Where's the maths? This is trigonometry — the cosine graph, used not as a triangle ratio but as a shape that waves up and down. A metal plate that is bowed or driven at just the right frequency does not wobble randomly: it settles into a standing pattern in which some parts swing up and down hard while other parts never move at all. The parts that never move are called nodal lines, and the height of the plate at the point (x, y) is described by a product of two cosines, one for each direction. A nodal line is simply the set of points where that expression is zero, so finding the pattern is the same job as solving cos(something) = 0 — a Sec 3/4 trigonometry question in two directions at once. Sand poured on the plate is thrown off wherever the plate moves and stays wherever it does not, so the sand is literally drawing the solution set of an equation. Change the two whole numbers m and n and you change the equation, and the sand redraws itself into a new figure.

Measure across the square plate with x and y both running from 0 to 1. The displacement of the plate — how far up or down it is at that point — is modelled by

u(x, y) = cos(mπx)·cos(nπy) ± cos(nπx)·cos(mπy)

with m and n whole numbers. Because mπx runs all the way from 0 to mπ, the angle goes far past 90°: this only works because cosine has been extended to obtuse and larger angles, which is exactly syllabus item 4.4. The graph of cos(mπx) is a cosine wave squashed by the factor m, so it crosses zero m times between the edges — that is the "periodicity" idea from the A-Math graphs topic.

Take the single term cos(mπx)cos(nπy) first. A product is zero when either factor is zero, so the nodal set is m straight lines running one way and n running the other: m + n straight lines, cutting the plate into (m + 1)(n + 1) vibrating regions — the sand-free tiles between the ridges. Set m = 4, n = 2 and count the sand ridges: 4 + 2 = 6.

Now add the swapped term. The plate is square, so if (m, n) is a possible pattern then so is (n, m), and any mixture of the two is possible as well. The + and − mixtures bend the straight lines into the diagonals, loops and stars that Chladni drew in 1787. Two things worth noticing: with − and m = n the two terms are identical and cancel completely, so there is no pattern at all; and swapping + for − turns each figure into a rotated or reflected version of the other.

The pitch you must play to get a figure grows with √(m² + n²), so the busier the figure, the higher the note. Sliding the frequency slider walks through the modes in that order — the same order in which a real plate finds them as you bow it faster.

Drag to orbit the plate, scroll or pinch to zoom. The m and n sliders set the mode; the frequency slider steps through every mode in order of pitch; Shake re-scatters the sand.