Two drums, one spectrum

Same notes, different weights

My model gives both drums the same 60 lowest frequencies. They agree to 14 significant digits; the largest relative gap is 1.7×10⁻¹⁴, which is floating-point rounding. What differs is how loud each note is.

Each vibration mode k has a shape φ_k. Push the whole skin evenly and mode k is excited in proportion to ∫φ_k, the net volume of air its shape sweeps. From across the room you hear it in proportion to ∫φ_k again. The notes line up exactly, but the stems below don’t match in height.

Push drum A and its strongest overtone is mode 2, at 1.200 times the fundamental, almost exactly a just minor third (6:5). Drum B barely sounds that note. It leans on mode 3 instead, at 1.428 times the fundamental, between a tritone and a fifth. That is the difference in the blind test.

Seven of the sixty modes (9, 21, 27, 38, 44, 55 and 60) have identical weights on both drums. In each one, a single triangle’s vibration is copied onto all seven triangles that make up the drum, so each drum is really doing the same thing.

Same frequency, different patterns

A mode’s nodal lines are the curves that stay still, where sand would collect on a Chladni plate. They cut the skin into nodal domains. For modes 1 to 3 both drums split into the same number of pieces. At mode 4, drum A has 3 domains and drum B has 4. Gnutzmann, Smilansky and Søndergaard suggested in 2005 that counting nodal domains could tell isospectral drums apart, and my counts agree.

Of the 48 modes whose counts held steady when I refined the mesh, 26 split differently on the two drums. In the other 12, two nodal lines nearly touch, and whether they join depends on the resolution, so I left those out of the tally.

They cool differently, but not at first

Warm both plates to the same temperature and hold their rims at zero. The heat left after time t is Q(t) = Σ_k e^−λ_kt (∫φ_k)². The decay rates λ_k match. The weights are the same (∫φ_k)² as above, and those don’t.

Until about t = 0.015 the two totals agree to 14 digits, the limit of double precision. That fits: the drums share their area, their perimeter and every corner angle (two each of 45°, 90°, 135° and 270°), so heat escaping near the rim behaves the same on both. The gap then switches on like e^−d²/4t with d between 1.1 and 1.2, about one triangle side. That is roughly how far heat has to spread before it can feel how the triangles are glued together. In the long run drum B holds 10.8% more heat, which is (∫φ₁^B / ∫φ₁^A)² − 1.

How this was computed

Each drum is seven right isosceles triangles with legs of length 1, for an area of 3.5. I cut every triangle into n² identical smaller triangles and solved −Δφ = λφ with linear finite elements, holding the rim fixed. Eigenpairs came from SciPy’s shift-invert Lanczos solver. Everything on this page uses n = 96, about 32,000 nodes per drum, except the heat curve, which needed every eigenvalue and uses n = 40.

Only one way of cutting works exactly

Each drum has two unit squares that can be split along either diagonal, which gives 4 cuttings per drum and 16 pairings. Only one pairing, the one drawn above, makes the two meshes isospectral to machine precision at every resolution. For every other pairing the gap shrinks like h², the ordinary discretization error.

Why the weights differ

The exact match comes from transplantation (Bérard; Buser, Conway, Doyle and Semmler). Any mode of drum A can be rebuilt as a mode of drum B by filling each B triangle with a signed sum of three A triangles. I recovered the recipe numerically from the per-triangle integrals of all 60 modes, and it holds to 10⁻¹¹. Every A triangle is used three times, but the signs don’t cancel evenly: the column sums are +1, −1, −1, +1, −1, −1 and −3. Because the total displacement gets reshuffled, ∫φ comes out different, and so do the thump and the cooling.

Accuracy checks

The lowest eigenvalue is 10.1624, 10.1576 and 10.1556 at n = 64, 96 and 128. It converges slowly because of the 270° corners and extrapolates to about 10.152. Mode 9 has an exact value, 5π² ≈ 49.348, because it is one triangle’s ground state copied seven times. At n = 96 the model gives 49.371, 0.05% high.

In the sound model, the fundamental is 98 Hz (G2), and each mode rings down with a time constant proportional to f^−0.9. A strike at point p gives mode k the amplitude ω_k φ_k(p) ∫φ_k: an impulse at p, heard as a small source set in a wall. Every sound is scaled to the same RMS loudness.

Sources

• Mark Kac, “Can one hear the shape of a drum?”, American Mathematical Monthly 73 (1966).
• C. Gordon, D. Webb, S. Wolpert, “One cannot hear the shape of a drum”, Bulletin of the AMS 27 (1992).
• T. Driscoll, “Eigenmodes of isospectral drums”, SIAM Review 39 (1997). Project page.
• S. Gnutzmann, U. Smilansky, N. Søndergaard, “Resolving isospectral ‘drums’ by counting nodal domains”, J. Phys. A 38 (2005).
• Cleve Moler, “Can one hear the shape of a drum?” (2012). The vertex coordinates used here come from this post, which also shows finite-difference versions are isospectral.