Simple Integration equation HELP

Question

sese 2013-8-4

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/83957-simple-integration-equation-help

Dear friends, i am really in trable, i am trying to find the specific attenuation due to rain but i have problem in my code and i am asking you your help. My problem is in the last command which is an integration operation, and i am sure you will do your best for helping me. The link below is for the equations and it's followed by the code.

http://www.mediafire.com/view/u7ktl5tj5jy700u/nnnn.png

radius = 1;
nMax = 40; % maximum mode number
No=8*10^3;     %m^ -4 
R1=140;        %rainfall rate=1mm/hr 
AS1=8.2*R1^-0.21; 
N1D=No*exp(-AS1*radius*1e-3);    
% mode numbers
mode = 1:nMax; 
frequency = 6e9;
% speed of light
c = 299792458.0;
lambda = c / ( frequency ) ;
for n=1:10;
n2 = (2*n+1);
% radian frequency
w = 2.0*pi*frequency;
% wavenumber
k = w/c;
% conversion factor between cartesian and spherical Bessel/Hankel function
s = sqrt(0.5*pi/(k*radius));
% compute spherical bessel, hankel functions
[J(mode)] = besselj(mode + 1/2, k*radius); J = J*s;
[H(mode)] = besselh(mode + 1/2, 2, k*radius); H = H*s;
[J2(mode)] = besselj(mode + 1/2 - 1, k*radius); J2 = J2*s;
[H2(mode)] = besselh(mode + 1/2 - 1, 2, k*radius); H2 = H2*s;
% derivatives of spherical bessel and hankel functions
% Recurrence relationship, Abramowitz and Stegun Page 361
kaJ1P(mode) = (k*radius*J2 - mode .* J );
kaH1P(mode) = (k*radius*H2 - mode .* H );
% Ruck, et. al. (3.2-1)
An = -((1i).^mode) .* ( J ./ H ) .* (2*mode + 1) ./ (mode.*(mode + 1));
% Ruck, et. al. (3.2-2), using derivatives of bessel functions 
Bn = ((1i).^(mode+1)) .* (kaJ1P ./ kaH1P) .* (2*mode + 1) ./ (mode.*(mode + 1));
Qt = (lambda^2/2*pi)*sum(n2).*sum(real(An + Bn));
end
Co1=4.343*No;

==============================================================

NOW HOW TO FIND THE INTEGRATION OF "k", where k=integration of (Qt*N1D)

7 个评论
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Mike Hosea 2013-8-6

This question is a repeat of

http://www.mathworks.com/matlabcentral/answers/83917-integration-help-mie-scattering

Jan 2013-8-6

I got 5 emails today with questions which belongs to the forum. Two of them were double posts also. It seems to be the day of the nervous askers.

请先登录，再进行评论。

请先登录，再回答此问题。

Answer 1

Walter Roberson 2013-8-5

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在 MATLAB Online 中打开

Change

n2 = (2*n+1);

to

n2(n) = (2*n+1);

Change

An = -((1i).^mode) .* ( J ./ H ) .* (2*mode + 1) ./ (mode.*(mode + 1));
% Ruck, et. al. (3.2-2), using derivatives of bessel functions
Bn = ((1i).^(mode+1)) .* (kaJ1P ./ kaH1P) .* (2*mode + 1) ./ (mode.*(mode + 1));
Qt = (lambda^2/2*pi)*sum(n2).*sum(real(An + Bn));
end

to

    An(n) = -((1i).^mode) .* ( J ./ H ) .* (2*mode + 1) ./ (mode.*(mode + 1));
    % Ruck, et. al. (3.2-2), using derivatives of bessel functions
    Bn(n) = ((1i).^(mode+1)) .* (kaJ1P ./ kaH1P) .* (2*mode + 1) ./ (mode.*(mode + 1));
end
Qt = (lambda^2/2*pi)*sum(n2 .* real(An + Bn));
end

Also, in theory you should change

for n=1:10;

to

for n=1:inf

but you are unlikely to have an infinite amount of memory or an infinite amount of time to wait.

Now in addition to all of this, you should not be setting "radius" to 1.0, or frequency to 6e9 or running mode over 1:40 : you should be bundling everything into a function that takes radius and lambda and the mode number as parameters. You would then be invoking that function from within integrate() or quadgk(), using parameterization to pass in one specific lambda and mode number, such as

this_frequency = 6e9;
this_lambda = c ./ this_frequency;
this_mode_number = 5;
low_r = 0;
high_r = 173;
k = integrate(@(r) compute_Qt(r, this_lambda, this_mode_number), [low_r, high_r]);

Please review the equations and notice that only one mode number (m) is used, not a range of mode numbers. If for some reason you want to compute for different mode numbers, that should be done by a series of integrals.