Sound absorption in sea. Fisher & Simmons equation

Mathematical definition

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

Notation	Description	Units	Limits
$\alpha$	absorption	$\text{dB/km}$
$T$	temperature	$^{\circ}\text{C}$
$D$	depth	$\text{m}$
$f$	frequency	$\text{kHz}$

$${A_1} = \left( {1.03 \times {{10}^{ - 8}}} \right) + \left( {2.36 \times {{10}^{ - 10}}} \right)T - \left( {5.22 \times {{10}^{ - 12}}} \right){T^2}$$

$${P_1} = 1$$

$${f_1} = \left( {1.32 \times {{10}^3}} \right){T_K}\exp \left( { - 1700/{T_K}} \right)$$

$${A_2} = \left( {5.62 \times {{10}^{ - 8}}} \right) + \left( {7.52 \times {{10}^{ - 10}}T} \right)$$

$${P_2} = 1 - \left( {10.3 \times {{10}^{ - 4}}} \right)P + \left( {3.7 \times {{10}^{ - 7}}} \right){P^2}$$

$${f_2} = \left( {1.55 \times {{10}^7}} \right){T_K}\exp \left( { - 3052/{T_K}} \right)$$

$${A_3} = \left( {55.9 - 2.37T + \left( {4.77 \times {{10}^{ - 2}}} \right){T^2} - \left( {3.48 \times {{10}^{ - 4}}} \right){T^3}} \right) \times {10^{ - 15}}$$

$${P_3} = 1 - \left( {3.84 \times {{10}^{ - 4}}} \right)P + \left( {7.57 \times {{10}^{ - 8}}} \right){P^2}$$

Notation	Description	Units
$T_{K}$	$=T + 273.1$, temperature in Kelvin	$\text{K}$
$f_{1}$	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_fisher(T,D,f)
% Inputs
%   T: temperature \ degree Celsius
%   D: depth \ m
%   f: frequency \ kHz \ 10 < f < 400
% Outputs
%   alpha: absorption of sound in seawater \ dB/km

    T_kel = 273.1 + T;
    P = D ./ 10.0;
    f = f * 1e3;

    % Boric
    A1 = (1.03e-8) + (2.36e-10)*T - (5.22e-12)*(T.^2);
    P1 = 1;
    f1 = (1.32e3)*T_kel.*exp(-1700./T_kel);
    Boric = (A1.*P1.*f1.*(f.^2))./((f.^2)+(f1.^2));
    Boric = Boric * 8686;

    % MgSO4
    A2 = (5.62e-8) + (7.52e-10)*T;
    P2 = 1 - (10.3e-4)*P + (3.7e-7)*(P.^2);
    f2 = (1.55e7)*T_kel.*exp(-3052./T_kel);
    MgSO4 = (A2.*P2.*f2.*(f.^2))./((f.^2)+(f2.^2));
    MgSO4 = MgSO4 * 8686;

    % H2O
    A3 = (55.9 - 2.37*T + (4.77e-2)*(T.^2) - (3.48e-4)*(T.^3))*(1e-15);
    P3 = 1 - (3.84e-4)*P + (7.57e-8)*(P.^2);
    H2O = A3.*P3.*(f.^2);
    H2O = H2O * 8686;

    % Total
    alpha = (Boric + MgSO4 + H2O);
end

Computational examples

References

1. Fisher, FH; Simmons, VP, "Sound absorption in sea water", 1977
2. Lyman, John; Fleming, Richard H, "Composition of sea water", 1940