Physical origins of speed control in swimming fishes.

The linear relationship between tail-beat frequency and swimming speed in fish, first described by Bainbridge in 1958, has since been confirmed in species as disparate as eels and tuna. Dimensional scaling has explained why linearity should emerge at high Reynolds numbers, but the frequency-speed relationship itself has remained empirical, with no physical framework describing its behavior across swimming regimes, and no account of what sets the two quantities that define it: the distance covered per tail beat and the frequency below which forward motion supposedly ceases. Here, we derive a relationship from a balance of fin thrust, generated by a leading-edge vortex, and drag due to the boundary layer on the body. This framework yields an explicit expression for speed in terms of midline kinematics and body morphology. This formulation shows that the maximum stride length is set by the undulatory wavelength and the tail-beat phase offset, rather than by the conventional scaling metric of body length, and that no minimum frequency is required for forward motion. The model reproduces kinematic measurements from freely swimming glass catfish, published simulations, and Bainbridge's original data. Without recalibration, it predicts the results of independent simulations spanning wide variations in wavelength, phase offset, and morphology to within ten percent. Linear scaling is thus one asymptotic limit of a more general propulsive relationship linking morphology and body-wave kinematics to swimming speed in fishes and bioinspired vehicles, where the body waveform, rather than the actuation frequency, emerges as the variable that sets performance.