Sound speed in sea. UNESCO equation

Mathematical definition

$$\boxed{C\left( {T,S,P} \right) = {C_W}\left( {T,P} \right) + A\left( {T,P} \right)S + B\left( {T,P} \right){S^{\frac{3}{2}}} + D\left( {T,P} \right){S^2}}$$

Notation	Description	Units	Limits	Conversion
$C$	sound speed	$\text{m/s}$
$T$	temperature	$^{\circ}\text{C}$	$0 < T < 40$
$S$	salinity	$\text{‰}$	$5 < S < 40$
$P$	pressure	$\text{kPa}$	$0 < P < 100000$	$P\left[ {{\rm{bar}}} \right] = P\left[ {{\rm{kPa}}} \right] \times {10^{ - 2}}$

$$\begin{array}{l} {C_W}\left( {T,P} \right) = \left( {{C_{00}} + {C_{01}}T + {C_{02}}{T^2} + {C_{03}}{T^3} + {C_{04}}{T^4} + {C_{05}}{T^5}} \right)\\ + \left( {{C_{10}} + {C_{11}}T + {C_{12}}{T^2} + {C_{13}}{T^3} + {C_{14}}{T^4}} \right)P\\ + \left( {{C_{20}} + {C_{21}}T + {C_{22}}{T^2} + {C_{23}}{T^3} + {C_{24}}{T^4}} \right){P^2}\\ + \left( {{C_{30}} + {C_{31}}T + {C_{32}}{T^2}} \right){P^3} \end{array}$$

$$\begin{array}{l} A\left( {T,P} \right) = \left( {{A_{00}} + {A_{01}}T + {A_{02}}{T^2} + {A_{03}}{T^3} + {A_{04}}{T^4}} \right)\\ + \left( {{A_{10}} + {A_{11}}T + {A_{12}}{T^2} + {A_{13}}{T^3} + {A_{14}}{T^4}} \right)P\\ + \left( {{A_{20}} + {A_{21}}T + {A_{22}}{T^2} + {A_{23}}{T^3}} \right){P^2}\\ + \left( {{A_{31}} + {A_{31}}T + {A_{32}}{T^2}} \right){P^3} \end{array}$$

$$B\left( {T,P} \right) = {B_{00}} + {B_{01}}T + \left( {{B_{10}} + {B_{11}}T} \right)P$$

$$D\left( {T,P} \right) = {D_{00}} + {D_{10}}P$$

Coefficient	Value	Coefficient	Value
$C_{00}$	$1402.388$	$A_{02}$	$7.166\times10^{-5}$
$C_{01}$	$5.03830$	$A_{03}$	$2.008\times10^{-6}$
$C_{02}$	$-5.81090\times10^{-2}$	$A_{04}$	$-3.21\times10^{-8}$
$C_{03}$	$3.3432\times10^{-4}$	$A_{10}$	$9.4742\times10^{-5}$
$C_{04}$	$-1.47797\times10^{-6}$	$A_{11}$	$-1.2583\times10^{-5}$
$C_{05}$	$3.1419\times10^{-9}$	$A_{12}$	$-6.4928\times10^{-8}$
$C_{10}$	$0.153563$	$A_{13}$	$1.0515\times10^{-8}$
$C_{11}$	$6.8999\times10^{-4}$	$A_{14}$	$-2.0142\times10^{-10}$
$C_{12}$	$-8.1829\times10^{-6}$	$A_{20}$	$-3.9064\times10^{-7}$
$C_{13}$	$1.3632\times10^{-7}$	$A_{21}$	$9.1061\times10^{-9}$
$C_{14}$	$-6.1260\times10^{-10}$	$A_{22}$	$-1.6009\times10^{-10}$
$C_{20}$	$3.1260\times10^{-5}$	$A_{23}$	$7.994\times10^{-12}$
$C_{21}$	$-1.7111\times10^{-6}$	$A_{30}$	$1.100\times10^{-10}$
$C_{22}$	$2.5986\times10^{-8}$	$A_{31}$	$6.651\times10^{-12}$
$C_{23}$	$-2.5353\times10^{-10}$	$A_{32}$	$-3.391\times10^{-13}$
$C_{24}$	$1.0415\times10^{-12}$	$B_{00}$	$-1.922\times10^{-2}$
$C_{30}$	$-9.7729\times10^{-9}$	$B_{01}$	$-4.42\times10^{-5}$
$C_{31}$	$3.8513\times10^{-10}$	$B_{10}$	$7.3637\times10^{-5}$
$C_{32}$	$-2.3654\times10^{-12}$	$B_{11}$	$1.7950\times10^{-7}$
$A_{00}$	$1.389$	$D_{00}$	$1.727\times10^{-3}$
$A_{01}$	$1.262\times10^{-2}$	$D_{10}$	$-7.9836\times10^{-6}$

Octave/Matlab implementation

function C = sound_speed_sea_unesco(T,S,P)
% Inputs
%   T: temperature \ degree Celsius \ 0 < T < 40 
%   S: salinity \ ppt \ 5 < S < 40
%   P: pressure \ kPa \ 0 < P < 100000 
% Outputs
%   C: speed of sound in seawater \ m/s

    C00=1402.388; C01=5.03830; C02=-5.81090e-2; C03=3.3432e-4; 
    C04=-1.47797e-6; C05=3.1419e-9;
    C10=0.153563; C11=6.8999e-4; C12=-8.1829e-6; C13=1.3632e-7;
    C14=-6.1260e-10; C20=3.1260e-5; C21=-1.7111e-6; 
    C22=2.5986e-8; C23=-2.5353e-10; C24=1.0415e-12; C30=-9.7729e-9;
    C31=3.8513e-10; C32=-2.3654e-12;
    A00=1.389; A01=-1.262e-2; A02=7.166e-5; A03=2.008e-6;
    A04=-3.21e-8;
    A10=9.4742e-5; A11=-1.2583e-5; A12=-6.4928e-8; A13=1.0515e-8;
    A14=-2.0142e-10;
    A20=-3.9064e-7; A21=9.1061e-9; A22=-1.6009e-10; A23=7.994e-12;
    A30=1.100e-10; A31=6.651e-12; A32=-3.391e-13;
    B00=-1.922e-2; B01=-4.42e-5; B10=7.3637e-5; B11=1.7950e-7;
    D00=1.727e-3;
    D10=-7.9836e-6;

    p = P*1e-2;

    Cw = (C00 + C01*T + C02*(T.^2) + C03*(T.^3) + C04*(T.^4) + C05*(T.^5)) ...
        + (C10 + C11*T + C12*(T.^2) + C13*(T.^3) + C14*(T.^4)).*p ...
        + (C20 + C21*T + C22*(T.^2) + C23*(T.^3) + C24*(T.^4)).*(p.^2) ...
        + (C30 + C31*T + C32*(T.^2)).*(p.^3);

    A = (A00 + A01*T + A02*(T.^2) + A03*(T.^3) + A04*(T.^4)) ...
        + (A10 + A11*T + A12*(T.^2) + A13*(T.^3) + A14*(T.^4)).*p ...
        + (A20 + A21*T + A22*(T.^2) + A23*(T.^3)).*(p.^2) ...
        + (A30 + A31*T + A32*(T.^2)).*(p.^3);

    B = B00 + B01*T + (B10 + B11*T).*p;

    D = D00+(D10.*p);

    C = Cw + A.*S + B.*(S.^(3/2)) + D.*(S.^2);
end

Computational examples

$D$\$T$	$0°\text{C}$	$10°\text{C}$	$20°\text{C}$	$30°\text{C}$	$40°\text{C}$
$10\ \text{m}$	$1442.62$	$1483.92$	$1516.06$	$1540.59$	$1558.51$
$1000\ \text{m}$	$1458.91$	$1500.37$	$1532.69$	$1557.44$	$1575.52$
$2000\ \text{m}$	$1475.80$	$1517.20$	$1549.55$	$1574.45$	$1592.71$
$5000\ \text{m}$	$1528.73$	$1568.88$	$1600.53$	$1625.45$	$1644.16$

References

Chen, Chen‐Tung; Millero, Frank J, "Speed of sound in seawater at high pressures", 1977
Millero, FJ; Li, X, "On equations for the speed of sound in seawater-comment", 1994