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

### Contents

#### Isospectrality

Kac's article is not actually about a drum, which is three-dimensional, but rather about the two-dimensional drum head, which is more like a tambourine or membrane. The vibrations are modeled by the partial differential equation $$ \Delta u + \lambda u = 0 $$ where $$ \Delta u(x,y) = \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} $$ The boundary conditions are the key. Requiring $u(x,y) = 0$ on the boundary of a region in the plane corresponds to holding the membrane fixed on that boundary. The values of $\lambda$ that allow nonzero solutions, the*eigenvalues*, are the squares of the frequencies of vibration, and the corresponding functions $u(x,y)$, the

*eigenfunctions*, are the modes of vibration. The MathWorks logo comes from this partial differential equation, on an L-shaped domain [2], [3], but this article is not about our logo. A region determines its eigenvalues. Kac asked about the opposite implication. If one specifies all of the eigenvalues, does that determine the region? In 1991, Gordon, Webb and Wolpert showed that the answer to Kac's question is "no". They demonstrated a pair of regions that had different shapes but exactly the same infinite set of eigenvalues [4]. In fact, they produced several different pairs of such regions. The regions are known as "isospectral drums". Wikipedia has a good background article on Kac's problem [5]. I am interested in finite difference methods for membrane eigenvalue problems. I want to show that the finite difference operators on these regions have the same sets of eigenvalues, so they are also isospectral. I was introduced to isospectral drums by Toby Driscoll, a professor at the University of Delaware. A summary of Toby's work is available at his Web site [6]. A reprint of his 1997 paper in SIAM Review is also available from his Web site [7]. Toby developed methods, not involving finite differences, for computing the eigenvalues very accurately. The isospectral drums are not convex. They have reentrant $270^\circ$ corners. These corners lead to singularities in most of the eigenfunctions -- the gradients are unbounded. This affects the accuracy and the rate of convergence of finite difference methods. It is possible that for convex regions the answer to Kac's question is "yes".

#### Vertices

I will look at the simplest known isospectral pair. The two regions are specified by the*xy*-coordinates of their 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};Let's first plot the regions.

clf shg set(gcf,'color','white') for d = 1:2 % Plot the region. vs = vertices{d}; subplot(2,2,d) plot(vs(1,:),vs(2,:),'k.-'); axis([-0.1 3.1 -0.1 3.1]) axis square title(sprintf('drum%d',d)); end

#### Finite difference grid

I want to investigate simple finite difference methods for this problem. The MATLAB function`inpolygon`determines the points of a rectangular grid that are in a specified region.

% Generate a coarse finite difference grid. ngrid = 5; h = 1/ngrid; [x,y] = meshgrid(0:h:3); % Loop over the two regions. for d = 1:2 % Determine points inside and on the boundary. vs = vertices{d}; [in,on] = inpolygon(x,y,vs(1,:),vs(2,:)); in = xor(in,on); % Plot the region and the grid. subplot(2,2,d) plot(vs(1,:),vs(2,:),'k-',x(in),y(in),'b.',x(on),y(on),'k.') axis([-0.1 3.1 -0.1 3.1]) axis square title(sprintf('drum%d',d)); grid{d} = double(in); end

#### Finite difference Laplacian

Defining the 5-point finite difference Laplacian involves numbering the points in the region. The`delsq`function generates a sparse matrix representation of the operator and a

`spy`plot of the nonzeros in the matrix shows its the band structure.

for d = 1:2 % Number the interior grid points. G = grid{d}; p = find(G); G(p) = (1:length(p))'; % Display the numbering. fprintf('grid%d =',d); minispy(flipud(G)) % Discrete Laplacian. A = delsq(G); % Spy plot subplot(2,2,d) markersize = 6; spy(A,markersize) title(sprintf('delsq(grid%d)',d)); end

grid1 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 . . . . . . . . . . . . . . 48 55 . . . . . . . . . . . . . 41 47 54 . . . . . . . . . . . . 35 40 46 53 . . . . . . . . . . . 30 34 39 45 52 60 63 65 66 . . . . . . 26 29 33 38 44 51 59 62 64 . . . . . . 18 25 28 32 37 43 50 58 61 . . . . . . 11 17 24 27 31 36 42 49 57 . . . . . . 5 10 16 23 . . . . . . . . . . . . 4 9 15 22 . . . . . . . . . . . . 3 8 14 21 . . . . . . . . . . . . 2 7 13 20 . . . . . . . . . . . . 1 6 12 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . grid2 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 . . . . . . . . . . . . . . . 56 62 . . . . . . . . . . . . . . 55 61 65 . . . . . . . . . . . . . 54 60 64 66 . . 5 11 18 26 30 34 38 42 46 50 53 59 63 . . . 4 10 17 25 29 33 37 41 45 49 52 58 . . . . 3 9 16 24 28 32 36 40 44 48 51 . . . . . 2 8 15 23 27 31 35 39 43 47 . . . . . . 1 7 14 22 . . . . . . . . . . . . . 6 13 21 . . . . . . . . . . . . . . 12 20 . . . . . . . . . . . . . . . 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

#### Compare eigenvalues

The Arnoldi method implemented in the`eigs`function readily computes the eigenvalues and eigenvectors. Here are the first twenty eigenvalues.

% How many eigenvalues? eignos = 20; % A finer grid. ngrid = 32; 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); 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 format long lambda = flipud(lambda)

lambda = 10.165879621248989 10.165879621248980 14.630600866993412 14.630600866993376 20.717633982094981 20.717633982094902 26.115126153750705 26.115126153750676 28.983478457829740 28.983478457829762 36.774063407607322 36.774063407607287 42.283017757114585 42.283017757114699 46.034233949715308 46.034233949715386 49.213425509524733 49.213425509524775 52.126973962396342 52.126973962396370 57.063486161172904 57.063486161172769 63.350675017756465 63.350675017756259 67.491111510445251 67.491111510445194 70.371453210957910 70.371453210957810 75.709992784621988 75.709992784621818 83.153242199788906 83.153242199788821 84.673734481953971 84.673734481953886 88.554340162610174 88.554340162610174 94.230337192953158 94.230337192953257 97.356922250794767 97.356922250794653

#### How about a proof?

Varying the number of eigenvalues,`eignos`, and the grid size,

`ngrid`, in this script provides convincing evidence that the finite difference Laplacians on the two domains are isospectral. But this is not a proof. For the continuous problem, Chapman [8] outlines an approach where any eigenfunction on one of the domains can be constructed from triangular pieces of the corresponding eigenfunction on the other domain. It is necessary to prove that these pieces fit together smoothly and that the differential equation continues to be satisfied across the boundaries. For this proof Chapman refers to a paper by Berard [9]. I will explore the discrete analog of these arguments in a later post.

#### References

- Marc Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), 1-23.
- 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).
- 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.
- 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.

*Numerical Computing with MATLAB*, and check out pdegui.m.

**Category:**- Eigenvalues

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