What Happened
A drumhead clamped vibrate in certain shapes, at certain frequencies. Those shapes and frequencies are the solutions of
Each solution u is a mode, a standing wave, and each λ gives a frequency proportional to √λ. eigenvalue problem, and for almost formula. So Eigendrum solves it numerically: it covers a mesh of triangles, builds the finite element stiffness and mass matrices, smallest eigenvalues of Kφ = λMφ.
A few shapes have spectra written down exactly, and the solver is tested against change. A circle's frequencies Bessel functions; a rectangle's are π²(m²/a² + n²/b²). The solver reproduces both to better than a tenth of a percent, and because a conforming finite element method minimises energy over a restricted space, its answers are guaranteed slight overestimates, never under. The measured “the numbers”.
Striking a spot drives proportion . Hit a line where a mode stands cannot excite . programmed in; projecting the mallet .
Why It Matters
So a strike : , in a mixture mallet landed. mixture, marked with a square mallet . Pressing a row instead plays that single mode alone - something no mallet can do, frequency of a shape actually sounds like.
Besides tracing an outline . r(t) gives the radius as t sweeps , so 1 + 0.3cos(5t) is a five-lobed flower; a parametric x(t), y(t) pair reaches the closed curves polar cannot, like a nephroid . a shortcut for drawing. It reaches shapes no hand traces accurately - eleven even lobes, a superellipse partway between a circle and a square - a shape something you vary: change one number .
A written shape travels . a formula holds the formula, something retype rather than a few hundred characters of encoded outline, and editing address bar works. Anything honestly is refused rather than answered, because a sliver would still return numbers .
exactly that in 1966. In 1992 Carolyn Gordon, Wolpert answered no, by building two different shapes with identical spectra. I and II. triangles, rearranged. They enclose perimeter, and every frequency matches. Switch between them and listen: the outlines are plainly different .
What Comes Next
The frequency ratios shapes are physics, fixed entirely by the outline. The absolute pitch depends tension, overtone fades depends on the material , sliders rather than silent assumptions. scaled before solving, .
The mallet is modelled too. a slider; its contact a few milliseconds, because no real beater is instantaneous equally hard. Both decide a mode a strike can reach, and neither can move a mode's frequency. Damping is Rayleigh damping, square of frequency: the high overtones , a drum darkens .
Eigendrum is hosted at eigendrum.com. address ; the older baselashraf81.github.io/eigendrum is a mirror that now redirects there.
For advertising or partnership enquiries, write to u2679054@uel.ac.uk. For anything interface, repository is better, because public.
application backend: the mesh, machine. The deployed site uses Vercel Analytics, Google Analytics and Google AdSense, domain . address #, which browsers a server, and analytics is configured not to record it. Details in the privacy notice. * by indestructible type*. After Kac, a Drum? (1966); Gordon, Webb and Wolpert (1992); and Driscoll, Eigenmodes of Isospectral Drums (1997), whose coordinates .
Source, including the solver against the closed-form spectra: github.com/BaselAshraf81/eigendrum
, with no account and nothing to install. something towards it, or would rather -supported: ko-fi.com/baselashraf
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