Sound absorption in sea. Francois & Garrison equation

Mathematical definition

$$\boxed{\alpha \left( {T,S,D,f,pH} \right) = \frac{{{A_1}{P_1}{f_1}{f^2}}}{{{f_1}^2 + {f^2}}} + \frac{{{A_2}{P_2}{f_2}{f^2}}}{{{f_2}^2 + {f^2}}} + {A_3}{P_3}{f^2}}$$

Notation	Description	Units	Limits for $f < 500 \; \text{kHz}$	Limits for $f > 500 \; \text{kHz}$
$\alpha$	absorption	$\text{dB/km}$
$T$	temperature	$^{\circ}\text{C}$	$-2 < T < 22$	$0 < T < 30$
$S$	salinity	$\text{‰}$	$30 < S < 35$	$0 < S < 40$
$D$	depth	$\text{m}$	$0 < D < 3500$	$0 < D < 10000$
$f$	frequency	$\text{kHz}$	$10^{ - 2} < f$	$f < {10^3}$
$pH$	acidity

$${A_1} = \left( {8.86/C} \right) \times {10^{0.78pH - 5}}$$

$${f_1} = 2.8\sqrt {S/35} \times {10^{\left( {4 - 1245/{T_K}} \right)}}$$

$${A_2} = 21.44\left( {S/C} \right)\left( {1 + 0.025T} \right)$$

$${P_2} = 1 - \left( {1.37 \times {{10}^{ - 4}}} \right)D + \left( {6.2 \times {{10}^{ - 9}}} \right){D^2}$$

$${f_2} = \frac{{8.17 \times {{10}^{\left( {8 - 1990/{T_K}} \right)}}}}{{1 + 0.0018\left( {S - 35} \right)}}$$

$${A_3} = \left\{ {\begin{array}{c} {\begin{array}{l} {\left( {4.937 \times {{10}^{ - 4}}} \right) - \left( {2.590 \times {{10}^{ - 5}}} \right)T}\\ { + \left( {9.11 \times {{10}^{ - 7}}} \right){T^2} - \left( {1.5 \times {{10}^{ - 8}}} \right){T^3}} \end{array}}&{T \le 20}\\ {\begin{array}{l} {\left( {3.964 \times {{10}^{ - 4}}} \right) - \left( {1.146 \times {{10}^{ - 5}}} \right)T}\\ { + \left( {1.45 \times {{10}^{ - 7}}} \right){T^2} - \left( {6.5 \times {{10}^{ - 10}}} \right){T^3}} \end{array}}&{T > 20} \end{array}} \right.$$

$${P_3} = 1 - \left( {3.83 \times {{10}^{ - 5}}} \right)D + \left( {4.9 \times {{10}^{ - 10}}} \right){D^2}$$

$$C = 1412 + 3.21T + 1.19S + 0.0167D$$

Notation	Description	Units
$T_{K}$	$=T + 273$, temperature in Kelvin	$\text{K}$
$f_{1}$	boric acid relaxation frequency	$\text{kHz}$
$f_{2}$	magnesium sulfate relaxation frequency	$\text{kHz}$
$C$	sound speed	$\text{m/s}$
$\frac{{{A_1}{P_1}{f_1}{f^2}}}{{{f_1}^2 + {f^2}}}$	boric acid contribution	$\text{dB/km}$
$\frac{{{A_2}{P_2}{f_2}{f^2}}}{{{f_2}^2 + {f^2}}}$	magnesium sulfate contribution	$\text{dB/km}$
${A_3}{P_3}{f^2}$	pure water contribution	$\text{dB/km}$

Octave/Matlab implementation

function [alpha, Boric, MgSO4, H2O, C] = sound_absorption_sea_francois ...
    (T,S,D,f,pH)
% Inputs
%   T: temperature \ degree Celsius \ 
%      -2 < T < 22 for 10 Hz < f < 500 kHz
%      0 < T < 30 for f > 500 kHz
%   S: salinity \ ppt \
%       30 < S < 35 for 10 Hz < f < 500 kHz
%       0 < S < 40 for f > 500 kHz
%   D: depth \ m \ 
%       0 < D < 3500 for 10 Hz < f < 500 kHz
%       0 < D < 10000 for f > 500 kHz
%   f: frequency \ kHz
%   pH: "potential of hydrogen"
% Outputs
%   alpha: absorption of sound in seawater \ dB/km

    T_kel = 273 + T;

    % Sound speed
    C = 1412 + 3.21*T + 1.19*S + 0.0167*D;

    % Boric
    A1 = (8.86./C).*10.^(0.78.*pH-5);
    P1 = 1;
    f1 = 2.8 * sqrt(S./35).*10.^(4-1245./T_kel);
    Boric = (A1.*P1.*f1.*(f.^2))./((f.^2)+(f1.^2));

    % MgSO4
    A2 = 21.44*(S./C).*(1+0.025*T);
    P2 = 1 - (1.37e-4)*D + (6.2e-9)*(D.^2);
    f2 = (8.17*(10.^(8-1990./T_kel)))./(1+0.0018*(S-35));
    MgSO4 = (A2.*P2.*f2.*(f.^2))./((f.^2) + (f2.^2));

    % H2O
    if T <= 20
        A3 = (4.937e-4) - (2.590e-5)*T ...
            + (9.11e-7)*(T.^2) - (1.5e-8)*(T.^3);
    else
        A3 = (3.964e-4) - (1.146e-5)*T ...
            + (1.45e-7)*(T.^2) - (6.5e-10)*(T.^3);
    end
    P3 = 1 - (3.83e-5)*D + (4.9e-10)*(D.^2);
    H2O = A3*P3*(f.^2);

    % Total
    alpha = Boric + MgSO4 + H2O;
end

Computational examples

References

Francois, RE; Garrison, GR, "Sound absorption based on ocean measurements: Part I: Pure water and magnesium sulfate contributions", 1982
Francois, RE; Garrison, GR, "Sound absorption based on ocean measurements. Part II: Boric acid contribution and equation for total absorption", 1982