Drumhead
z = ∫₀^π cos(r · sin(u)) du/π · cos(t)
Open in the app The dials and keys named below are the app's.
What it draws
The height is Bessel's integral, computed at every mesh point: z = (1/π) ∫₀^π cos(r · sin u) du · cos t. The integral is J₀(r), the Bessel function, so the sheet is J₀(r) · cos t, a function of the distance r = hypot(x, y) from the centre alone. At the centre J₀(0) = 1. Out from it the function falls through zero at r ≈ 2.405, bottoms out at −0.403 near r ≈ 3.83, and crosses zero again at r ≈ 5.520 and at r ≈ 8.654. On a sheet of half-width 8 the first two circles are whole and the third clips the corners.
The factor cos t makes a standing wave. Every point rises and falls in step, the centre between +1 and −1, while the zero circles never move. The sheet flips sign every π ≈ 3.14 seconds at Speed 1.
Why the integral
cos(r · sin u) is a plane wave of wavelength 2π travelling in the direction u. Averaging over u adds up equal plane waves from every direction, and the sum is a circular ripple, J₀(r): at the centre every wave has a crest, along the zero circles they cancel.
The drum
A membrane stretched over a circular frame obeys the wave equation, and its circular standing waves are exactly J₀(k · r) · cos(ω · t), with the rim pinned where J₀ is zero. Cut this sheet at r = 2.405 and it is the fundamental of a drum of that radius, one dome breathing up and down. Cut it at r = 5.520 and it is the drum's second circular mode, the dome ringed by a trough with a still circle between. The zeros of J₀ are not evenly spaced, so the frequencies of these modes are not multiples of the fundamental: the second is 5.520/2.405 ≈ 2.295 times the first. A string's overtones are exact multiples, hence a string's pitch and a drum's thud.
History
Leonhard Euler met this function in 1764, in the vibrations of a stretched membrane. Friedrich Bessel studied the functions systematically in 1824, in the perturbations of planetary orbits, and the integral on screen is his.
Try
- Top: the still circles keep the ground's colour while the rings between them swap sign with the clock.
- Add a diameter, integral(cos(r·sin(u) − u), u, 0, π)/π · cos(θ) · cos(t): the integral is now J₁(r), the drum's first lopsided mode, its halves rising against each other across a still line along the y axis.
- Double the wavenumber, integral(cos(2·r·sin(u)), u, 0, π)/π · cos(t): that is J₀(2r), every circle at half its radius, the first at r ≈ 1.202.
- Side: the profile is J₀ itself, the trough only 0.403 of the centre dome.