figure on the right), the transer function can be used to
relate the states between glottis and the radiate sound:

$$T(\omega)=T_1(\omega)*T_2(\omega)*T_3(\omega)*....*T_n(\omega)$$

and the overall equation for vocal tract becomes:

$$\begin{bmatrix}
P_g \\
U_g\\
\end{bmatrix}
=T(\omega)* \begin{bmatrix}
P_r \\
U_r \\
\end{bmatrix}, where P_r=U_r*Z_{rad}$$

In this equation, Z~rad~ is the radiation impedance. Up to a frequency
of about 6000 Hz, the acoustic radiation impedance can be written
approximately as:

$$Z_{rad}=\rho*c/A_m*(\pi*f^2/c^2*A_m)*K_s(f)+j*2*\pi*f*\rho*(0.8a)/A_m$$

where A is the area of the mouth opening, a is the effective radius, and
K~s~(f)is adimensionless frequency dependant factor that accounts for
baffling effect of the head.

The Transfer function of the system can be calculated as fallows:

$$H(\omega)=U_r/U_g$$

Therefore the equations result in:

$$P_r(\omega)=U_g(\omega)*H(\omega)*Z_{rad}(\omega)$$

As can be seen, the equation above expresses the pressure in front of
the mouse with a source, filter and radiation characteristics of the
mouse. This equation describes the source filter theory mentioned in the
first section.

## Effects of vocal tract wall and other losses

In previous section vocal tract is modeled as a system without losses,
except the termination impedance term. However, There are some other
second order effects that are necessary for precise modelling, such as
wall effects, heat conduction and viscosity, glottal opening. These
loses can change band-widths of the resonances frequencies. Also, they
can change or shift resonance frequences.

## Resonant frequencies of air in a tube

The relationship between vocal tract shape and transfer function is
complex -we will consider the simple case of a uniform tube. The vocal
tract in a vowel can be approximated by a tube which is closed at one
end (the glottis) and open at the other (the lips). For a relatively
unconstricted vocal tract, the resonances of a 17 cm vocal tract occur
at the following frequencies:

`f= n * c / 4 * L for n = 1, 3, 5, ...`

f = formant frequency in Hz c = speed of sound 34,000 cm/s L= length of
vocal tract in cm

So the lowest formant frequency in a 17 cm. vocal tract is:

f = c / 4 \* L

# 34,000 / 4 \* 17

500 Hz

And the spacing between formants is: f = 2 \*c / 4 \* L

# c / 2 \* L (always twice the lowest f)

1000 Hz

Therefore, formant frequencies are: F~1~=500, F~2~=1500, F~3~=2500,
F~4~=3500.

## Two Tube Vocal Tract Models of Vowels

!Two tube model for vowel
/a/\|left\|300px
!Two tube model for vowel
/i/\|right\|300px

Two resonators or uniform tubes of different cross sectional areas can
be connected to approximate some vowels or consonants. In this case,
natural frequencies of the whole system are not simply the frequencies
for each tube because of acoustic coupling. figures() shows different
configuration of tubes to simulate vowels /a/, /i.

Typical values (for an adult male vocal tract for vowel /a/) are l1 =
8 cm, l2 = 9 cm with A1 = 5 cm2, A2 = 0.5 cm2. Acoustic theory predicts
that there will be resonances at 944 Hz, 1063 Hz, 2833 Hz. The narrow
and wide tubes can be considered as separate tubes with resonance
frequencies obeying the one stated in previous section for a tube.
However, the acoustic impedance at the boundary between two tubes is not
zero, thus effects the natural frequencies of the tubes. The natural
frequencies of the combined system are the frequencies for which the sum
of the reactances at the junction is zero that is:

$$-\rho*c/A_1 cot(KL_1)+\rho*c/A_2tan(KL_2)=0$$

It should be noted that when the natural frequencies of the tubes are
remote from one another, the influence of coupling is small.

Typical values for vowel /i/ in the human vocal tract are l1 = 9 cm, l2
= 8 cm, A1 = 5 cm2, A2 = 0.5 cm2. Thus, in theory, F1 = 202 Hz, F2 =
1890 Hz, F3 = 2125 Hz.

## Four Tube Vocal Tract Models of Vowels

!four tube
model\|400px

Four tube models of vowels provide a much better estimate of formant
frequencies for a wider range of vowels than do two tube models and so
are more a more popular method of modeling vowels. Such models consist
of a lip tube (tube 1) a tongue constriction tube (tube 3) and
unconstricted tubes either side of the constriction tube. This model is
controlled by three parameters. They are i) the position of the centre
of tube 3, ii) the cross-sectional area of tube 3, and iii) the ratio of
the length to the cross-sectional area at the lip section. For extreme
back constrictions tube 4 disappears whilst for extreme front
constrictions tube 2 disappears.

Calculations of resonance frequencies using the 4 tube model are quite
complex and so Fant (1960) supplied a (fairly complex) graphical
representation of the relationship between the three parameters and the
resultant formant frequencies. These graphical representations are
called nomograms. The original
versions of these nomograms supply, for a continuous range of x
constriction positions (i.e. distance from the centre of the tongue
constriction to the glottis) a continuous range of resultant F1 to F5
values. The original nomograms do this for 5 values of lip area (A1) and
for two values of tongue constriction cross-sectional area (A3). For
different vocal tract lengths, different nomograms need to be computed.

The four tube, three parameter, model provides a sufficiently accurate
prediction of most vowel sounds, but cannot model nasalisation of
vowels.

## References

1- Kenneth N. Stevens,2000, Acoustic phonetics, The MIT Press.

2- Kinsler *et al.* 2000,Fundamentals of Acoustics, John Wiley & Sons.

3- Titze, I.R. (1994). Principles of Voice Production, Prentice Hall
(currently published by NCVS.org),
.

4- James L. Flangam and Lawrence R. Rabiner,1973, Speech synthesis.


# Engineering Acoustics/Moving Coil Loudspeaker

# Moving Coil Transducer

The purpose of the acoustic transducer is to convert electrical energy
into acoustic energy. Many variations of acoustic transducers exist,
although the most common is the moving coil-permanent magnet transducer.
The classic loudspeaker is of the moving coil-permanent magnet type.

The classic electrodynamic loudspeaker driver can be divided into three
key components:

1\) The Magnet Motor Drive System

2\) The Loudspeaker Cone System

3\) The Loudspeaker Suspension

```{=html}
<center>
```
![](loud_speaker.gif "loud_speaker.gif")

```{=html}
</center>
```
```{=html}
<center>
```
Figure 1 Cut-away of a moving coil-permanent magnet loudspeaker

```{=html}
</center>
```
## The Magnet Motor Drive System

The main purpose of the Magnet Motor Drive System is to establish a
symmetrical magnetic field in which the voice coil will operate. The
Magnet Motor Drive System is comprised of a front focusing plate,
permanent magnet, back plate, and a pole piece. In figure 2, the
assembled drive system is illustrated. In most cases, the back plate and
the pole piece are built into one piece called the yoke. The yoke and
the front focusing plate are normally made of a very soft cast iron.
Iron is a material that is used in conjunction with magnetic structures
because the iron is easily saturated when exposed to a magnetic field.
Notice in figure 2, that an air gap was intentionally left between the
front focusing plate and the yoke. The magnetic field is coupled through
the air gap. The magnetic field strength (B) of the air gap is typically
optimized for uniformity across the gap. \[1\]

```{=html}
<center>
```
Figure 2 Permanent Magnet Structure

```{=html}
</center>
```
When a coil of wire with a current flowing is place inside the permanent
magnetic field, a force is produced. B is the magnetic field strength, l
is the length of the coil, and I is the current flowing through the
coil.

$F = Bli$

```{=html}
<center>
```
![](Magnet2.gif "Magnet2.gif")

```{=html}
</center>
```
```{=html}
<center>
```
Figure 3 Voice Coil Mounted in Permanent Magnetic Structure

```{=html}
</center>
```
The coil is excited with the AC signal that is intended for sound
reproduction, when the changing magnetic field of the coil interacts
with the permanent magnetic field then the coil moves back and forth in
order to reproduce the input signal. The coil of a loudspeaker is known
as the voice coil.

```{=html}
<center>
```
Figure 4 Photograph - Voice Coil

```{=html}
</center>
```
## The Loudspeaker Cone System

On a typical loudspeaker, the cone serves the purpose of creating a
larger radiating area allowing more air to be moved when excited by the
voice coil. The cone serves a piston that is excited by the voice coil.
The cone then displaces air creating a sound wave. In an ideal
environment, the cone should be infinitely rigid and have zero mass, but
in reality neither is true. Cone materials vary from carbon fiber,
paper, bamboo, and just about any other material that can be shaped into
a stiff conical shape. The loudspeaker cone is a very critical part of
the loudspeaker. Since the cone is not infinitely rigid, it tends to
have different types of resonance modes form at different frequencies,
which in turn alters and colors the reproduction of the sound waves. The
shape of the cone directly influences the directivity and frequency
response of the loudspeaker. When the cone is attached to the voice
coil, a large gap above the voice coil is left exposed. This could be a
problem if foreign particles make their way into the air gap of the
voice coil and the permanent magnet structure. The solution to this
problem is to place what is known as a dust cap on the cone to cover the
air gap. Below a figure of the cone and dust cap are shown.

```{=html}
<center>
```
![](loud_cone.gif "loud_cone.gif")

```{=html}
</center>
```
```{=html}
<center>
```
Figure 6 Cone and Dust Cap attached to Voice Coil

```{=html}
</center>
```
## The Loudspeaker Suspension

Most moving coil loudspeakers have a two piece suspension system, also
known as a flexure system. The combination of the two flexures allows
the voice coil to maintain linear travel as the voice coil is energized
and provide a restoring force for the voice coil system. The two piece
system consists of large flexible membrane surrounding the outside edge
of the cone, called the surround, and an additional flexure connected
directly to the voice coil, called the spider. The surround has another
purpose and that is to seal the loudspeaker when mounted in an
enclosure. Commonly, the surround is made of a variety of different
materials, such as, folded paper, cloth, rubber, and foam. Construction
of the spider consists of different woven cloth or synthetic materials
that are compressed to form a flexible membrane. The following two
figures illustrate where the suspension components are physically at on
the loudspeaker and how they function as the loudspeaker operates.

```{=html}
<center>
```
![](loud_suspension.gif "loud_suspension.gif")

```{=html}
</center>
```
```{=html}
<center>
```
Figure 7 Loudspeaker Suspension System

```{=html}
</center>
```
```{=html}
<center>
```
![](loudspk.gif "loudspk.gif")

```{=html}
</center>
```
```{=html}
<center>
```
Figure 8 Moving Loudspeaker

```{=html}
</center>
```
## Modeling the Loudspeaker as a Lumped System

Before implementing a loudspeaker into a specific application, a series
of parameters characterizing the loudspeaker must be extracted. The
equivalent circuit of the loudspeaker is key when developing enclosures.
The circuit models all aspects of the loudspeaker through an equivalent
electrical, mechanical, and acoustical circuit. Figure 9 shows how the
three equivalent circuits are connected. The electrical circuit is
comprised of the DC resistance of the voice coil, Re, the imaginary part
of the voice coil inductance, Le, and the real part of the voice coil
inductance, Revc. The mechanical system has electrical components that
model different physical parameters of the loudspeaker. In the
mechanical circuit, Mm, is the electrical capacitance due to the moving
mass, Cm, is the electrical inductance due to the compliance of the
moving mass, and Rm, is the electrical resistance due to the suspension
system. In the acoustical equivalent circuit, Ma models the air mass and
Ra models the radiation impedance\[2\]. This equivalent circuit allows
insight into what parameters change the characteristics of the
loudspeaker. Figure 10 shows the electrical input impedance as a
function of frequency developed using the equivalent circuit of the
loudspeaker.

```{=html}
<center>
```
![](Eq_circuit.gif "Eq_circuit.gif")

```{=html}
</center>
```
```{=html}
<center>
```
Figure 9 Loudspeaker Analogous Circuit

```{=html}
</center>
```
```{=html}
<center>
```
![](Freq_resp.gif "Freq_resp.gif")

```{=html}
</center>
```
```{=html}
<center>
```
Figure 10 Electrical Input Impedance

```{=html}
</center>
```
## References

\[1\] The Loudspeaker Design Cookbook 5th Edition; Dickason, Vance.,
Audio Amateur Press, 1997. \[2\] Beranek, L. L. Acoustics. 2nd ed.
Acoustical Society of America, Woodbridge, NY. 1993.


# Engineering Acoustics/Microphone Design and Operation

![](Mic_Title.jpg "Mic_Title.jpg")

## Introduction

Microphones are devices which convert pressure fluctuations into
electrical signals. Two main methods of achieving this are used in the
mainstream entertainment industry today - **dynamic microphones** and
**condenser microphones**. Piezoelectric
transducers
can also be used as microphones but they are not commonly used in the
entertainment industry.

## Dynamic microphones

Dynamic microphones utilise \'Faraday's Law\'. The principle states that
when an electrical conductor is moved through a magnetic field, an
electrical current is induced within the conductor. In these microphones
the magnetic field comes from permanent magnets. There are two common
arrangements for the conductor.

![](Moving_coil.JPG "Moving_coil.JPG"){width="300"}

Figure 1: Sectional View of Moving-Coil Dynamic Microphone

The first conductor arrangement has a moving coil of wire. The wire is
typically copper and is attached to a circular membrane or piston
usually made from lightweight plastic or occasionally aluminum. The
impinging pressure fluctuation on the piston causes it to move in the
magnetic field and thus creates the desired electrical current.

![](Ribbon.jpg){width="300"}

Figure 2: Dynamic Ribbon Microphone

The second conductor arrangement is a ribbon of metallic foil suspended
between magnets. The metallic ribbon moves in response to a pressure
fluctuation and an electrical current is produced. In both
configurations, dynamic microphones follow the same principals as
acoustical
transducers.

## Condenser Microphones

Condenser microphones convert pressure fluctuations into electrical
potentials by changes in electrical capacitance, hence they are also
known as capacitor microphones. An electrical capacitor consists of two
charged electrical conductors placed at some relatively small distance
to each other. The basic relation that describes capacitors is:

:   **$Q=C\times V$**

where Q is the electrical charge of the capacitor's conductors, C is the
capacitance, and V is the electric potential between the capacitor's
conductors. If the electrical charge of the conductors is held at a
constant value, then the voltage between the conductors will be
inversely proportional to (a) the capacitance and (b) the distance
between the conductors.

![](Condenser_schema.jpg "Condenser_schema.jpg"){width="600"}

Figure 3: Sectional View of Condenser Microphone

The capacitor in a condenser microphone consists of two parts: the
diaphragm and the backplate. The diaphragm moves due to impinging
pressure fluctuations and the backplate is held in a stationary
position. When the diaphragm moves closer to the backplate, the
capacitance increases and a change in electric potential is produced.
The diaphragm is typically made of metallic coated Mylar. The assembly
that houses both the backplate and the diaphragm is commonly referred to
as a capsule.

To keep the diaphragm and backplate at a constant charge, an electric
potential must be presented to the capsule. There are various ways this
can be achieved. The first uses a battery to supply the needed DC
potential to the capsule (figure 4). The resistor across the leads of
the capsule is very high, in the range of 10 mega ohms, to keep the
charge on the capsule close to constant.

![](Voltage.JPG "Voltage.JPG"){width="500"}

Figure 4: Internal Battery Powered Condenser Microphone

An alternative technique for providing a constant charge on the
capacitor is to supply a DC electric potential through the microphone
cable that carries the microphones output signal. Standard microphone
cable is known as XLR cable and is terminated by three pin connectors.
Pin one connects to the shield around the cable. The microphone signal
is transmitted between pins two and three.

![](XLR.jpg "XLR.jpg"){width="700"}

Figure 5: Dynamic Microphone Connected to a Mixing Console via XLR Cable

### 48V Phantom Powering

The most popular method of providing a DC potential through a microphone
cable is to supply +48 V to both of the microphone output leads, pins 2
and 3, and use the shield of the cable, pin 1, as the ground to the
circuit. Because pins 2 and 3 see the same potential, any fluctuation of
the microphone powering potential will not affect the microphone signal
seen by the attached audio equipment. The +48 V will be stepped down at
the microphone using a transformer and provide the potential to the
backplate and diaphragm in a similar fashion as the battery solution.

![](Powering.jpg "Powering.jpg"){width="600"}

Figure 6: Condenser Microphone Powering Techniques

### 12V T-Powering

A less popular method of running the potential through the cable is to
supply 12 V between pins 2 and 3. This method is referred to as
T-powering. The main problem with T-powering is that potential
fluctuation in the powering of the capsule will be transmitted into an
audio signal because the audio equipment analyzing the microphone signal
will not see a difference between a potential change across pins 2 and 3
due to a pressure fluctuation and one due to the power source electric
potential fluctuation.

### Electret Condenser Microphones

Finally, the diaphragm and backplate can be manufactured from a material
that maintains a fixed charge, known as \'electret\' (from
electric+magnet, because these materials can be seen as the electric
equivalents of permanent magnets). As a result, these microphones are
termed electret condenser microphones (ECM). In early electret designs,
the charge on the material tended to become unstable over time. Advances
in science and manufacturing have effectively eliminated this problem in
present designs.

## Conclusion

Two types of microphones are used in the entertainment industry.

-   Dynamic microphones, which are found in the moving-coil and ribbon
    configurations. The movement of the conductor in dynamic microphones
    induces an electric current which is then transformed into the
    reproduction of sound.
-   Condenser microphones which utilize the properties of capacitors.
    The charge on the capsule of condenser microphones can be
    accomplished by battery, by phantom powering, by T-powering, and by
    using \'electrets\' - materials with a fixed charge.

## References

-Sound Recording Handbook. Woram, John M. 1989.

-Handbook of Recording Engineering Fourth Edition. Eargle, John. 2003.

## Microphone Manufactures Links

AEA

AKG

Audio
Technica

Audix

Bruel & Kjaer

1

Neumann

Rode

Shure

sE Electronics

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Acoustics