Fluke Strouhal number, drag power and the blubber fuel budget on a Hawaii to Alaska swim. Drag the 3D view to orbit.
Energy cost per kilometre against swimming speed (marker = now, dashed = closed-form optimum)
The science
Fluke propulsion. The Strouhal number St = f A / U (beat frequency, peak-to-peak tail amplitude, swimming speed) tells how a flapping foil sheds vortices. Swimming and flying animals cruise at 0.2 < St < 0.4 where efficiency peaks (Triantafyllou 1993; Taylor et al. 2003). The model uses a log-normal bell eta(St) = 0.75 exp(-ln(St/0.30)^2 / (2 * 0.40^2)): the peak value and width are an assumed illustration of that published band, not a measurement of one whale.
Drag. D = 1/2 rho U^2 S Cf FF k with water rho = 1025 kg/m3, wetted area S = 0.09 m^(2/3) (allometric assumption), Prandtl-Schlichting skin friction Cf = 0.455 / (log10 Re)^2.58, Re = U L / nu, nu = 1.19e-6 m2/s, form factor FF = 1 + 1.5 (d/L)^1.5 + 7 (d/L)^3 (Hoerner, d/L = 0.24) and an active-swimming enhancement k = 1.5 (assumed; real values are debated).
Energy. Useful power P = D U; metabolic power P_met = D U / (eta(St) eta_muscle) + x B with eta_muscle = 0.25 and the Kleiber resting rate B = 3.39 m^0.75 W times an assumed activity multiplier x. Fuel is blubber at 37 MJ/kg, with a reserve of 30 percent of body mass at departure. Swim time uses ground speed U + current.
Cheapest speed. Energy per distance is E/d = a U^n + P_rest / U. Setting the derivative to zero gives U* = (P_rest / ((n) a))^(1/(n+1)). With the 1/5-power friction law n = 1.8. Swimming faster than U* wastes energy to drag, slower wastes it to the metabolic clock.
Simplifications. Steady swimming only (no breaching, lunge feeding, diving or calf drafting), constant mass, a single tail-beat model with a travelling body wave of one body length, no weather, no feeding en route. Real humpbacks may feed and lose more mass in a season: treat percentages as a teaching model, not as a prediction for one animal. Typical values (mass, speed, fluke frequency) should be spot-checked against the literature before quoting.