UNDER THE SURFACE / FURTHER READING
The math behind the ride.
A few starting points for the equations illustrated on our site. Voyager adapts these principles into a simplified, tunable game simulation.
Crossing swells & wave motion
η(x,t) = Σ Aᵢ cos(kᵢ · x − ωᵢt + φᵢ)
ω² = gk · tanh(kh); in deep water, ω² ≈ gk
- MIT — Waves and linear superposition
How traveling wave components combine into a surface. - MIT — Water Waves
Linear wave theory, deep-water behavior, and the limits of those approximations.
Lift, speed & angle of attack
L = ½ρv²SCL
- NASA Glenn — The lift equation
Dynamic pressure, surface area, and the lift coefficient. - MIT — Maneuvering and Control of Surface and Underwater Vehicles (PDF)
Deeper reading on lifting surfaces, finite-wing behavior, and marine vehicle forces.
NASA explains these relationships using wings in air. For a hydrofoil, the model uses water density and velocity relative to the water. Coefficients and operating conditions still matter.
Wing shape & the price of lift
CDi = CL² / (π e AR)
CLα ≈ 2π / (1 + 2π / (π e AR))
- NASA Glenn — Induced drag coefficient
Wingtip vortices, aspect ratio, efficiency, and drag due to lift.
The illustrated lift-slope expression is a simplified finite-wing correction, with angles in radians. It describes the pre-stall trend; it is not a universal formula for every foil or flow condition.