# Engineering Acoustics/Outdoor Sound Propagation

 Edit this template Part 1: Lumped Acoustical Systems – 1.1 – 1.2 – 1.3 – 1.4 – 1.5 – 1.6 – 1.7 – 1.8 – 1.9 – 1.10 – 1.11 Part 2: One-Dimensional Wave Motion – 2.1 – 2.2 – 2.3 Part 3: Applications – 3.1 – 3.2 – 3.3 – 3.4 – 3.5 – 3.6 – 3.7 – 3.8 – 3.9 – 3.10 – 3.11 – 3.12 – 3.13 – 3.14 – 3.15 – 3.16 – 3.17 – 3.18 – 3.19 – 3.20 – 3.21 – 3.22 – 3.23 – 3.24

## 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[1]

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

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

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

This equation can be simplified for an ideal gas leading to

${\displaystyle c^{2}=\gamma {\frac {P_{0}}{\rho _{0}}}}$

where ${\displaystyle \gamma }$ is the ratio of heat capacities

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

${\displaystyle c_{0}=331.5\ {\mbox{m/s}}}$

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

${\displaystyle c_{0}=343\ {\mbox{m/s}}}$

An equivalent expression for the speed of sound in terms of the temperature in Kelvin ${\displaystyle T_{K}}$ is

${\displaystyle c^{2}=\ \gamma rT_{K}}$

where ${\displaystyle r}$ is the specific gas constant.

## Decibel scale[2]

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 ${\displaystyle IL}$ of a sound intensity ${\displaystyle I}$ is defined by

${\displaystyle IL=10\log _{10}{\bigg (}{\frac {I}{I_{ref}}}{\bigg )}{\mbox{ dB}}\,}$

where ${\displaystyle I_{ref}}$ 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

${\displaystyle SPL=10\log _{10}{\bigg (}{\frac {{P_{e}}^{2}}{{P_{ref}}^{2}}}{\bigg )}=20\log _{10}{\bigg (}{\frac {P_{e}}{P_{ref}}}{\bigg )}{\mbox{ dB}}\,}$

where ${\displaystyle P_{e}}$ is the measured effective pressure amplitude of the sound wave and ${\displaystyle P_{ref}}$ 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. ${\displaystyle SPL}$ is also called sound level ${\displaystyle L_{p}}$.

For air, the pressure reference is taken to be

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

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

Figure 1 - Plot of decibel and inverse

## Sound attenuation[3]

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 ${\displaystyle A_{T}}$ in dB between the source and the receiver. The total attenuation ${\displaystyle A_{T}}$ can be expressed as

${\displaystyle A_{T}=10\log _{10}{\bigg (}{\frac {I_{ps}}{I_{pr}}}{\bigg )}=20\log _{10}{\bigg (}{\frac {P_{s}}{P_{r}}}{\bigg )}{\mbox{ dB}}\,}$

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

The total attenuation is defined as the sum of the attenuation due to geometric spreading ${\displaystyle A_{s}}$, the attenuation due to atmospheric absorption ${\displaystyle A_{a}}$, and the excess attenuation due to all other effects ${\displaystyle A_{e}}$, namely

${\displaystyle A_{T}=A_{s}+A_{a}+A_{e}{\mbox{ dB}}\,}$

The excess attenuation ${\displaystyle A_{e}}$ can include attenuation from the ground in a homogeneous atmosphere ${\displaystyle A_{g}}$, 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.

The geometric spreading loss ${\displaystyle A_{s}}$, in dB, between two points at a distance ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$ from a source can be expressed as

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

where ${\displaystyle g}$ is a constant given by the geometry of the problem. ${\displaystyle g=0}$ for plane wave propagation (uniform pipe), ${\displaystyle g=0.5}$ for cylindrical propagation from a line source, and ${\displaystyle g=1}$ 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 (${\displaystyle r_{2}=2r_{1}}$) corresponds to a loss of ${\displaystyle 6{\mbox{dB}}}$.

## Attenuation by atmospheric absorption[5] [6] [7]

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 ${\displaystyle A_{a}}$, in dB due to atmospheric absorption can be expressed as

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

where ${\displaystyle r}$, in meters, is the path length of the traveling wave, ${\displaystyle P(r)}$ is the sound pressure after traveling the distance ${\displaystyle r}$, ${\displaystyle P(0)}$ is the initial sound pressure at ${\displaystyle r=0}$, ${\displaystyle \alpha }$ is the attenuation coefficient in Nepers per meter, and ${\displaystyle a}$ 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 ${\displaystyle a}$ for pure tone frequencies can be expressed as

${\displaystyle {\frac {a}{p_{s}}}={\frac {20}{\ln 10}}\ {\frac {F^{2}}{p_{s0}}}\left\{1.84\times 10^{-11}\left({\frac {T}{T_{0}}}\right)^{1/2}+\left({\frac {T}{T_{0}}}\right)^{-5/2}\left[0.01278{\frac {e^{-2239.1/T}}{F_{r,O}+F^{2}/F_{r,O}}}+0.1068{\frac {e^{-3352/T}}{F_{r,N}+F^{2}/F_{r,N}}}\right]\right\}\quad {\frac {\mathrm {dB} }{\mathrm {m} \cdot \mathrm {atm} }}\,}$

with ${\displaystyle F=f/p_{s}}$, ${\displaystyle F_{r,O}=f_{r,O}/p_{s}}$, and ${\displaystyle F_{r,N}=f_{r,N}/p_{s}}$, and where ${\displaystyle f}$ is the acoustic frequency in Hz, ${\displaystyle p_{s}}$ is the atmospheric pressure, ${\displaystyle p_{s0}}$ is the reference atmospheric pressure (1 atm), ${\displaystyle T}$ is the atmospheric temperature in K, ${\displaystyle T_{0}}$ is the reference temperature (293.15 K), ${\displaystyle f_{r,O}}$ is the relaxation frequency of molecular oxygen and ${\displaystyle f_{r,N}}$ is the relaxation frequency of molecular nitrogen. Scaled relaxation frequencies for oxygen and nitrogen formulas from experimental measurements are given by

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

and

${\displaystyle F_{r,N}={\frac {1}{p_{s0}}}\left({\frac {T_{0}}{T}}\right)^{1/2}\left(9+280h\times \exp \left\{-4.17\left[\left({\frac {T_{0}}{T}}\right)^{1/3}-1\right]\right\}\right)\quad {\frac {\mathrm {Hz} }{\mathrm {atm} }}}$

where ${\displaystyle h}$ is the molar concentration of water vapor (absolute humidity) in percent. ${\displaystyle h}$ is calculated from the relative humidity ${\displaystyle h_{r}}$ as follows

${\displaystyle h=p_{s0}\left({\frac {h_{r}}{p_{s}}}\right)\left({\frac {p_{sat}}{p_{s0}}}\right)\quad \%}$

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

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

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

The formulas, are valid for a pressure under 2 atm, a temperature under 330 K 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 ${\displaystyle \alpha }$ 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.

Figure 2 - Attenuation coefficient for atmospheric absorption per atmosphere as a function of frequency and relative humidity, for air at 20°C.[8] [9]

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

Table 1 - Atmospheric attenuation coefficient ${\displaystyle a}$ (dB/km) at selected frequencies at 1 atm[10]

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

[dubious ]

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.[11] Some atmospheric attenuation also occurs in fog and precipitation,[12] in dust in air,[13] and at frequencies below 10 Hz due to electromagnetic radiation of moist air molecules.[14]

## Reflection from the surface of a solid[15]

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.

Figure 3 - Angle of incidence for a reflected sound wave

## Attenuation over the ground[16]

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[17][18]

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.

Figure 4 - Outdoor sound refraction caused by temperature gradients

## Diffraction[19][20]

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.

Figure 5 - Different paths of sound propagation behind a barrier

Figure 6 - Outdoor sound diffraction behind a small opening

## 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.

Figure 7 - Different types of noise barriers
Figure 8 - Vertical vegetation walls

## Useful Websites

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, 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

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)*(0.01278*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

#!/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(T0 / T, 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)

if __name__ == '__main__':
## 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()


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