Analytical insights into optimality and resonance in fish swimming

Abstract

This paper provides analytical insights into the hypothesis that fish exploit resonance to reduce the mechanical cost of swimming. A simple body–fluid fish model, representing carangiform locomotion, is developed. Steady swimming at various speeds is analysed using optimal gait theory by minimizing bending moment over tail movements and stiffness, and the results are shown to match with data from observed swimming. Our analysis indicates the following: thrust–drag balance leads to the Strouhal number being predetermined based on the drag coefficient and the ratio of wetted body area to cross-sectional area of accelerated fluid. Muscle tension is reduced when undulation frequency matches resonance frequency, which maximizes the ratio of tail-tip velocity to bending moment. Finally, hydrodynamic resonance determines tail-beat frequency, whereas muscle stiffness is actively adjusted, so that overall body–fluid resonance is exploited.