Question

starting from this code:

    clc, clear all, close all
tic

k1 = 0.01:0.1:100;
k2 = 0.01:0.1:100;
k3 = 0.01:0.1:100;

k = sqrt(k1.^2 + k2.^2 + k3.^2);

c = 1.476;
gamma = 3.9;

colors = {'cyan'};
Ek = (1.453*k.^4)./((1 + k.^2).^(17/6));
E = @(k) (1.453*k.^4)./((1 + k.^2).^(17/6));
E_int = zeros(1,numel(k1));
E_int(1) = 1.5;

for i = 2:numel(k)
    E_int(i) = E_int(i-1) - integral(E,k(i-1),k(i));
end

beta = c*gamma./(k.*sqrt(E_int));


F_11 = zeros(1,numel(k1));
F_22 = zeros(1,numel(k1));
F_33 = zeros(1,numel(k1));

count = 0;
for i = 1:numel(k1)
    count = count + 1;
    phi_11 = @(k2,k3) phi_11_new(k1,k2,k3,beta,i);
    phi_22 = @(k2,k3) phi_22_new(k1,k2,k3,beta,i);
    phi_33 = @(k2,k3) phi_33_new(k1,k2,k3,beta,i);
    F_11(count) = integral2(phi_11,-100,100,-100,100);
    F_22(count) = integral2(phi_22,-100,100,-100,100);
    F_33(count) = integral2(phi_33,-100,100,-100,100);
end

figure
hold on
plot(k1,F_11,'b')
plot(k1,F_22,'cyan')
plot(k1,F_33,'magenta')
hold off

where

function phi_11 = phi_11_new(k1,k2,k3,beta,i)
k = sqrt(k1(i).^2 + k2.^2 + k3.^2);
k30 = k3 + beta(i).*k1(i);
k0 = sqrt(k1(i).^2 + k2.^2 + k30.^2);
E_k0 = 1.453.*k0.^4./((1 + k0.^2).^(17/6));
C1 = (beta(i).*k1(i).^2).*(k1(i).^2 + k2.^2 - k3.*k30)./(k.^2.*(k1(i).^2 + k2.^2));
C2 = k2.*k0.^2./((k1(i).^2 + k2.^2).^(3/2)).*atan2((beta(i).*k1(i).*sqrt(k1(i).^2 + k2.^2)),(k0.^2 - k30.*k1(i).*beta(i)));
xhsi1 = C1 - k2./k1(i).*C2;
xhsi1_q = xhsi1.^2;
phi_11 = E_k0./(4.*pi.*k0.^4).*(k0.^2 - k1(i).^2 - 2.*k1(i).*k30.*xhsi1 + (k1(i).^2 + k2.^2).*xhsi1_q);
end

function phi_22 = phi_22_new(k1,k2,k3,beta,i)
k = sqrt(k1(i).^2 + k2.^2 + k3.^2);
k30 = k3 + beta(i).*k1(i);
k0 = sqrt(k1(i).^2 + k2.^2 + k30.^2);
E_k0 = 1.453.*k0.^4./((1 + k0.^2).^(17/6));
C1 = (beta(i).*k1(i).^2).*(k1(i).^2 + k2.^2 - k3.*k30)./(k.^2.*(k1(i).^2 + k2.^2));
C2 = k2.*k0.^2./((k1(i).^2 + k2.^2).^(3/2)).*atan2((beta(i).*k1(i).*sqrt(k1(i).^2 + k2.^2)),(k0.^2 - k30.*k1(i).*beta(i)));
xhsi2 = k2./k1(i).*C1 + C2;
xhsi2_q = xhsi2.^2;
phi_22 = E_k0./(4.*pi.*k0.^4).*(k0.^2 - k2.^2 - 2.*k2.*k30.*xhsi2 + (k1(i).^2 + k2.^2).*xhsi2_q);
end

function phi_33 = phi_33_new(k1,k2,k3,beta,i)
k = sqrt(k1(i).^2+k2.^2+k3.^2);
k30 = k3 + beta(i).*k1(i);
k0 = sqrt(k1(i).^2+k2.^2+k30.^2);
E_k0 = (1.453.*k0.^4./((1+k0.^2).^(17/6)));
phi_33 = (E_k0./(4*pi.*(k.^4))).*(k1(i).^2+k2.^2);
end

This procedure is leading me to results not matching some others coming from a study. The results I should match are posted below:

whereas mine look like these

It’s quite easy to esteem how only the comp w match the theoretical results; therefore, I believe that the flaw may reside in the definition of beta outside the function phi_11_new (and phi_22_new).

May any of you suggest how to calculate beta within phi_11_new(and phi_22_new) instead than the way I currently do?

I thank you all in advance for supporting.

Best Regards,
fpe

Answer 1

I have improved the interpolation so that it no longer breaks down for small values. It also returns more correct values since it now interpolates the logarithms of the values.

Here is the code, as it is now.

function test15()

[k1,k2,k3] = deal(0.01:0.1:400);

k = sqrt(k1.^2 + k2.^2 + k3.^2);

c = 1.476;
gamma = 3.9;

Ek = (1.453*k.^4)./((1 + k.^2).^(17/6));
E_int = 1.5-cumtrapz(k,Ek);
beta = c*gamma./(k.*sqrt(E_int));

[F_11,F_22,F_33] = deal(zeros(1,numel(k1)));

k_vec = k;
beta_vec = beta;

kLim = 100;

for ii = 1:numel(k1)
    phi_11 = @(k2,k3) phi_11_new(k1(ii),k2,k3,k_vec,beta_vec);
    phi_22 = @(k2,k3) phi_22_new(k1(ii),k2,k3,k_vec,beta_vec);
    phi_33 = @(k2,k3) phi_33_new(k1(ii),k2,k3,k_vec,beta_vec);
    F_11(ii) = quad2d(phi_11,-kLim,kLim,-kLim,kLim);
    F_22(ii) = quad2d(phi_22,-kLim,kLim,-kLim,kLim);
    F_33(ii) = quad2d(phi_33,-kLim,kLim,-kLim,kLim);
end

figure
loglog(k1,F_11,'b')
hold on
loglog(k1,F_22,'cyan')
loglog(k1,F_33,'magenta')
hold off
grid on

end

function phi_11 = phi_11_new(k1,k2,k3,k_vec,beta_vec)
k = sqrt(k1^2 + k2.^2 + k3.^2);

log_beta_vec = interp1(log(k_vec),log(beta_vec),log(k(:)),'linear','extrap');
log_beta = reshape(log_beta_vec,size(k));
beta = exp(log_beta);

k30 = k3 + beta*k1;
k0 = sqrt(k1^2 + k2.^2 + k30.^2);
E_k0 = 1.453*k0.^4./((1 + k0.^2).^(17/6));
C1 = (beta*k1^2).*(k1^2 + k2.^2 - k3.*k30)./(k.^2.*(k1^2 + k2.^2));
C2 = k2.*k0.^2./((k1^2 + k2.^2).^(3/2)).*atan2((beta*k1.*sqrt(k1^2 + k2.^2)),(k0.^2 - k30*k1.*beta));
xhsi1 = C1 - (k2/k1).*C2;
xhsi1_q = xhsi1.^2;
phi_11 = E_k0./(4.*pi.*k0.^4).*(k0.^2 - k1^2 - 2*k1*k30.*xhsi1 + (k1^2 + k2.^2).*xhsi1_q);
end

function phi_22 = phi_22_new(k1,k2,k3,k_vec,beta_vec)
k = sqrt(k1^2 + k2.^2 + k3.^2);

log_beta_vec = interp1(log(k_vec),log(beta_vec),log(k(:)),'linear','extrap');
log_beta = reshape(log_beta_vec,size(k));
beta = exp(log_beta);

k30 = k3 + beta*k1;
k0 = sqrt(k1^2 + k2.^2 + k30.^2);
E_k0 = 1.453*k0.^4./((1 + k0.^2).^(17/6));
C1 = (beta*k1^2).*(k1^2 + k2.^2 - k3.*k30)./(k.^2.*(k1^2 + k2.^2));
C2 = k2.*k0.^2./((k1^2 + k2.^2).^(3/2)).*atan2((beta*k1.*sqrt(k1^2 + k2.^2)),(k0.^2 - k30.*k1.*beta));
xhsi2 = (k2/k1).*C1 + C2;
xhsi2_q = xhsi2.^2;
phi_22 = E_k0./(4.*pi.*k0.^4).*(k0.^2 - k2.^2 - 2.*k2.*k30.*xhsi2 + (k1^2 + k2.^2).*xhsi2_q);
end

function phi_33 = phi_33_new(k1,k2,k3,k_vec,beta_vec)
k = sqrt(k1^2+k2.^2+k3.^2);

log_beta_vec = interp1(log(k_vec),log(beta_vec),log(k(:)),'linear','extrap');
log_beta = reshape(log_beta_vec,size(k));
beta = exp(log_beta);

k30 = k3 + beta*k1;
k0 = sqrt(k1^2+k2.^2+k30.^2);
E_k0 = (1.453*k0.^4./((1+k0.^2).^(17/6)));
phi_33 = (E_k0./(4*pi*(k.^4))).*(k1^2+k2.^2);
end

The figure seems to agree with the original result quite well. Even if there still are some differences.

Side note: Since a k-value of 100 is set as an upper limit in the simulation the values greater than this in the figure are incorrect. They are calculated without using all values in the full (k2,k3)-“circle”. We can also see a deviation for these values.