Wave Equation Explorer

This interactive 3D visualization allows you to explore various wave equations in physics and mathematics.

Features

• Multiple wave equations: Helmholtz, membrane waves, Schrödinger equation, and wave packets
• Interactive controls: Adjust wave parameters in real-time
• 3D visualization: Explore the wave behavior from any angle

Wave Equations

Helmholtz Equation

The Helmholtz equation is used in physics to describe standing waves:

∇²ψ + k²ψ = 0

Where ∇² is the Laplacian operator and k is the wavenumber.

Membrane Wave

A classical wave equation representing vibration of a membrane:

∂²ψ/∂t² = c² (∂²ψ/∂x² + ∂²ψ/∂y²)

Schrödinger Equation

The time-dependent Schrödinger equation describes how the quantum state of a physical system changes:

iħ ∂ψ/∂t = -ħ²/(2m) ∇²ψ + Vψ

Wave Packet

A wave packet combines multiple waves to create a localized wave:

ψ(x,t) = A e^(-(x-x₀)²/(2σ²)) e^(i(kx-ωt))

Usage

• Use the dropdown menu to select different wave equations
• Adjust the wavenumber slider to change the wave frequency
• Modify the animation speed with the speed slider
• Click and drag to rotate the view
• Scroll to zoom in and out

Technical Details

This visualization uses Three.js for 3D rendering and implements numerical solutions to the various wave equations. The wave amplitude at each point is represented by both the size and color of the spheres.

Performance Note

The visualization creates a full 3D grid of spheres (10x10x10 by default), which may affect performance on less powerful devices. If you experience lag, try reducing the grid size in the config object.