# Mathematical expressions

In the case of multiple layers, Miki noticed that the real part of the surface impedance when computed with the original Delany-Bazley model sometimes becomes negative at low frequencies denoting a non-physical result. From Delany and Bazley measurements data realized 20 years before on fibrous materials with porosities close to 1.00, Miki proposes to use the following expressions for the wave number and characteristic impedance (a $+j\omega t$ time convention is assumed):

$Z_{c} = \rho_{0} c_{0} \Bigg[ 1+5.50 \left(10^3\frac{f}{\sigma}\right)^{-0.632} - j 8.43 \left(10^3\frac{f}{\sigma}\right)^{-0.632} \Bigg]$
$k = \displaystyle{\frac{\omega}{c_{0}}} \Bigg[ 1 + 7.81 \left(10^3\frac{f}{\sigma}\right)^{-0.618} - j 11.41 \left(10^3\frac{f}{\sigma}\right)^{-0.618} \Bigg]$

The elapsed time between the original Delany-Bazley model and Miki modifications of it explains, partly, the extensive use of Delany-Bazley expressions although Miki corrections are preferable to use.

The boundaries, proposed by Delany and Bazley, for the validity of their original expressions are:

$0.01 < \displaystyle{\frac{f}{\sigma}} < 1.00$

Miki takes care not to extrapolate these boudaries even if he had observed his revised expressions are well behaved in a larger frequency range, in particular for $f/\sigma < 0.01$

# Script

Check this basic online Delany-Bazley-Miki simulation or follow the link comparison_DB70_DBM90.m to download a Matlab/GNU Octave script which compares the model by Delany-Bazley and the model by Delany-Bazley-Miki:

% comparison of models by Delany-Bazley
% and Delany-Bazley-Miki
% for normal sound absorption coefficient
%
% M. E. Delany and E. N. Bazley,
% Acoustical properties of fibrous absorbent materials,
% Applied Acoustics (3), 1970, pp. 105-116
%
% Y. Miki
% Acoustical properties of porous materials
% - modifications of Delany-Bazley models -
% J. Acoust. Soc. Jpn. (E), 11 (1), 1990, pp. 19-24
%
% Copyleft 2006 luc.jaouen@matelys.com
% cf. APMR on the web,
% PropagationModels/MotionlessSkeleton/DelanyBazleyMikiModel.html

close all
clear all

f = [200:5:8000];
omega = 2*pi*f;

rho_0 = 1.213;      % [Kg.m-3] density at rest of air at 18C, 1atm
c_0   = 342.2;      % [m.s-1] speed of sound in air at 18C, 1atm
P_0   = 1.0132e+05; % [N.m-2] atmospheric pressure at 18C, 1atm
sigma = 10000       % [N.s.m-4] static air flow resistivity of material
h     = 0.05        % [m] thickness of material

%%%%%
%%%%% Compute variable X and print frequency
%%%%% limits of validity for the two models
%%%%%
X = f/sigma;
f_min = 0.01*sigma
f_max = 1.00*sigma

%%%%%
%%%%% Delany and Bazley model
%%%%% (NB: gamma = alpha + j beta = j k )
%%%%%

Z_DB70 = rho_0*c_0*( 1 + 9.08*(X*1000).^(-0.75) ...
- i*11.9*(X*1000).^(-0.73) );

k_DB70 = omega/c_0 .* (-i) .* ( 10.3*(X*1000).^(-0.59) ...
+ i* ( 1 + 10.8*(X*1000).^(-0.70) ) );

K_DB70 = Z_DB70.*omega./k_DB70;
rho_DB70 = k_DB70.*Z_DB70./omega;

%%%%%
%%%%% Revised expressions of Delany and Bazley model by Miki
%%%%% (NB: gamma = alpha + j beta = j k )
%%%%%

Z_DB70_Mik90 = rho_0*c_0*( 1 + 5.50*(X*1000).^(-0.632) ...
- i*8.43*(X*1000).^(-0.632) );

k_DB70_Mik90 = omega/c_0 .* (-i) .* ( 11.41*(X*1000).^(-0.618) ...
+ i* (1 + 7.81*(X*1000).^(-0.618) ) );

K_DB70_Mik90 = Z_DB70_Mik90.*omega./k_DB70_Mik90;
rho_DB70_Mik90 = k_DB70_Mik90.*Z_DB70_Mik90./omega;

%%%%%
%%%%% Compute normal sound absorption using the two
%%%%% models for a sample of thickness d backed by a
%%%%% rigid and impervious wall under at room
%%%%% temperature and pressure conditions
%%%%%

Z = -j.*Z_DB70./tan(k_DB70*h);
alpha_DB70 = 1 - ( abs( (Z-rho_0*c_0)./(Z+rho_0*c_0) ) ).^2;

Z = -j.*Z_DB70_Mik90./tan(k_DB70_Mik90*h);
alpha_DB70_Mik90 = 1 - ( abs( (Z-rho_0*c_0)./(Z+rho_0*c_0) ) ).^2;

%%%%%
%%%%% Compare results
%%%%%

figure(1)
set(gca,'FontSize',16)
plot(f,real(rho_DB70)/rho_0,'k-','LineWidth',2)
hold on
plot(f,imag(rho_DB70)/rho_0,'b-','LineWidth',2)
plot(f,real(rho_DB70_Mik90)/rho_0,'r--','LineWidth',2)
plot(f,imag(rho_DB70_Mik90)/rho_0,'m--','LineWidth',2)
xlabel('Frequency (Hz)')
ylabel('Normalized dynamic density')
legend('DB70 (Re)','DB70 (Im)','DB70+Mik90 (Re)','DB70+Mik90 (Im)',4)

figure(2)
set(gca,'FontSize',16)
plot(f,real(K_DB70)/P_0,'k-','LineWidth',2);
hold on
plot(f,imag(K_DB70)/P_0,'b-','LineWidth',2);
plot(f,real(K_DB70_Mik90)/P_0,'r--','LineWidth',2);
plot(f,imag(K_DB70_Mik90)/P_0,'m--','LineWidth',2);
xlabel('Frequency (Hz)')
ylabel('Normalized dynamic bulk modulus')
legend('DB70 (Re)','DB70 (Im)','DB70+Mik90 (Re)','DB70+Mik90 (Im)',4)

figure(3)
set(gca,'FontSize',16)
plot(f,alpha_DB70,'k-','LineWidth',2);
hold on
plot(f,alpha_DB70_Mik90,'r--','LineWidth',2);
xlabel('Frequency (Hz)')
ylabel('Normal sound absorption coefficient')
legend('DB70','DB70+Mik90',4)