Sound speed in pure water. Belogol'skii, Sekoyan et al equation

Mathematical definition

$$\boxed{\begin{array}{l} C\left( {T,P} \right) = C\left( {T,0} \right) + {M_1}\left( T \right)\left( {P - 0.101325} \right)\\ + {M_2}\left( T \right){\left( {P - 0.101325} \right)^2} + {M_3}\left( T \right){\left( {P - 0.101325} \right)^3} \end{array}}$$

Notation	Description	Units	Limits	Conversion
$C$	sound speed	$\text{m/s}$
$T$	temperature	$^{\circ}\text{C}$	$0 < T < 40$
$P$	pressure	$\text{kPa}$	$100 < P < 60000$	$P\left[ {{\rm{MPa}}} \right] = P\left[ {{\rm{kPa}}} \right] \times {10^{ - 3}}$

$$C\left( {T,0} \right) = {a_{00}} + {a_{10}}T + {a_{20}}{T^2} + {a_{30}}{T^3} + {a_{40}}{T^4} + {a_{50}}{T^5}$$

$${M_1}\left( T \right) = {a_{01}} + {a_{11}}T + {a_{21}}{T^2} + {a_{31}}{T^3}$$

$${M_2}\left( T \right) = {a_{02}} + {a_{12}}T + {a_{22}}{T^2} + {a_{32}}{T^3}$$

$${M_3}\left( T \right) = {a_{03}} + {a_{13}}T + {a_{23}}{T^2} + {a_{33}}{T^3}$$

Coefficient	Value	Coefficient	Value
$a_{00}$	$+1402.38744$	$a_{02}$	$+4.31532833 \times 10^{-3}$
$a_{10}$	$+5.03836171$	$a_{12}$	$-2.938590293 \times 10^{-4}$
$a_{20}$	$-5.81172916 \times 10^{-2}$	$a_{22}$	$+6.822485943 \times 10^{-6}$
$a_{30}$	$+3.34638117 \times 10^{-4}$	$a_{32}$	$-6.674551162 \times 10^{-8}$
$a_{40}$	$-1.48259672 \times 10^{-6}$	$a_{03}$	$-1.852993525 \times 10^{-5}$
$a_{50}$	$+3.16585020 \times 10^{-9}$	$a_{13}$	$+1.481844713 \times 10^{-6}$
$a_{01}$	$+1.49043589$	$a_{23}$	$-3.940994021 \times 10^{-8}$
$a_{11}$	$+1.077850609 \times 10^{-2}$	$a_{33}$	$+3.939902307 \times 10^{-10}$
$a_{21}$	$-2.232794656 \times 10^{-4}$
$a_{31}$	$+2.718246452 \times 10^{-6}$

Octave/Matlab implementation

function C = sound_speed_water_belogolskii(T, P)
% Inputs
%    T: temperature \ degree Celsius \ 0 < T < 40
%    P: pressure \ kPa \ 100 < P < 60000
% Outputs
%    C: speed of sound in pure water \ m/s

    P = P*1e-3;

    a00 = +1402.38744;
    a10 = +5.03836171;
    a20 = -5.81172916e-2;
    a30 = +3.34638117e-4;
    a40 = -1.48259672e-6;
    a50 = +3.16585020e-9;
    a01 = +1.49043589;
    a11 = +1.077850609e-2;
    a21 = -2.232794656e-4;
    a31 = +2.718246452e-6;
    a02 = +4.315328330e-3;
    a12 = -2.938590293e-4;
    a22 = +6.822485943e-6;
    a32 = -6.674551162e-8;
    a03 = -1.852993525e-5;
    a13 = +1.481844713e-6;
    a23 = -3.940994021e-8;
    a33 = +3.939902307e-10;

    C0 = a00 + a10*T + a20*(T.^2) + a30*(T.^3) + a40*(T.^4) + a50*(T.^5);
    M1 = a01 + a11*T + a21*(T.^2) + a31*(T.^3);
    M2 = a02 + a12*T + a22*(T.^2) + a32*(T.^3);
    M3 = a03 + a13*T + a23*(T.^2) + a33*(T.^3);
    C = C0 + M1.*(P - 0.101325) + M2.*((P - 0.101325).^2) ...
        + M3.*((P - 0.101325).^3);
end

Computational examples

References

1. Belogol'Skii, VA; Sekoyan, SS; Samorukova, LM; Stefanov, SR; Levtsov, VI, "Pressure dependence of the sound velocity in distilled water", 1999