PDEs / Vibrations / Finite Difference

Wave Equation Studio

This simulation solves the two-dimensional damped wave equation on a square membrane with fixed edges. Click or drag to strike the surface, seed standing modes, or switch on random raindrops to watch superposition, reflection, and numerical stability play out in real time.

← Back to Mathematical Simulations

Live State

Peak |u| 0

RMS Amplitude 0

Energy Proxy 0

Center u 0

Courant λ 0

FPS 0

How To Read It

Warm ridges are positive displacement, cool troughs are negative displacement, and the glossy lighting comes from the local surface slope of the membrane.

With the boundary clamped to zero, waves reflect off the square edge, interfere with one another, and can lock into standing patterns when the initial shape matches an eigenmode.

Use drag to add impulses and scroll on the canvas to change brush size.

Presets

Initial Conditions

These presets emphasize different wave behaviors: localized impulses, mode shapes, and continuous random forcing.

Simulation

Time-Stepping Controls

Grid Resolution 168 x 168 Higher grids resolve tighter wavelengths but require smaller effective steps for the same stability margin.

Courant Number λ 0.46 This is the dimensionless ratio c Δt / Δx. In 2D with the five-point Laplacian, stability requires λ ≤ 1 / √2.

Damping 0.006 Damping drains energy from the velocity term, mimicking friction or air losses on the membrane.

Steps Per Frame 3 Multiple updates per frame let the oscillation move faster without raising the display frame rate.

Forcing

Impulses And Continuous Excitation

Brush Radius 10 px The brush applies a smooth Gaussian displacement profile instead of a hard-edged square impulse.

Impulse Strength 0.80 Positive values push the membrane up; negative values pluck it downward.

Rain Rate 0.18 Controls the probability of a random raindrop impulse on each update step when rain mode is enabled.

Random Rain Forcing

Glow Lighting

Mesh Overlay

Readout

Diagnostics

Current PresetCenter Pulse

Mean Velocity Proxy0.0000

Edge Amplitude0.0000

Centerline Nodes0

Painted Impulses0

FPS0.0

Center Displacement History

Method

Discrete Wave Equation

This page solves a damped version of the membrane equation with fixed Dirichlet boundary conditions, using an explicit second-order finite-difference update on a square lattice.

u_tt = c^2 (u_xx + u_yy) - gamma u_t

lambda = c Delta t / Delta x

u_next = (2 - gamma) u - (1 - gamma) u_prev + lambda^2 Laplacian(u)

u = 0 on the boundary

Finite difference stencil. The Laplacian uses the standard five-point stencil, so each interior cell reacts only to its four cardinal neighbors during an update.

Standing modes. The mode presets seed sinusoidal shapes that are eigenfunctions of the clamped square membrane, so the pattern oscillates in time while the nodal lines stay fixed.

Stability condition. In two space dimensions, the explicit scheme becomes unstable once the Courant number exceeds roughly 1 / √2, which is why the λ slider is bounded below that edge.

Accuracy note: this is a clean square membrane model with unit lattice spacing, fixed boundaries, and simple linear damping. It is not trying to reproduce the full material physics of a drumhead.