6 modes of a drum, and their frequency ratios

At its defaults it draws 6 modes of a drum, and their frequency ratios. Nodal-line diagrams for 6 modes of a circular membrane, each labelled with its frequency as a multiple of the lowest mode's. A mode (m, n) has m nodal diameters and n − 1 nodal circles, and the circles are drawn at the radii where the computed radial function J_m(j(m,n)·r/R) crosses zero — not at guessed fractions of the radius. The two tints are the two directions the head is moving in at that instant, and the lines between them are the parts of it that never move. The ratios are 1.000, 1.593, 2.136, 2.295, 2.653, 2.917: each one is a quotient of two zeros of Bessel functions, computed here from the power series and checked against their published values to 4.4e-7. Not one is a whole number, which is why a drum has no harmonic series and no pitch in the sense a string has one — and why these same ratios belong to every circular membrane ever stretched, whatever it is made of and however tightly it is pulled.

membrane-modes is one function in lib/figures/waves.js — travelling, standing, adding and shifting. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

Nodal-line diagrams for 6 modes of a circular membrane, each labelled with its frequency as a multiple of the lowest mode's. A mode (m, n) has m nodal diameters and n − 1 nodal circles, and the circles are drawn at the radii where the computed radial function J_m(j(m,n)·r/R) crosses zero — not at guessed fractions of the radius. The two tints are the two directions the head is moving in at that instant, and the lines between them are the parts of it that never move. The ratios are 1.000, 1.593, 2.136, 2.295, 2.653, 2.917: each one is a quotient of two zeros of Bessel functions, computed here from the power series and checked against their published values to 4.4e-7. Not one is a whole number, which is why a drum has no harmonic series and no pitch in the sense a string has one — and why these same ratios belong to every circular membrane ever stretched, whatever it is made of and however tightly it is pulled.

The number of vibration modes of a square with wavenumber below k, counted exactly — every eigenvalue of this region is a closed form, so the staircase is the true count and not an estimate. There are 265 of them below k = 60. The smooth curves are what Weyl's law predicts. The upper one is the leading term alone, the area times k² over 4π, and it is too high by 21.5 modes at the right-hand edge; the lower one subtracts the perimeter term, the perimeter times k over 4π, and is out by 2.4. The content of the law is that the count depends on the region through its area and its perimeter and — to this order — through nothing else at all: not through its shape, not through where its corners are, not through whether it is convex. The staircase's steps are the individual modes, and they cluster where two different pairs of indices give the same wavenumber. That the count is smooth in the large while being a staircase in the small is what makes a mode count usable in thermodynamics, where it appears as a density of states and never as a list.

Nodal-line diagrams for 6 modes of a circular membrane, each labelled with its frequency as a multiple of the lowest mode's. A mode (m, n) has m nodal diameters and n − 1 nodal circles, and the circles are drawn at the radii where the computed radial function J_m(j(m,n)·r/R) crosses zero — not at guessed fractions of the radius. The two tints are the two directions the head is moving in at that instant, and the lines between them are the parts of it that never move. The ratios are 1.000, 1.593, 2.136, 2.295, 2.653, 2.917: each one is a quotient of two zeros of Bessel functions, computed here from the power series and checked against their published values to 4.4e-7. Not one is a whole number, which is why a drum has no harmonic series and no pitch in the sense a string has one — and why these same ratios belong to every circular membrane ever stretched, whatever it is made of and however tightly it is pulled.

A square, a right triangle and an oblong four times as long as it is wide, all of exactly the same area, with the leading term of Weyl's law subtracted from each one's exact mode count. If the area were the whole story the three residuals would sit on top of one another. They do not: each falls along a straight line of its own, and the measured slopes are -0.323, -0.387, -0.384 against the −L/4π the law predicts, -0.318, -0.384, -0.398. The three shapes have perimeters 4.000, 4.828, 5.000 and that is exactly what separates them. So a spectrum reports two things about a region without being told either: its area, from how fast the count grows, and its perimeter, from how far the count falls behind. The square is the shape of least perimeter here and therefore the one with the most modes below any given k. What the spectrum does not report is the shape itself — two regions of the same area and perimeter can have different spectra, and famously two different shapes can have the same one.

The number of modes below a wavenumber for a string, a square membrane and a cubical box, all of unit size, on logarithmic axes. The three staircases straighten to slopes of 1.02, 2.04, 3.08, measured over the top fifth of the range and approaching one, two and three from above — the edge and surface corrections are deficits that shrink relative to the leading term, so a count is always still catching up. The count goes as the volume times k to the power of the dimension, and the constant in front is the volume of the unit ball divided by the appropriate power of 2π. That single fact is the origin of several results that look unrelated. The density of modes per unit frequency in three dimensions goes as ω², which is the factor in front of the Planck distribution and the reason the classical equipartition argument produced an ultraviolet catastrophe. The same ω² is what makes a solid's heat capacity rise as T³ at low temperature. And in one dimension the count is linear, which is why a string's modes are evenly spaced and a room's are not. Counting is the step that turns a wave equation into a thermodynamic quantity, and it is almost always done by this estimate rather than by listing anything.

The lowest 6 modes of a circular membrane and the first 6 harmonics of a string, on one axis of frequency, both in units of their own lowest note. The string's are the whole numbers, because a one-dimensional boundary admits whole numbers of half-wavelengths. The drum's are 1.000, 1.593, 2.136, 2.295, 2.653, 2.917 — the zeros of Bessel functions divided by the first zero of J₀ — and they interleave with the harmonics without ever landing on one. The closest approach on this drawing is mode (1,2) at 2.917, which is 2.8% below the 3rd harmonic, and the loudest partial, (2,1) at 2.136, misses the octave by 6.8%. A quarter of a semitone is about 1%, so these are not small errors that an ear rounds off: they are intervals with no name. The scale is set separately, and only the scale depends on the drum: at R = 0.325 m, T = 1900 N/m and σ = 0.262 kg/m² the lowest mode is 100.3 Hz and the rest follow it at these ratios.

physicscheck asserts something about membrane-modes that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.