WingHopper · Designer · Guide · Sections · Reef · Methods · Pricing · FAQ
Pumping and gliding pull a wing in different directions, and the published fluid dynamics says fairly clearly which way each one pulls. This article collects what the literature actually supports, with sources, and points at the WingHopper numbers that measure each effect.
Every claim below carries a citation. Where the research does not settle a question, it says so instead of guessing.
Gliding is steady-state. The wing holds your weight at some speed and you want the least energy lost doing it, which is the classic lifting-line problem: for a fixed lift and span, induced drag is minimised by an elliptical spanwise load distribution, and it falls as span (and so aspect ratio) rises Prandtl, NACA TR 116, 1923. That result is a century old and still the reason downwind wings are long and thin.
Pumping is unsteady propulsion. The rider oscillates the foil vertically and it generates thrust, which puts it in the same family as a flapping fin rather than a fixed wing. The relevant theory is Theodorsen's unsteady airfoil solution (NACA TR 496, 1935) extended to propulsion by Garrick, who derived the propelling force on an airfoil oscillating in vertical flapping, torsional pitching, or aileron motion (NACA TR 567, 1936). The forces have a circulatory part that lags the motion and an inertial (added-mass) part that does not.
A wing optimised purely for one job is compromised at the other. That tension is real physics, not a marketing distinction.
For oscillating propulsion the governing dimensionless group is the Strouhal number, St = f·A/U, with f the stroke frequency, A the amplitude, and U the forward speed.
Triantafyllou and colleagues showed that thrust from an oscillating foil comes from a jet-like mean flow that is convectively unstable over a narrow band of frequencies, and that peak efficiency coincides with the frequency of maximum amplification: St ≈ 0.25 to 0.35 (Triantafyllou et al., Journal of Fluids and Structures 7:205, 1993).
Biology converges on the same band. Across dolphins, sharks, bony fish, birds, bats and insects, cruising kinematics cluster in 0.2 < St < 0.4, which is the interval where propulsive efficiency peaks (Taylor, Nudds & Thomas, Nature 425:707, 2003).
The practical consequence: your stroke has a correct shape for a given speed, and it is not "harder". Frequency, amplitude and speed trade against each other inside a fixed St band. Pumping at 4 m/s with a small stroke sits well below the efficient band; the same stroke at 2 m/s can land inside it. WingHopper's Pump analysis reports St directly and labels whether the stroke you dialed in is inside 0.2 to 0.4.
In a heaving-and-pitching foil the phase between the two motions matters as much as either amplitude. Read, Hover and Triantafyllou measured propulsive efficiency against heave amplitude, Strouhal number, angle of attack and heave-pitch phase, and found best thrust performance at a phase angle of roughly 90 to 100 degrees, with efficiencies up to about 71% at a 15 degree maximum angle of attack and much larger thrust (planform thrust coefficient 2.4) at 35 degrees (Read, Hover & Triantafyllou, Journal of Fluids and Structures 17:163-183, 2003).
Two things fall out of that for a foil designer. Efficiency wants a modest angle-of-attack swing; raw thrust wants a big one. And a rider who shifts fore and aft while heaving (changing pitch as well as height) is doing something the pure-heave model does not capture.
Heave acceleration is resisted by the added mass of fluid the wing drags with it, an inertial term independent of lift and present in Garrick's formulation above. It scales with chord squared times span, so a fat-chord wing carries an inertial penalty that a static polar cannot see at all. Two wings with identical lift curves can feel different to pump for this reason alone.
Treating each instant of the stroke as a steady flow is a useful approximation, and it is what WingHopper's pump model does, reading lift off the wing's own VLM polar at the instantaneous effective angle of attack. It stops being trustworthy when the wake no longer has time to settle. Theodorsen and Garrick's theory exists precisely because the circulatory response lags the motion, so any quasi-steady number at high stroke frequency should be read as indicative. WingHopper flags results above 2.5 Hz for that reason.
A pumped hydrofoil is not a foil in unbounded fluid. Rozhdestvensky's model of a pumped hydrofoil reduces the vertical motion to a linear oscillator excited by the rider's swinging mass, and identifies the restoring term as the automatic stabilisation of a shallowly submerged foil: lift falls as the foil approaches the free surface and rises as it moves away (Rozhdestvensky, JMSE 11(5):913, 2023).
That effect is strong and it is not small print. Lift and drag on a hydrofoil near the surface depend on submergence depth, and the theory and towing-tank measurements for it are long established (Daskovsky, Ocean Engineering 27(10):1129-1159, 2000). A wing evaluated only in deep water is being evaluated in conditions you do not ride in during the part of the stroke that matters most.
Riding shallow also risks ventilation: air pulled down from the surface along the strut or foil, which collapses lift. A towing-tank study of surface-piercing hydrofoils mapped the onset against depth-based Froude number and angle of attack, over Fr 0.5 to 2.5 at aspect ratios 1.0 and 1.5, and identified three distinct trigger mechanisms (nose, tail and base). Nose ventilation dominated at low Fr, tail ventilation at higher Fr, and the boundary of the globally stable region extended to notably higher angle of attack than previously estimated (Ferreira et al., Journal of Fluid Mechanics 1028:A25, 2026). Aggressive high-angle pumping close to the surface is the corner of the envelope where that lives.
The glide question is usually posed as "maximise L/D", but for staying up with the least rider input the right objective is minimum sink rate, which is minimum power required, not minimum drag. Minimum sink occurs where the quantity CL^1.5/CD is maximised, and it happens at a lower speed and a lower L/D than best glide: L/D at minimum sink is roughly 0.86 of L/D max (Princeton MAE331, Gliding, Climbing, and Turning Flight Performance; Virginia Tech AOE3104, Gliding Flight).
This is why area matters and not only aspect ratio. Sink rate per unit weight is V/(L/D), and substituting V = √(2(W/S)/(ρ·CL)) gives √(2(W/S)/ρ)·CD/CL^1.5. Two levers lower it: more area (lower wing loading, so a slower speed carries you) and better L/D at high lift. A pure aspect-ratio score sees only the second. WingHopper's pump score is built on this minimum power quantity for exactly that reason, so wing area moves it.
Foil sections run at chord Reynolds numbers where laminar separation bubbles are a first-order effect, not a refinement. Lissaman's review of the regime identifies the bubble, not viscous drag in general, as the dominant mechanism airfoil designers have to work around below roughly Re 10^5 to 10^6, since it can change the effective camber and move separation forward of where an inviscid or turbulent-boundary-layer calculation would place it (Lissaman, Annual Review of Fluid Mechanics 15:223-239, 1983).
Two implications. Thin, sharp-entry sections that look good on an inviscid polar can behave badly in practice, and a panel-method result (including WingHopper's) has no bubble and no real stall model. Treat computed CL max as an upper bound.
Ordered by how well the literature supports it:
governing group for pumping (Triantafyllou 1993; Taylor 2003). Decide the speed you want to pump at, then check the wing can hold the resulting angle-of-attack swing without stalling.
aspect ratio (Prandtl 1923); added mass rises with chord squared (Garrick 1936). This is the core trade and it has a real sign.
count through wing loading (Princeton MAE331).
efficient Strouhal band at realistic pumping speeds demands large angle-of-attack swings, and a wing that stalls inside that swing cannot use the band at all. This is the tension WingHopper's stall flag surfaces.
forces (Daskovsky 2000) and is the restoring force pumping relies on (Rozhdestvensky 2023), with ventilation waiting at high angle of attack (Ferreira et al., JFM 1028).
(Lissaman 1983).
for a mechanically actuated foil (Read et al. 2003). Nobody has published, as far as this article's sources go, the equivalent for a human on a board who couples pitch through fore-aft weight shift.
There is no published measurement set of real pumping power for foilers to calibrate against, so treat the absolute number as a comparator between wings, not a physiological figure.
exciting mass; the sports-science side has begun describing the motion itself (Zöllner, Krause & Gemeinhardt, iWOAR 2024, Similarities of Motion Patterns in Skateboarding and Hydrofoil Pumping), but the two literatures have not been joined into a design tool.
Method notes for WingHopper's own numbers: how the numbers are computed.
Next: Structure