# 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()
```

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