Sound speed in air. Cramer equation

Mathematical definition

$$\boxed{\begin{array}{l} C\left( {T,P,{X_w},{X_c}} \right) = {a_0} + {a_1}T + {a_2}{T^2} + \left( {{a_3} + {a_4}T + {a_5}{T^2}} \right){X_w}\\ + \left( {{a_6} + {a_7}T + {a_8}{T^2}} \right)P + \left( {{a_9} + {a_{10}}T + {a_{11}}{T^2}} \right){X_c}\\ + {a_{12}}{X_w}^2 + {a_{13}}{P^2} + {a_{14}}{X_c}^2 + {a_{15}}{X_w}P{X_c} \end{array}}$$

Notation Description Units Limits Conversion
$C$ sound speed $\text{m/s}$
$T$ temperature $^{\circ}\text{C}$ $0 < T < 30$
$P$ pressure $\text{kPa}$ $75 < P < 102$ $P\left[ {{\rm{Pa}}} \right] = P\left[ {{\rm{kPa}}} \right] \times {10^3}$
$X_{w}$ water vapor mole fraction $X_{w} < 0.06$
$X_{c}$ carbon dioxide mole fraction

$${p_{sv}} = 1{\rm{Pa}} \times \exp \left( {A{T_K}^2 + B{T_K} + C + \frac{D}{{{T_K}}}} \right)$$

$$f = \alpha + \beta P + \gamma {T^2}$$

$${x_v} = hf\left( {P,T} \right)\frac{{{p_{sv}}\left( T \right)}}{P}$$

Notation Description Units
$p_{sv}$ saturation vapour pressure of moist air $\text{Pa}$
$f$ enhancement factor
$x_{v}$ the mole fraction of water vapour
$h$ relative humidity $\text{\%}$
Coefficient Value Coefficient Value
$a_{1}$ $+331.5024$ $a_{9}$ $-85.20931$
$a_{2}$ $+0.603055$ $a_{10}$ $-0.228525$
$a_{3}$ $-0.000528$ $a_{11}$ $+5.91 \times 10^{-5}$
$a_{4}$ $+51.471935$ $a_{12}$ $-2.835149$
$a_{5}$ $+0.1495874$ $a_{13}$ $-2.15 \times 10^{-13}$
$a_{6}$ $-0.000782$ $a_{14}$ $+29.179762$
$a_{7}$ $-1.82 \times 10^{-7}$ $a_{15}$ $+0.000486$
$a_{8}$ $+3.73 \times 10^{-8}$
Coefficient Value Units
$A$ $+1.2378847 \times 10^{-5}$ $\text{K}^{-2}$
$B$ $-1.9121316 \times 10^{-2}$ $\text{K}^{-1}$
$C$ $+33.93711047$
$D$ $-6.3431645 \times 10^{3}$ $\text{K}$
$\alpha$ $+1.00062$
$\beta$ $+3.14 \times 10^{-8}$ $\text{Pa}^{-1}$
$\gamma$ $+5.6 \times 10^{-7}$ $\text{K}^{-2}$

Octave/Matlab implementation

function C = sound_speed_air_cramer(T,RH,P)
% Inputs
%    T: temperature \ degree Celsius \ 0 < T < 30
%   RH: relative humidity \ percentage
%    P: pressure \ kPa \ 75 < P < 102
% Outputs
%    C: speed of sound in air \ m/s

    T_kel = T + 273.15;
    P = P*1e3;

    A = +1.2378847e-5;
    B = -1.9121316e-2;
    C = +33.93711047;
    D = -6.3431645e3;

    alpha = 1.00062;
    beta = 3.14e-8;
    gamma = 5.6e-7;

    Psv = exp(A*(T_kel.^2) + B*T_kel + C + D./T_kel);
    F = alpha + beta*P + gamma*(T.^2);
    H = RH.*F.*Psv./P;
    Xw = H/100.0;
    Xc = 400e-6;

    a0 = 331.5024; a1 = 0.603055; a2 = -0.000528; a3 = 51.471935;
    a4 = 0.1495874; a5 = -0.000782; a6 = -1.82e-7; a7 = 3.73e-8;
    a8 = -2.93e-10; a9 = -85.20931; a10 = -0.228525; a11 = 5.91e-5;
    a12 = -2.835149; a13 = -2.15e-13; a14 = 29.179762; a15 = 0.000486;

    C = a0 + a1*T + a2*(T.^2) + (a3 + a4*T + a5*(T.^2)).*Xw ...
        + (a6 + a7*T + a8*(T.^2)).*P + (a9 + a10*T+a11*(T.^2)).*Xc ...
        + a12*Xw.^2 + a13*(P.^2) + a14*(Xc.^2) + a15.*Xw.*P.*Xc;
end

Computational examples

References

  1. Cramer, Owen, "The variation of the specific heat ratio and the speed of sound in air with temperature, pressure, humidity, and CO2 concentration", 1993
  2. Davis, Richard S, "Equation for the determination of the density of moist air (1981/91)", 1992
  3. Giacomo, P, "Equation for the determination of the density of moist air (1981)", 1982