Capillary-gravity waves propagating at the free surface of a liquid are driven by a balance between the liquid inertia and its tendency, under the action of gravity and surface tension forces, to return to a state of stable equilibrium. For an inviscid liquid of infinite depth, the dispersion relation relating the angular frequency ω to the wave number k is given by ω² = gk + γk³∕ρ where ρ is the liquid density, γ the liquid-air surface tension, and g the acceleration due to gravity. The above equation may also be written as a dependence of wave velocity c = ω(k)∕k on wave number c(k) = (g∕k + γk ∕ρ)

^1∕2. The dispersive nature of capillary-gravity waves is responsible for the complicated wave pattern generated at the free surface of a still liquid by a moving disturbance such as a partially immersed object (e.g. a boat or an insect) or an external surface pressure source. The propagating waves generated by the moving disturbance continuously remove energy to infinity. Consequently, the disturbance will experience a drag, R_w, called the wave resistance. In the case of boats and large ships, this drag is known to be a major source of resistance and important efforts have been devoted to the design of hulls minimizing it. The case of objects small relative to the capillary length κ^-1 = (γ∕(ρg ))

^1∕2 has only recently been considered.

Figure 1:

Dependence of the wave resistance R_w as a function of the reduced velocity V∕c_min =

Ω∕c_min for different ratios between the trajectory radius

, and the object size b, as predicted by the theory). The red curve (presenting many oscillations) corresponds to

∕b = 100, while the black one (with fewer oscillations) corresponds to

∕b = 10. The green curve displaying a typical discontinuity at V = c_min is the wave drag for a straight uniform motion with velocity V . The object size, b, was set to b = 0.1 κ^-1.

It has been shown that in the case of a disturbance moving at constant velocity

, the wave resistance R_w cancels out for V < c_min where V stands for the magnitude of the velocity and c_min = (4gγ∕ρ)^1∕4 is the minimum of the wave velocity) [1][2]. For water with γ = 73 mN ⋅ m^-1 and ρ = 10³ kg ⋅ m^-3, one has c_min = 0.23 m ⋅ s^-1 (room temperature). This striking behavior of R_w around c_min is similar to the well-known Cerenkov radiation emitted by a charged particle, and has been recently studied experimentally. We have demonstrated that just like accelerated charged particles radiate electromagnetic waves even while moving slower than the speed of light, an accelerated disturbance experiences a non-zero wave resistance R_w even when propagating below c_min. We have considered the special case of a uniform circular trajectory, a situation of particular importance for the study of whirligig beetles whose characteristic circular motion might facilitate the emission of surface waves that may be used for echolocation.

Figure 2:

A whirligig beetle at the surface of water.

We have shown theoretically that a disturbance moving along a circular trajectory experienced a wave drag even at angular velocities corresponding to V < c_min, where c_min is the minimum phase velocity of capillary-gravity waves [3]. Our prediction is supported by experimental observation of a long distance wake for V∕c_min as low as 0.6. For V∕c_min > 0.8, we observed by naked eye Archimedean spiral shaped crests, in good agreement with theory. These results should be important for a better understanding of the propulsion of water-walking insects where accelerated motions frequently occurs (e.g when hunting a prey or escaping a predator).

[3] A. Chepelianskii, F. Chevy and E. Raphaël, Phys. Lett. Lett. submitted , (2007).