function main()

   k = 2; % k-th asimuthal number and bessel function
   p = 2; % p-th bessel root

 

   % Get a grid
   R1=linspace(0.0, 1.0, N); 
   Theta1=linspace(0.0, 2*pi, N);
   [R, Theta]=meshgrid(R1, Theta1);
   X=R.*cos(Theta);
   Y=R.*sin(Theta);

   T=linspace(0.0, 2*pi/q, N); T=T(1:(N-1));

   for iter=1:length(T);
      
      t = T(iter);
      Z=sin(q*t)*besselj(k, q*R).*cos(k*Theta);

      figure(1); clf; 
      surf(X, Y, Z);
      caxis([-1, 1]);
      shading faceted;
      colormap autumn;

      % viewing angle
      view(108, 42);
      
      axis([-1, 1, -1, 1, -1, 1]);
      axis off;

      H=text(0, -0.3, 1.4, sprintf('(%d, %d) mode', k, p), 'fontsize', 25);

      
      file=sprintf('Frame%d.png', 1000 iter);
      disp(sprintf('Saving to %s', file));
      print('-dpng',  '-zbuffer',  '-r100', file);

      pause(0.1);
   end

   % converted to gif with the command 
   % convert -antialias -loop 10000 -delay 10  -scale 50% Frame10* Drum_vibration_mode22.gif

function r = find_pth_bessel_root(k, p)

   % a dummy way of finding the root, just get a small interval where the root is
   
   X=0.5:0.5:(10*p 1); Y = besselj(k, X);
   [a, b] = find_nthroot(X, Y, p);

   X=a:0.01:b; Y = besselj(k, X);
   [a, b] = find_nthroot(X, Y, 1);

   X=a:0.0001:b; Y = besselj(k, X);
   [a, b] = find_nthroot(X, Y, 1);

   r=(a b)/2;
   
function [a, b] = find_nthroot(X, Y, n)

   l=0;

   m=length(X);
   for i=1:(m-1)
      if ( Y(i) >= 0  & Y(i 1) <= 0 ) | ( Y(i) <= 0  & Y(i 1) >= 0 )
	 l=l 1;
      end

      if l==n
	 a=X(i); b=X(i 1);

	 %disp(sprintf('Error in finding the root %0.9g', b-a));
	 return;
      end
   end

   disp('Root not found!');