# Chladni Cymatics Plate

Interactive 2D Chladni plate vibration patterns — sand particles accumulate on nodal lines of plate eigenmodes, revealing beautiful geometric patterns governed by the Helmholtz equation

> Canonical page: https://elysiatools.com/en/visualizations/chladni-cymatics-plate

- **Category:** Physics

## Overview

Interactive 2D Chladni plate vibration pattern visualization. Sand particles accumulate on nodal lines of plate eigenmodes, governed by the Helmholtz equation ∇²u + k²u = 0. Three plate geometries: Square (sin/cos modes), Circle (Bessel function J_m(kr)·cos(mθ) modes), Triangle (superposition modes). Two boundary conditions: Fixed Edge (u=0 on boundary, sin/cos modes) and Free Edge (du/dn=0, cos modes). Sand particle system simulates real Chladni physics: particles drift toward nodal lines via gradient descent on |u(x,y)|², with thermal noise and damping. Three visualization panels: (1) Main Chladni plate with animated sand particles accumulating on nodal lines and mode amplitude background; (2) Vibration mode shape |u(x,y)| with hot colormap showing amplitude distribution; (3) Cross-section plot at y=0.5 with zero-crossing markers for nodal points. Adjustable parameters: mode numbers (n,m) 1-10, particle count 500-8000, animation speed, geometry (square/circle/triangle), boundary condition (fixed/free), driving frequency 50-2000 Hz. Six presets: Star Pattern (2,3), Diamond (3,3), Concentric Rings (circle, 3,0), Cross (1,1), Complex (5,7 free), Circle Mode (2,3 circle). Real-time statistics: mode number, computed eigenfrequency, theoretical wavelength, nodal line count, geometry, boundary type. Educational content covers Chladni's 1787 discovery, Helmholtz equation eigenmodes, Bessel function circular modes, sand migration physics, and applications in violin making, structural engineering, MEMS, concert hall acoustics, cymatics, and quantum mechanics analogies. Multi-language support (zh, en, es, fr, de, ru, pt).

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