Image:Drum vibration mode21.gif

function main()
 
   k = 2; % k-th asimuthal number and bessel function
   p = 1; % p-th bessel root
 
   q=find_pth_bessel_root(k, p); 
 
   N=20; % used for plotting
 
   % 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_mode21.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!');