Sound absorption in sea. Ainslie & McColm 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
$\alpha$	absorption	$\text{dB/km}$
$T$	temperature	$^{\circ}\text{C}$	$-6 < T < 35$
$S$	salinity	$\text{‰}$	$5 < S < 50$
$D$	depth	$\text{m}$	$0 < D < 7000$
$f$	frequency	$\text{kHz}$	$10^{-1} < f < {10^3}$
$pH$	acidity		$7.7 < pH < 8.3$

$${A_1} = 0.106 \times \exp \left( {\left( {pH - 8} \right)/0.56} \right)$$

$${P_1} = 1$$

$${f_1} = 0.78\sqrt {S/35} \exp \left( {T/26} \right)$$

$${A_2} = 0.52\left( {S/35} \right)\left( {1 + T/43} \right)$$

$${P_2} = \exp \left( { - D/6} \right)$$

$${f_2} = 42\exp \left( {T/17} \right)$$

$${A_3} = 0.00049\exp \left( { - \left( {T/27 + D/17} \right)} \right)$$

$${P_3} = 1$$

Notation	Description	Units
$f_{1}$	boric acid relaxation frequency	$\text{kHz}$
$f_{2}$	magnesium sulphate relaxation frequency	$\text{kHz}$
$\frac{{{A_1}{P_1}{f_1}{f^2}}}{{{f_1}^2 + {f^2}}}$	boric acid contribution	$\text{dB/km}$
$\frac{{{A_2}{P_2}{f_2}{f^2}}}{{{f_2}^2 + {f^2}}}$	magnesium sulphate contribution	$\text{dB/km}$
${A_3}{P_3}{f^2}$	pure water contribution	$\text{dB/km}$

Octave/Matlab implementation

function [alpha, Boric, MgSO4, H2O] = sound_absorption_sea_ainslie(T,S,D,f,pH)
% Inputs
%   T: temperature \ degree Celsius
%   S: salinity \ ppt
%   D: depth \ m
%   f: frequency \ kHz
%   pH: "potential of hydrogen"
% Outputs
%   alpha: absorption of sound in seawater \ dB/km

    D = D*1e-3;

    % Boric
    A1 = 0.106*exp((pH-8)./0.56);
    P1 = 1;
    f1 = 0.78*sqrt(S./35).*exp(T./26);
    Boric = (A1.*P1.*f1*(f.^2))./((f.^2)+(f1.^2));

    % MgSO4
    A2 = 0.52*(S./35).*(1+T./43);
    P2 = exp(-D./6);
    f2 = 42*exp(T./17);
    MgSO4 = (A2.*P2.*f2*(f.^2))./((f.^2)+(f2.^2));

    % H2O
    A3 = 0.00049*exp(-(T./27+D./17));
    P3 = 1;
    H2O = A3.*P3.*(f.^2);

    % Total
    alpha = Boric + MgSO4 + H2O;
end

Computational examples

References

1. Ainslie, Michael A; McColm, James G, "A simplified formula for viscous and chemical absorption in sea water", 1998