Helmholtz Wave Equation Visualization

This visualization demonstrates the Helmholtz wave equation in 3D space. The equation is a partial differential equation that describes wave propagation:

∇²A + k²A = 0

Where:

• ∇² is the Laplacian operator
• A is the wave amplitude
• k is the wave number (related to frequency)

Visualization Details

In this visualization:

• Each sphere represents a point in 3D space
• The size/radius of each sphere represents the wave amplitude at that point
• The color spectrum (blue to red) also indicates the wave value
• The animation shows how the wave evolves over time

Mathematical Implementation

The visualization uses solutions to the Helmholtz equation in spherical coordinates:

• Spherical Bessel functions for the radial component
• Spherical harmonics for the angular components
• Time-dependent oscillation factor

These components are combined to create a physically accurate representation of standing waves in 3D space.

Controls

• Use your mouse to rotate the view
• Scroll to zoom in and out
• Click and drag to pan