Cleve's Corner

This is the second part of a series of posts about Marc Kac's 1966 paper in the American Mathematical Monthly [1]. This part is devoted to contour plots of the eigenfunctions.

Contents

Eigenfunctions

I described the isospectral drums in part 1. Contour plots of the eigenfunctions are beautiful. Here are the first twenty. The detail increases as the frequency increases. Notice the triangles created by the ninth eigenfunction. These triangles play a central role in the next part of this article.

   % Vertices
   drum1 = [0 0 2 2 3 2 1 1 0
            0 1 3 2 2 1 1 0 0];
   drum2 = [1 0 0 2 2 3 2 1 1
            0 1 2 2 3 2 1 1 0];
   vertices = {drum1,drum2};

   % Number of eigenvalues

   eignos = 20;

   % Grid size
   ngrid = 32;

   % Compute the eigenvalues and eigenfunctions

   h = 1/ngrid;
   [x,y] = meshgrid(0:h:3);

   inpoints = (7*ngrid-2)*(ngrid-1)/2;
   lambda = zeros(eignos,2);
   V = zeros(inpoints,eignos,2);

   % Loop over the two drums

   for d = 1:2
      vs = vertices{d};
      [in,on] = inpolygon(x,y,vs(1,:),vs(2,:));
      in = xor(in,on);

      % Number the interior grid points.

      G = double(in);
      p = find(G);
      G(p) = (1:length(p))';
      grid{d} = G;

      % The discrete Laplacian

      A = delsq(G)/h^2;

      % Sparse matrix eigenvalues and vectors.

      [V(:,:,d),E] = eigs(A,eignos,0);
      lambda(:,d) = diag(E);
   end

   % Plot the eigenfunctions.

   for d = 1:2
      for k = 1:eignos
         figure(ceil(k/2))
         set(gcf,'color','white')
         subplot(2,2,2*mod(k-1,2)+d)

         % Insert the k-th eigenvector in the grid interior.

         G = grid{d};
         p = find(G);
         u = zeros(size(G));
         u(p) = V(:,eignos+1-k,d);

         % Make first eigenvector positive so its color matches the others.

         if k == 1
            u = -u;
         end

         % Insert NaN's to make the exterior disappear.

         vs = vertices{d};
         [in,on] = inpolygon(x,y,vs(1,:),vs(2,:));
         u(~in) = NaN;

         % A filled contour plot with a line on the boundary.

         s = max(abs(u(:)));
         contourf(x,y,u,s*(-1:1/4:1))
         line(vs(1,:),vs(2,:),'color','black','linewidth',2)
         title(num2str(k))
         axis([-0.1 3.1 -0.1 3.1])
         axis square off
      end
   end

Continuous solution for ninth eigenfunction.

The ninth eigenfunction of either region is the first eigenfunction of the isosceles triangle subregion, reflected to fill out the entire region. The corresponding eigenvalue of the continuous problem is $5 \pi^2$.

$$ v_9 = \sin{2 \pi x} \sin{\pi y} - \sin{\pi x} \sin{2 \pi y} $$

   v9continuous = @(x,y) sin(2*pi*x).*sin(pi*y) - sin(pi*x).*sin(2*pi*y);

   figure(gcf+1)
   set(gcf,'color','white')
   for d = 1:2
      u = v9continuous(x,flipud(y));
      subplot(2,2,d)
      vs = vertices{d};
      [in,on] = inpolygon(x,y,vs(1,:),vs(2,:));
      u(~in) = 0.5*u(~in);
      s = max(abs(u(:)));
      contourf(x,y,u,s*(-1:1/4:1))
      line(vs(1,:),vs(2,:),'color','black','linewidth',3)
      title('Continuous v9')
      axis([-0.1 3.1 -0.1 3.1])
      axis square off
   end

References

Get the MATLAB code

Published with MATLAB® 7.14

Published with MATLAB® 7.14