# Can One Hear the Shape of a Drum? Part 2, Eigenfunctions 1

Posted by **Cleve Moler**,

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

- Marc Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), 1-23.
- Carolyn Gordon, David Webb, Scott Wolpert, One cannot hear the shape of a drum, Bull. Amer. Math. Soc. 27 (1992), 134-138.
- Wikipedia, Hearing the shape of a drum.
- Cleve Moler, The MathWorks logo is an eigenfunction of the wave equation (2003).
- Lloyd N. Trefethen and Timo Betcke, Computed eigenmodes of planar regions (2005).
- Tobin Driscoll, Isospectral Drums.
- Tobin Driscoll, Eigenmodes of isospectral drums, SIAM Review 39 (1997), 1-17.
- S. J. Chapman, Drums that sound the same, Amer. Math. Monthly 102 (1995), 124-138.
- Pierre Berard, Transplantation et isospectralite, Math. Ann. 292 (1992), 547-559.

Get the MATLAB code Published with MATLAB® 7.14

**Category:**- Eigenvalues,
- Graphics

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