1. Physics-Driven Animation Foundations in JavaScript
Modern Web UI/UX is evolving beyond static CSS transitions and linear tweens. To replicate natural fluid behaviors like the Ripple Effect, falling raindrops, or interactive ripples, developers must adopt Physics-based Simulation.
The Ripple Effect simulates surface wave propagation triggered by external forces. As waves propagate, surface height displacement bends light rays via Light Refraction, creating dynamic visual distortion of the background texture.
Two primary mathematical approaches exist:
- Approach 1: 2D Finite Difference Wave Simulation (CPU Canvas 2D) — Solves the continuous wave partial differential equation (PDE) over a discrete pixel grid, enabling natural multi-source wave interference.
- Approach 2: Sombrero Trigonometric Coordinate Perturbation (GPU WebGL) — Leverages closed-form traveling wave equations
cos(k·r - ω·t)inside GLSL Fragment Shaders for ultra-fast procedural ripples.
2. Approach 1: 2D Finite Difference Wave Simulation on Canvas 2D
2.1. Continuous Wave Equation
The water surface is modeled as a scalar heightfield u(x, y, t) governed by the 2D wave PDE:
∂²u / ∂t² = c² · ∇²u = c² · ( ∂²u/∂x² + ∂²u/∂y² )2.2. Discretization via Central Differences
Using 5-point Laplacian spatial approximations and temporal central differences, the discretized velocity formulation updates heightfield buffers across discrete frame intervals.
2.3. Courant Stability & Hugo Elias Algorithm
By enforcing Courant stability λ = c²Δt²/Δx² = 1/2, the update equation simplifies to u_{t+1} = (1/2) · Σneighbors − u_{t-1}, implemented via double-buffered typed arrays.
3. Approach 2: GPU Trigonometric Perturbation on WebGL
Leverages GLSL Fragment Shaders executing radial wave functions cos(k·r - ω·t) * A to offset UV sampling coordinates directly on the GPU hardware pipeline.
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