# Chladni Pattern Simulator - Vibrating Plate Sand Figures, Drum Bessel Modes

> Chladni pattern simulator: sweep the frequency and watch sand gather on nodal lines, play the tone, animate drum Bessel modes and mix degenerate modes.

- URL: https://qilingtools.com/en/physics/chladni
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

The Chladni plate tab simulates a square or rectangular metal plate covered in fine sand. Sweep the drive frequency through the resonances: sand is kicked away where the plate moves and collects on the still nodal lines, drawing the classic Chladni figures. Frequencies come from the thin-plate bending formula using the material, side length and thickness, and you can play the tone in your browser. The circular drum tab computes the Bessel functions J_m and their zeros numerically, shows each (m, n) mode as a 3D surface, heat map or sand figure, and lists the non-integer frequency ratios. The rectangular membrane tab shows how (n,m) and (m,n) become degenerate on a square and how the nodal lines morph as you mix them.

## How to use

1. On the Chladni plate tab, choose the material, side and thickness, then drag the drive frequency or tap a mode button to jump to a resonance
2. Watch the sand settle onto the nodal lines; switch to the heat map or 3D view, and tap Play to hear the pitch
3. On the circular drum tab, pick a drum preset or enter tension and areal density, then choose a mode in the m × n grid
4. On the rectangular membrane tab, keep the sides equal, pick a degenerate group and drag the mixing angle α

## Features

- Sand animation where bounce scales with amplitude, so grains collect on the nodal lines
- Plate resonances from E, ρ, ν, thickness and size, with a frequency sweep and the nearest resonant mode
- Numerical Bessel functions J_m and zeros j_mn, 3D drum-mode animation and a frequency-ratio table
- Rectangular-membrane modes grouped by frequency, with degeneracies marked and continuous mixing
- Three views: sand, amplitude map with nodal lines, and a lit 3D surface
- Web Audio sine tone that only plays when you tap and stops when you leave the page

## FAQ

### How do Chladni patterns form?

At a resonant frequency the plate vibrates as a standing wave, and some lines on it never move: the nodal lines. Sand where the plate moves strongly keeps getting bounced around at random, but near a nodal line nothing kicks it, so over time all of it piles up there and traces the skeleton of the standing wave. Ernst Chladni first showed these figures in the late 18th century by bowing the edge of a metal plate.

### Why do plate frequencies rise so quickly with mode number?

On a string or membrane the wave speed is fixed, so frequency is proportional to the wavenumber k. Bending waves in a thin plate obey ω = k²√(D/ρh), so frequency grows with k². The (n,m) plate modes therefore climb roughly as n² + m², and more intricate patterns sit at much higher pitches. A thicker or stiffer plate (larger D) rings higher; a denser one rings lower.

### Why doesn't a drum have a clear pitch?

A string's overtones are whole-number multiples of the fundamental (1, 2, 3…), and the ear fuses them into one note. A circular membrane's mode frequencies follow the Bessel zeros j_mn, giving ratios of 1, 1.59, 2.14, 2.30, 2.65… that aren't whole numbers, so the sound is closer to noise. Timpani get a clear pitch because air loading from the kettle and the strike position pull the (1,1), (2,1) and (3,1) modes close to a 2:3:4 ratio.

### What are degenerate modes?

On a square membrane the (1,2) and (2,1) modes have exactly the same frequency; they are degenerate. Any mix of the two is then also a mode at that frequency, so the nodal pattern isn't unique. Either one alone has a nodal line parallel to a side, while an equal mix turns it into a diagonal. Make the sides slightly different and the degeneracy breaks, the frequencies split and the pattern locks in. The ± Chladni figures of a square plate come from exactly this kind of mixing.

### How does the simulation differ from a real experiment?

A real free-edged square plate has no simple closed-form solution. This tool uses the classic cosine-superposition approximation and an empirical split between the + and − combinations, so plate frequencies and figures are approximate. The drum and rectangular membrane use the ideal membrane equation (no air loading or bending stiffness), so their frequency ratios are exact theory. The sand animation is qualitative: it only encodes that grains bounce harder where the amplitude is larger.

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