What is the 2D wave equation?

The 2D wave equation ∂²u/∂t² = c²(∂²u/∂x² + ∂²u/∂y²) governs transverse displacement u(x,y,t) of a membrane under tension. The constant c = √(T/ρ) is the wave speed (T=tension, ρ=surface mass density). Solutions with fixed boundary conditions u=0 on the boundary are superpositions of normal modes u_{mn}(x,y,t) = sin(mπx/W)·sin(nπy/H)·cos(ω_{mn}t) with frequencies ω_{mn}=πc√(m²/W²+n²/H²).

What are Chladni patterns?

Chladni patterns (Ernst Chladni, 1787) are the nodal line figures that appear when a vibrating plate is resonated at one of its natural frequencies and fine sand is sprinkled on it. Sand migrates away from antinodes (maximum vibration) toward nodal lines (zero vibration), making the invisible standing-wave pattern visible. The same principle underlies acoustic levitation, cymatic art, and structural vibration analysis.

How stable is the FDTD scheme?

The explicit finite-difference time-domain (FDTD) scheme is conditionally stable. The Courant-Friedrichs-Lewy (CFL) condition requires r = c·Δt/Δx ≤ 1/√2 ≈ 0.707. This simulator uses c=1, Δt=0.40, Δx=1, giving r=0.40, well within the stability limit. Exceeding the CFL limit causes exponential growth of numerical errors — a genuine instability, not a physical phenomenon.

Can I draw waves by clicking the canvas?

Yes. Click or drag on the canvas to inject a Gaussian displacement pulse at that location. This is equivalent to plucking the membrane at a chosen point and is useful for exploring how waves reflect off the boundary, how interference patterns build up from multiple excitations, and how different initial positions excite different combinations of normal modes.

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🧪 2D Wave Equation & Chladni Patterns

Interactive 2D wave equation simulator: FDTD finite-difference solver on a 220×220 grid. Visualise rectangular membrane drumhead modes, Chladni nodal patterns, two-source interference, and Gaussian pluck propagation. Fixed boundary conditions, adjustable damping and speed.

Physics & Mechanics2DModerate60 FPS

⚙ Under the hood

This simulation explores the 2D wave equation and its application in generating Chladni patterns on a drumhead. Users can observe how vibrations create intricate geometric patterns based on resonance.

Wave EquationChladni PatternsDrumhead Vibration

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install