Fig. 3.

Proprioceptive swimming. (A) Families of self-propelled feedback-driven solutions as functions of $\mathit{\delta }$ and $\mathit{\chi }$, in the proximity of the natural resonance frequencies (passive sheet, red line), obtained by solving Eqs. 15–17 with $m_f\ne 0$ (Eq. 9), $m_a=0$, ${\rho }_f$ = ${\rho }_s$ = 1,000 kg/m³, $R{e}^b$ = 3.5$\cdot {10}^4$, $\mathit{\rho }$ = 300. For each resonance peak, a representative proprioceptive swimmer’s midline trace is reported. (B) The locomotion velocity u and (C) the tail amplitude $\mathit{\lambda }$ as functions of $\mathit{\delta }$ and $\mathit{\chi }$. In A–C the blue, green, and orange surfaces are associated with the first, second, and third resonance peak. The maximum velocity and tail amplitude are located at $\omega \delta =\pi /2$.