 # ChamAllard1991.m

### ChamAllard1991.m

Matlab code to calculate the normalised complex impedance and wavenumber for sound propagation in a porous material using the model proposed by the Champoux and Allard in their J. Appl. Phys. paper in 1991 [Journal of Applied Physics 70, 1975 (1991)]. Also calculates the following parameters for a hard-backed layer of thickness d: surface impedance (Zs), reflection coefficient (rfl), absorption coefficient (alpha), dynamic density (Rp) and complex compressibility (Cp).

Input parameters:
freq = array of frequency points [Hz];
sigma = viscous flow resistivity [Pa s /m2];
sigmaprime = thermal flow resistivity [Pa s /m2]
om = open porosity [-]
q = tortuosity (sqrt(alpha_inf)) [-]
Lam, = viscous length [m]
Lamprime = thermal characteristic lengths [m]
d = layer thickness [m]

```% function Zs, impd, wnmb, rfl, alpha, Rp, Cp] = ChamAllard1991(freq, sigma, sigmaprime, om, q, Lam, Lamprime, d) % Used to calculate the normalised impedance and the wavenumber % for propagation inside a porous material by the Champoux and Allard % model as described in JAP 1991 paper. % On Entry: % freq = frequency points; % sigma = viscous flow resistivity [N s /m4]; % sigmaprime = thermal flow resistivity [N s /m4] % om = porosity % q = tortuosity (sqrt(alpha_inf)) % Lam, Lamd = characteristic lengths, m % d - layer thickness [m]. % Define the key parameters: %%%%% %%%%% Room conditions %%%%% T = 20; % Temperature in Celsius degrees P0 = 101320; % [N.m-2] atmospheric pressure % Compute air parameters related to room conditions % (acknowledgment goes to Dominic Pilon, % dominic.pilon@metafoam.com, for gathering almost all these % expression from the references below) % % References: % Lide, D. R. and Kehiaian H. V., % CRC. Handbook of Thermophysical and Thermochemical Data, % CRC. Press Inc, 1994 % % Touloukian, Y. S. and Makita, T., % Specific Heat - Non metallic Liquids and gases, % The TPRC Data Series, Volume 6, IFI/PLENUM, 1970 % % Pierce, A. D., % Acoustics, An Introduction to Its Physical Principles and Applications % Acoustical Society of America, 2nd edition, 1989 % Convert temperature from Celsius to Kelvin T = 273.16+T; % Universal gas constant (J.K-1.kg-1) [also 8.314 J.mol-1.K-1] R = 287.031; % Specific heat at constant pressure (J.kg-1.K-1; 260 K < T < 600 K Cp = 4168.8*(0.249679-7.55179e-5*T+1.69194e-7*T^2-6.46128e-11*T^3); % Specific heat at constant volume (J.kg-1.K-1; 260 K < T < 600 K Cv = Cp-R; % Dynamic viscosity (N.s.m-2; 100 K < T < 600 K eta = 7.72488e-8*T-5.95238e-11*T^2+2.71368e-14*T^3; % Ratio of specific heats gam = Cp/Cv; % Density of air (kg.m-3) rho0 = P0/(R*T); % Velocity of sound (m.s^-1) c0 = sqrt(gam*R*T); % Thermal conductivity (W.m-1.K-1) - cf A. D. Pierce p 513 kappa = 2.624e-02 * ( (T/300)^(3/2) * (300+245.4*exp(-27.6/300))/(T+245.4*exp(-27.6/T)) ); % Prandtl's number Npr = eta*Cp/kappa; omega = 2*pi*freq; q2 = q^2; ko = eta/sigma; a = sqrt(8*q2*eta/sigma/om); F = q2/om; %Lam = sqrt(4*F*ko); %Lamd = om*b/(3*(1-om)); %Rd = 8*q2*eta/Lamd^2; % Complex tortuosity: alp = q2 - i*sigma*om/rho0./omega.*sqrt(1 + 4*i*q2^2*eta*rho0*omega/(sigma*Lam*om)^2); Rp = alp*rho0/om; %Cp = om*(gam - (gam-1)./(1-(i*sigmaprime*om./(rho0*q2*Npr*omega).*... % sqrt(1+4*i*q2^2*eta*rho0*Npr*omega/(sigmaprime*Lamprime*om)^2)))); Cp = om*(gam - (gam-1)./(1-(i*sigmaprime*om./(rho0*Npr*omega).*... sqrt(1+4*i*eta*rho0*Npr*omega/(sigmaprime*Lamprime*om)^2))))/gam/P0; Adm = sqrt(Cp./Rp); wnmb = sqrt(Cp.*Rp).*omega; impd = 1./Adm; Zs = impd.*coth(i*wnmb*d)/rho0/c0; rfl = abs((Zs - 1)./(Zs + 1)).^2; alpha = 1 - rfl; Cp = Cp*gam*P0; %disp('Acoustic characteristics done..');```