Wave drag of a membrane

Abstract

The hydromechanical efficiency of swimming styles is of major interest in the theory of submarine animal locomotion. For larger species, a complication arises when the forward motion occurs near the surface where gravity waves are generated. Near-surface activities of aquatic and other animals, including man, can be difficult to model successfully; for example, an anguilliform motion with an associated divergent wave system appears computationally unpromising. However, for many seagoing creatures the forward thrust is derived from dorsiventral undulations of the caudal region, as in the Cetacea, and these movements are simulated by the motions of a shallowly submerged horizontal membrane. Based upon a two-dimensional linearized inviscid theory, an expression is found for the drag for a general flexing of the membrane with a given frequency. Some numerical results are presented graphically for threshing flexures with exaggerated tail movements intended to partly imitate carangiform or ostraciiform propulsion. The excitation of transverse waves might be expected to impede the forward propulsion, suggesting that maximum efficiency would occur when wave making is least. It is shown that this is not necessarily true because the longer waves supply a forward thrust. For a given maximum amplitude of flexure this is often a weaker force, but in terms of hydrodynamic efficiency it can rival the thrust from the wake at higher frequencies. A corresponding formula for the power is also derived; in particular this shows that energy may be extracted from a mainstream flow with a free surface by oscillations of a hydrofoil that generates shorter waves.