Sound absorption in air. ISO 9613-1 equation

Mathematical definition

$$\boxed{\alpha = {\alpha _{cl}} + {\alpha _{rot}} + {\alpha _{vib,O}} + {\alpha _{vib,N}}}$$

Notation Description Units Limits Conversion
$\alpha$ pure-tone sound attenuation coefficient $\text{dB/m}$
$T$ ambient atmospheric temperature $^{\circ}\text{C}$ $-20 < T < +50$ $T\left[ {\rm{K}} \right] = T\left[ {{\rm{^\circ C}}} \right] + 273.15$
$h_{r}$ relative humidity $\text{\%}$ $10 < h_{r} < 100$
$p_{a}$ atmosperica pressure $\text{kPa}$
$f$ frequency $\text{kHz}$ $0.05 < f < 10$

$${\alpha _{cr}} = {\alpha _{cl}} + {\alpha _{rot}} = \frac{{1.60 \times {{10}^{ - 10}}{{\left( {T/{T_0}} \right)}^{\frac{1}{2}}}{f^2}}}{{{p_a}/{p_r}}}$$

$${\alpha _{vib,O}} = {\left( {\alpha \lambda } \right)_{max,O}}\left( {f/c} \right)\left( {2\left( {f/{f_{rO}}} \right){{\left( {1 + {{\left( {f/{f_{rO}}} \right)}^2}} \right)}^{ - 1}}} \right)$$

$${\alpha _{vib,N}} = {\left( {\alpha \lambda } \right)_{max ,N}}\left( {f/c} \right)\left( {2\left( {f/{f_{rN}}} \right){{\left( {1 + {{\left( {f/{f_{rN}}} \right)}^2}} \right)}^{ - 1}}} \right)$$

$${\left( {\alpha \lambda } \right)_{max,O}} = \left( {2\pi /35} \right)\left( {10\lg \left( {{e^2}} \right)} \right){X_O}{\left( {{\theta _O}/T} \right)^2}\exp \left( { - {\theta _O}/T} \right)$$

$${\left( {\alpha \lambda } \right)_{max,N}} = \left( {2\pi /35} \right)\left( {10\lg \left( {{e^2}} \right)} \right){X_N}{\left( {{\theta _N}/T} \right)^2}\exp \left( { - {\theta _N}/T} \right)$$

$$c = 343.2{\left( {T/{T_0}} \right)^{\frac{1}{2}}}$$

$${f_{rO}} = \frac{{{p_a}}}{{{p_r}}}\left( {24 + 4.04 \times {{10}^4}h\frac{{0.02 + h}}{{0.391 + h}}} \right)$$

$${f_{rN}} = \frac{{{p_a}}}{{{p_r}}}{\left( {\frac{T}{{{T_0}}}} \right)^{ - \frac{1}{2}}}\left( {9 + 280h\exp \left( { - 4.170\left( {{{\left( {\frac{T}{{{T_0}}}} \right)}^{ - \frac{1}{3}}} - 1} \right)} \right)} \right)$$

$$h = {h_r}\left( {{p_{sat}}/{p_r}} \right)/\left( {{p_a}/{p_r}} \right)$$

$${p_{sat}}/{p_r} = {10^{ - 6.8346\left( {{T_{01}}/T} \right)}} + 4.615$$

Notation Description Units
$a_{cl}$ "classical" absorption caused by transport processes $\text{dB/m}$
$a_{rot}$ molecular absorption caused by rotational relaxation $\text{dB/m}$
$a_{vib,O}$ molecular absorption caused by vibrational relaxation of oxygen $\text{dB/m}$
$a_{vib,N}$ molecular absorption caused by vibrational relaxation of nitrogen $\text{dB/m}$
$\left( {\alpha \lambda } \right)_{max}$ maximum attenuation caused by vibrational relaxation $\text{dB}$
$c$ speef of sound $\text{m/s}$
$f_{rO}$ relaxation frequency of oxygen $\text{Hz}$
$f_{rN}$ relaxation frequency of nitrogen $\text{Hz}$
$h$ molar concentration of water vapour $\text{\%}$
$p_{sat}$ saturation vapour pressure $\text{kPa}$
$p_{a}$ atmospheric pressure $\text{kPa}$
$p_{r}$ reference ambient atmospheric pressure $\text{kPa}$
Notation Description Value Units
$\Theta_{O}$ characteristic vibrational temperature for oxygen $2239.1$ $\text{K}$
$\Theta_{N}$ characteristic vibrational temperature for nitrogen $3352.0$ $\text{K}$
$X_{O}$ fractional molar concentration for oxygen $0.209$
$X_{N}$ fractional molar concentration for nitrogen $0.781$

Octave/Matlab implementation

function alpha = sound_absorption_air_iso_annex(T,RH,P,f)
% Inputs
%   T: temperature \ degree Celsius \ -20 < T < +50
%   RH: relative humidity \ percentage \ 10 < RH < 100
%   P: pressure \ kPa
%   f: frequency \ kHz \ 0.05 < f < 10
% Outputs
%   alpha: absorption of sound in air \ dB/m

    Kelvin = 273.15;
    T_ref = Kelvin + 20;
    T_kel = Kelvin + T;
    T_rel = T_kel / T_ref;
    T_01 = Kelvin + 0.01;
    Tho = 2239.1;
    Thn = 3352.0;
    P_ref = 101.325;
    P_rel = P / P_ref;    
    Xo = 0.209;
    Xn = 0.781;

    P_sat_P_ref = 10.^(-6.8346*(T_01./T_kel).^(1.261)+4.6151);
    H = RH.*(P_sat_P_ref./P_rel);

    Fro = P_rel.*(24+(4.04e4)*H.*(0.02+H)./(0.391+H));
    Frn = P_rel.*(T_rel.^(-1/3)).*(9+280*H*exp(-4.170*(T_rel.^(-1/3)-1)));

    C = 343.2*(T_rel.^(1/2));
    acr = (1.60e-10)*(T_rel.^(1/2)).*(f.^2)./P_rel;

    amaxO = (2*pi/35)*(10*log10(exp(1)^2)) ...
        *Xo*((Tho./T_kel).^2).*exp(-Tho./T_kel);
    avibO = amaxO.*(f/C)*2.*(f./Fro)./(1+(f./Fro).^2);

    amaxN = (2*pi/35)*(10*log10(exp(1)^2)) ...
        *Xn*((Thn./T_kel).^2).*exp(-Thn./T_kel);
    avibN = amaxN.*(f/C)*2.*(f./Frn)./(1+(f./Frn).^2);

    alpha = acr + avibO + avibN;
end

Computational examples

References

  1. ISO 9613-1 : Acoustics-Attenuation of sound during propagation outdoors-Part 1: Calculation of the absorption of sound by the atmosphere, (1993).