Sound absorption coefficient of membrane backed with air cavity

I am trying to learn the membrane backed by air cavity in the following paper, figure 2. My code result does not match the paper result. can somebody point out the error?

I Have used equation no 3, 4, 5, and 9 to obtain figure 2 (a)

Article Membrane sound absorber with a granular activated carbon infill

clc
clear all;
f= (100:2:1600);
omega= 2*pi*f;
rho_s=0.265; T=76.53*(1+1j*0.005); D=0.1; a=0.05; % Fig 2 a
c=343; rho_0=1.213; Z_0 = rho_0*c;
k0=omega/c;
km=omega.*sqrt(rho_s/T);
%% Impedance of membrane with air cavity only
Z_m = (1j*omega*rho_s)./(1-((2./km*a).*(besselj(1, km*a)./besselj(0,km*a))));
Z_w = -1j*Z_0*cot(k0*D);
Z_s = Z_m + Z_w;
Z_s =Z_s/Z_0;
R = (Z_s - 1)./(Z_s + 1);
alpha_1 = 1 - ((abs(R)).^2);
figure(1)
set(gca,'FontSize',16)
plot(f,alpha_1);
xlabel('Frequency (Hz)')
ylabel('Sound absorption coefficient')
grid on
grid minor
ylim([0 1])
set(gca, 'XScale', 'log')
set(gca, 'YScale', 'log')

Answer 1

[VBBV]

clc
clear all;
f= (100:2:1600);
omega= 2*pi*f;
rho_s=0.265; 
T=76.53*(1+1j*0.005);
D=0.1; 
a=0.05; % Fig 2 a
c=343; 
rho_0=1.213;
Z_0 = rho_0*c;
k0=omega/c;
km=omega.*sqrt(rho_s/T);
%% Impedance of membrane with air cavity only
% Z_m = (1j*omega*rho_s)./(1-((2./km*a).*(besselj(1, km*a)./besselj(0,km*a))));
Z_m = (1j*omega*rho_s)./(((besselj(0, km*a)./besselj(2,km*a))));
Z_w = -1j*Z_0*cot(k0*D);
Z_s = Z_m + Z_w;
Z_s = Z_s/Z_0;
R = (Z_s - 1)./(Z_s + 1);
alpha_1 = 1 - ((abs(R)).^2);
figure(1)
set(gca,'FontSize',16)
plot(f,alpha_1);  % check using semilogx
%xticklabels(xL)
xlabel('Frequency (Hz)')
ylabel('Sound absorption coefficient')
grid on
grid minor
ylim([0 1])
set(gca, 'XScale', 'log')

The plot was drawn using the below equation

Z_m = (1j*omega*rho_s)./(((besselj(0, km*a)./besselj(2,km*a))));  % Eq (3) where he writes as it
 % can also be written as,

and not with equation you used. Convert the log representation of xlabels using xticklabels

[VBBV]

The y axis is not plotted in log scale. Check also by changing different values for

a = 0.05  % use different values, see the response

[RAJA KUMAR]

First of all sir, I thank you for your valuable comment. I had not used the second equation as it had said that the both the equations are same. I will work on your suggestions and update you.

The paper had said that both equations give the same result but now it can be clearly seen that the both the equations are not even approximately same right ?

[VBBV]

Apparently, the results show that they are different equations, applied for a specific context in their study. if it solved your problem pls accept the answer,