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.

2D Wave Equation & Chladni Patterns

FDTD drumhead simulator — click the membrane to pluck it, or choose a standing-wave mode

Preset: —

Frame: 0

Energy: 0

∂²u/∂t² = c²∇²u

Speed 1.0×

Display Amplitude 1.0

Damping γ 0.0030

The 2D Wave Equation

A thin elastic membrane stretched over a rigid frame satisfies the 2D wave equation:

∂²u/∂t² = c²(∂²u/∂x² + ∂²u/∂y²) − γ·∂u/∂t where: u(x,y,t) = transverse displacement c = √(T/ρ) wave speed (T = tension, ρ = surface mass density) γ = damping coefficient (energy dissipation) Fixed BCs: u = 0 on all four edges (Dirichlet)

FDTD discretisation: The finite-difference time-domain (FDTD) update rule is: u_new[i,j] = (2·u[i,j] − u_old[i,j]·(1−γΔt) + r²·Lap[i,j]) / (1+γΔt) where r = cΔt/Δx = 0.40 and Lap[i,j] = u[i+1,j]+u[i-1,j]+u[i,j+1]+u[i,j-1]−4u[i,j]. The CFL stability condition r ≤ 1/√2 ≈ 0.707 is satisfied.

Normal Modes of a Rectangular Membrane

For a rectangular membrane of width W and height H with fixed boundaries, the exact normal mode solutions are:

u_{mn}(x,y,t) = A·sin(mπx/W)·sin(nπy/H)·cos(ω_{mn}t + φ) Angular frequencies: ω_{mn} = πc·√(m²/W² + n²/H²) Natural frequencies: f_{mn} = (c/2)·√(m²/W² + n²/H²) For a square membrane (W=H): f_{mn} = (c/2W)·√(m² + n²) Examples (c=1, W=H=1): f₁₁ = c√2 / 2W (fundamental) f₂₁ = f₁₂ = c√5 / 2W (degenerate: two modes at same frequency!) f₂₂ = c√8 / 2W = 2f₁₁ (first overtone at exactly 2×fundamental) f₃₁ = f₁₃ = c√10 / 2W f₃₂ = f₂₃ = c√13 / 2W

Degeneracy: On a square membrane, modes with the same value of m²+n² are degenerate — they share the same frequency but have different spatial patterns. A square drum has many more degenerate pairs than a rectangular drum, making its timbre richer.

Chladni Patterns

Ernst Chladni (1756–1827) famously scattered sand on vibrating metal plates and excited them with a bow. The sand migrated to the nodal lines — curves where u = 0 at all times — forming beautiful geometric patterns. In the simulation the white regions near zero displacement visualise exactly these nodal figures.

Mode (m,n)	Frequency ratio to f₁₁	Nodal lines
(1,1) Fundamental	1.00	None (entire membrane moves in phase)
(2,1) / (1,2)	√(5/2) ≈ 1.58	1 vertical or 1 horizontal nodal line
(2,2)	2.00	1 vertical + 1 horizontal (cross pattern)
(3,1) / (1,3)	√5 ≈ 1.78	2 parallel nodal lines
(3,2) / (2,3)	√(13/2) ≈ 1.80	2 + 1 = 3 nodal lines (Chladni figure)
(3,3)	3.00	2 + 2 = 4 nodal lines (checker pattern)

Applications

Musical acoustics: Membrane modes determine the harmonic content (timbre) of drums and tympani
Structural engineering: Vibration modes of bridges and aircraft panels identified by Chladni-like scanning
Acoustic levitation: Standing-wave antinodes support small objects — used in space manufacturing
Seismology: Normal modes of the Earth (spheroidal and toroidal free oscillations) are the 3D analogue
Quantum mechanics: The Schrödinger equation for a particle-in-a-box in 2D has the identical eigenmode structure
Cymatics: Art and science of making sound waves visible via Chladni figures, Faraday waves, and Kundt's tube

Frequently Asked Questions

Why does the square membrane have degenerate modes?

Two modes (m,n) and (n,m) have the same frequency ω = πc√(m²+n²)/W on a square membrane whenever m ≠ n. Because both satisfy the wave equation and boundary conditions at the same frequency, any linear combination is also a valid solution. In practice this means the shape of the pattern depends on the symmetry of the driving force — exciting from the corner favours symmetric combinations, from the edge asymmetric ones. On a non-square rectangle there is no such degeneracy (generically), explaining why rectangular drums sound simpler.

What does energy represent in the HUD?

The HUD Energy readout is proportional to the sum of squared displacements Σu² across all grid cells — a proxy for the potential energy of the membrane (true potential energy is ½T·Σ|∇u|² and kinetic energy is ½ρ·Σ(∂u/∂t)²; their sum is conserved in undamped motion). In the Pluck Centre preset with zero damping, watch the energy oscillate as it converts between potential (displacement) and kinetic (velocity) form, just like a spring-mass system.

How is this related to the quantum particle in a 2D box?

Mathematically identical. The Schrödinger equation for a free particle in a 2D infinite square well is −ℏ²/(2m)·∇²ψ = E·ψ, which has eigenfunctions ψ_{mn} = sin(mπx/L)·sin(nπy/L) and energies E_{mn} = ℏ²π²(m²+n²)/(2mL²). These are exactly the spatial factors of the drumhead normal modes, with E_{mn} ∝ ω²_{mn}. The zero-displacement boundary condition (fixed drum edge) maps to the infinite-potential-well boundary condition (ψ=0 at walls). Quantum mechanics and classical wave mechanics share this deep structural similarity.

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