Base wave: y₀ = A₀ sin(2π f₀ x + φ)
Normal vector: N = (-y₀′, 1) / √(1 + y₀′²)
Arc-length coordinate: s(x) ≈ Σ √((Δx)² + (Δy₀)²)
Layered normal displacement: m(x,t) = Σ mᵢ(x,s,t), where each layer can be sin or cos, can use x or s, and can travel independently.
Final curve: x₁ = x - m y₀′ / √(1+y₀′²), y₁ = y₀ + m / √(1+y₀′²) + k.
| On | Type | Path | Amplitude | Frequency | Phase | Speed | Normal mix | Remove |
|---|
Path x gives the flatter early ripple behaviour from the simpler labs. Path s gives the true arc-length travel from the later cosine versions. Normal mix lets a layer partially flatten back toward ordinary vertical addition.