%%Attempt to solve a multiple scattering problem with a TRIANGULAR LATTICE

%Angular Momentum value                    %Paramaters for crystal
nmax=4;                                     f=.906;                  %filling factor
nvals=[-nmax:nmax];                         a=.2;                   %radius of cylinder (m)
                                            pho_scat=7.85;          %density of steel (g/cm^3)
%Wavelength,Lattice constants 
                                            pho_air=1.29*10^-3;     %density of air (g/cm^3)
lat_a=sqrt(2*pi*a^2/(sqrt(3)*f));           g_i=pho_scat/pho_air;   %density ratio
lambda=1.54*lat_a;                          cl_scat=4.51e3;         %long. sound vel. in steel (m/s)
kwav=2*pi/lambda;                           cl_air=343;             %long. sound vel. in air (m/s)
                                            h_i=cl_scat/cl_air;     %long.sound vel. ratio

%Set up for the cylinders----source is at rx=0,ry=0 -----lattice constant a
%This is a triangular lattice so that every other column is offset by a/2
%I is number of columns          J is the number of rows
%Total number of cylinders=floor(I/2)*J+ceil((I-1)/2)*(J-1)
I=8;                         J=11;                 
N=ceil(I/2)*J+ceil((I-1)/2)*(J-1);
%Lower Left Corner
xo=.5;  
%Values for the columns
index=(1:N);    col1=ones(J,1);             col2=zeros(J-1,1);
subcols=cat(1,col1,col2);
rx=xo+(sqrt(3)/2)*lat_a*(2*ceil(index/(2*J-1))'-repmat(subcols,I/2,1));
%Values for the Rows;
row1=(lat_a)*(-floor(J/2):floor(J/2));      row2=row1(1:J-1)+lat_a/2;
subrows=cat(1,row1',row2');
ry=repmat(subrows,I/2,1);
%Boundaries for Field
xright=ceil(max(rx)+5*lat_a);
ytop=ceil(max(ry)+2*lat_a);
ybot=floor(min(ry)-2*lat_a);
pause

%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

ri=sqrt(rx(n).^2+ry(n).^2);                                      %Get Magnitude of ri
phi_ri=atan2(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=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;

clear('T_i_n','Gamma_i_n_mat','Gamma_i_n','Gijln_temp','Gijln','del_i_j','phi_ri_rj',...
      'phi_ri_rj_temp','J','K','j','k');

%%Assuming I didn't screw up big time

revx=[-1:0.01:xright];    revy=[ybot:0.01:ytop];
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);

%Use for loop this time because computer can't handle the memory storage
term_sum=zeros(size(revX));
inside_cyls=[];

for in=1: num_columns
  
    if mod(in,2*nmax+1)==1
        r_ri=sqrt((revX-rx(n(in))).^2+(revY-ry(n(in))).^2);
        phi_ri=zeros(size(r_ri));
        phi_ri=atan2((revY-ry(n(in))),(revX-rx(n(in))));
        %Don't care about values inside cyls
        inside_cyls=cat(1,inside_cyls,find(r_ri <=a));
        outside_cyls=find(r_ri >a); 
        phi_ri(outside_cyls)=atan2((revY(outside_cyls)-ry(n(in))),...
                                   (revX(outside_cyls)-rx(n(in))));

    end
    
    term_sum(outside_cyls)=term_sum(outside_cyls)+...
                           i*pi.*Ain(in).*besselh(l_lr(in),kwav*r_ri(outside_cyls)).*...
                           exp(i.*l_lr(in).*phi_ri(outside_cyls));
end

pho_r=pho_source_r+term_sum;
pho_r(inside_cyls)=-.5;

%For pretty picture purposes---make length x width x 3 true color image
inside_cyls=repmat(inside_cyls,[1 1 3]);
pho_r_scaled=(real(pho_r)-min(real(pho_r(:))))/(max(real(pho_r(:)))-min(real(pho_r(:))));
pho_r3_scaled=repmat(pho_r_scaled,[1 1 3]);
pho_r3_scaled(inside_cyls(:,:,1))=1;
pho_r3_scaled(inside_cyls(:,:,2))=0;
pho_r3_scaled(inside_cyls(:,:,3))=0;

imagesc([minrx,maxrx],[minry,maxry],real(pho_r3_scaled));

%Attempt at a movie?
tmax=400;       omega=.65;
set(gcf,'doublebuffer','on');
for t=0:tmax
    pho_movie=exp(-i*omega*t).*pho_r;
    pho_movie(inside_cyls)=-.5;
    imagesc([minrx,maxrx],[minry,maxry],real(pho_movie),[-6 6]);colormap(gray);
    pause(0.1);
end