What the Simulation Represents (Musical and Physical Meaning)

Your simulation represents Chladni figures — the standing wave patterns formed on a vibrating surface. It visually models how particles (like sand) settle into nodal lines, where no vibration occurs.

🎼 Relation to Music

Just like a vibrating string or air column produces musical notes through harmonics, a flat plate (such as in your simulation) produces complex modal patterns when excited at different frequencies.

• Each pattern corresponds to a physical resonant mode — the 2D equivalent of a harmonic.
• These modes are defined by:
  • n: the number of angular nodes (symmetrical slices, like overtones)
  • m: the number of radial nodes (concentric circles of no vibration)
• As frequency increases, both n and m increase, creating more intricate figures — just like higher harmonics on a musical instrument.

What you see is a direct spatial analogy of a musical spectrum. The plate becomes a visual instrument: sound shapes geometry.

🧠 Physical Background

• Resonance frequencies are computed using plate theory:

  fₙₘ = (αₙₘ² / 2π) × √(D / ρh) × 1 / R²

• Where D is flexural rigidity, ρ is density, h is thickness, and R is plate radius.
• The shapes are derived using Bessel functions (for radial behavior) and trigonometric harmonics (for angular behavior).

By mapping sound to vibration to visual pattern, this simulation beautifully links physics, mathematics, and music.