Dr. Loveless Curiosity Lab

2D Wave Equation Explorer

Fixed Boundaries, Fourier Modes, Vibrating Regions, and Geometry

Start with a shape. Choose an initial disturbance. Watch geometry determine the vibration.

This first version focuses on rectangular regions. The right side is reserved for supporting spectrum notes now and the future circular model, where the sine modes will be replaced by radial and angular patterns.

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Rectangle First: A Visual Fourier Lab

The top graph is the fast guided demonstration. The lower graph is the interactive pluck lab. Both use fixed boundary conditions, so the surface is pinned to zero along all four edges.

Rectangular Frequency Demonstration
A guided graph for eigenmodes and preset initial conditions on a rectangular membrane.
\(L_x\) = 2.00
\(L_y\) = 3.00
Changing the side lengths changes the mode frequencies. Larger rectangles vibrate more slowly in that direction.
Eigenmode demo: click any mode while the animation keeps playing.
0.0s
Loading rectangular frequency demonstration...
Interactive Rectangle Pluck Lab
Use the pluck sliders or drag the red pluck point on the surface, choose the region size and accuracy, then press play.
\(L_x\) = 2.00
\(L_y\) = 3.00
\(T\) = 0.0s
\(a_p\) = 0.68
\(b_p\) = 0.62
\(H_p\) = -1.95
More modes give a more accurate Fourier approximation, but may animate more slowly.

3×3 Harmonic Spectrum for This Pluck

Use the sliders or drag the red point in the graph. This live spectrum estimates which rectangular modes are most excited by that pluck position.

Loading interactive rectangle graph...

Where It Started

This project grew from trying to make the wave equation visible, first in one dimension and then across a two-dimensional region.

The rectangle is the natural first step because its eigenfunctions are products of familiar sine waves.

Going Further

The circular case replaces rectangular sine products with radial and angular modes.

Eventually this leads toward vibrating drums, Chladni patterns, eigenvalue problems, and the famous question: can you hear the shape of a drum?

Links & Resources