%%Attempt to solve a multiple scattering problem 

%Source term, num cylinders                 %Paramaters for crystal
kwav=10;                                     a=.47;                 %radius of cylinder (m)
N=2;                                        pho_scat=7.85;          %density of steel (g/cm^3)
nmax=4;                                     pho_air=1.29*10^-3;     %density of air (g/cm^3)
nvals=[-nmax:nmax];                         g_i=pho_scat/pho_air;   %density ratio
                                            cl_scat=4.51e3;         %long. sound vel. in steel (m/s)
                                            cl_air=343;             %long. sound vel. in air (m/s)
                                            h_i=cl_scat/cl_air;     %long.sound vel. ratio

%The Matrix will have   Columns------ N*(2*nmax+1) 
%                       Rows--------- N*(2*nmax+1)
num_columns=N*(2*nmax+1);
j=[1:num_columns];
k=[1:num_columns];

%Index each column by l and each row by n
l_lr=repmat(nvals,1,N);                     l_ud=l_lr';             n_ud=l_ud;
n=ceil(j/(2*nmax+1))';                      
[J,I]=meshgrid(n,n);                        [l_ind,n_ind]=meshgrid(l_lr,l_ud);
l_minus_n=l_ind-n_ind;                      del_i_j=I~=J;

%Arguments of the bessel function
ka=kwav.*a.*ones(num_columns,1);
kah=ka/h_i;
gh=g_i*h_i*ones(num_columns,1);         %Assumes that all cylinders are the same

%Set up for the cylinders----source is at rx=0,ry=0
rx=[2*ones(N,1)];           ry=[2*round([1:N]-((N+1)/2)),0]';          %Define Positions
%rx=[2,3]';                      ry=[0,0]';
ri=sqrt(rx(n).^2+ry(n).^2);                                      %Get Magnitude of ri
phi_ri=atan2(ry(n),rx(n));                                        %Get Angle of ri   
%phi_ri=atan(ry(n)./rx(n));                                       %Get Angle of ri   
ri_rj=sqrt((rx(I)-rx(J)).^2+(ry(I)-ry(J)).^2);                   %Calculate Mag Differences
phi_ri_rj_temp=atan2(del_i_j.*(ry(I)-ry(J)),(rx(I)-rx(J)));       %Get angles
%phi_ri_rj_temp=atan(del_i_j.*(ry(I)-ry(J))./(rx(I)-rx(J)));      %Get angles
phi_ri_rj=zeros(num_columns);                                    %Create array for phi vals
phi_ri_rj(find(del_i_j==1))=phi_ri_rj_temp(find(del_i_j==1));    %Keep i!=j

%Here are the coefficients for the matrix multiplication to solve
%     i  i      N       nmax      i,j  j     i
%gamma  A    - sum     sum      G     A   = T
%     n  n      j,j~=i  l=-nmax   l,n  l     nr
Gamma_i_n=(besselh(l_ud,ka).*.5.*(besselj(l_ud-1,kah)-besselj(l_ud+1,kah))-...
           gh.*.5.*(besselh(l_ud-1,ka)-besselh(l_ud+1,ka)).*besselj(l_ud,kah))./...           
          (gh.*.5.*(besselj(l_ud-1,ka)-besselj(l_ud+1,ka)).*besselj(l_ud,kah)-...
           besselj(l_ud,ka).*.5.*(besselj(l_ud-1,kah)-besselj(l_ud+1,kah)));
       
Gamma_i_n_mat=diag(Gamma_i_n);

T_i_n=besselh(-n_ud,kwav*ri).*exp(-i.*n_ud.*phi_ri);

Gijln_temp=besselh(l_minus_n,kwav.*ri_rj).*exp(i.*l_minus_n.*phi_ri_rj);
Gijln=zeros(num_columns);
Gijln(find(del_i_j==1))=Gijln_temp(find(del_i_j==1));    %Keep i!=j

%%Solve for the Coeeficients Alj
Ain=(Gamma_i_n_mat-Gijln)^-1*T_i_n;
pause;
clear('T_i_n','Gamma_i_n_mat','Gamma_i_n','Gijln_temp','Gijln','del_i_j','phi_ri_rj',...
      'phi_ri_rj_temp');

%%Assuming I didn't screw up big time

revx=[0:0.01:4];          revy=[-3:0.01:3];
maxrx=max(revx);          maxry=max(revy); 
minrx=min(revx);          minry=min(revy);

[revX,revY]=meshgrid(revx,revy);
rev=(sqrt(revX.^2+revY.^2));
pho_source_r=i*pi*besselh(0,rev);

%Create the 3 dimensional array to get this done.  Summing Ani terms
% i designates num of cyliner       n designates order of hankel function
[length,width]=size(revX);
i_3rd_dim=zeros(1,1,num_columns);         i_3rd_dim(1,1,:)=n;
i3d=repmat(i_3rd_dim,[length width 1]);

n_3rd_dim=zeros(1,1,num_columns);         n_3rd_dim(1,1,:)=l_lr;
n3d=repmat(n_3rd_dim,[length width 1]);

Ain_3rd_dim=zeros(1,1,num_columns);       Ain_3rd_dim(1,1,:)=Ain;
Ain_3d=repmat(Ain_3rd_dim,[length width 1]);

revX3d=repmat(revX,[1 1 num_columns]);    revY3d=repmat(revY,[1 1 num_columns]);
r_ri_3d=sqrt((revX3d-rx(i3d)).^2+(revY3d-ry(i3d)).^2);
phi_ri_3d=atan2((revY3d-ry(i3d)),(revX3d-rx(i3d)));
%phi_ri_3d=atan((revY3d-ry(i3d))./(revX3d-rx(i3d)));
[val,uniqueis]=unique(n);
inside_cyls=find(r_ri_3d(:,:,uniqueis) <= a);
[rcylx,rcyly,icyl]=ind2sub(size(phi_ri_3d),inside_cyls);
rcyls=sub2ind(size(revX),rcylx,rcyly);
pause
clear('icyl','i3d','revY3d','revX3d','revx','revy','revX','revY','rclyx','rcyly','icyl');

term_sum=sum(i*pi.*Ain_3d.*besselh(n3d,kwav*r_ri_3d).*exp(i.*n3d.*phi_ri_3d),3);
pho_r=pho_source_r+term_sum;
pho_r(rcyls)=-.5;
imagesc([minrx,maxrx],[minry,maxry],real(pho_r));