Capillary-Gravity Waves Generated by a Slow Moving Object

We investigate theoretically and experimentally the capillary-gravity waves created by a small object moving steadily at the water-air interface along a circular trajectory. It is well established that, for straight uniform motion, no steady waves appear at velocities below the minimum phase velocity $c_{\min }=23\mathrm{cm}{\mathrm{s}}^{-1}$. We demonstrate that no such velocity threshold exists for a steady circular motion, for which, even for small velocities, a finite wave drag is experienced by the object. This wave drag originates from the emission of a spiral-like wave pattern. Our results are in good agreement with direct experimental observations of the wave pattern created by a circularly moving needle in contact with water. Our study leads to new insights into the problem of animal locomotion at the water-air interface.

Figure 1

Plot of the wave resistance $R_w$ in units of $p_0^2\kappa /\gamma$, as a function the reduced velocity $V/c_{\min }=\mathcal{R}\Omega /c_{\min }$ for different ratios between the trajectory radius $\mathcal{R}$, and the object size $b$, as predicted by Eq. (3). The red or gray curve (presenting many oscillations) corresponds to $\mathcal{R}/b=100$, while the black one (with fewer oscillations) corresponds to $\mathcal{R}/b=10$. The green or light gray curve displaying a typical discontinuity at $V=c_{\min }$ is the wave drag for a straight uniform motion with velocity $V$ [8]. The object size $b$ was set to $b=0.1{\kappa }^{-1}$.

Figure 2

Wave radiation for $V\approx 21\mathrm{cm}/\mathrm{s}\approx 0.9c_{\min }$ with a radius $\mathcal{R}\approx 2.7\mathrm{cm}\approx 9{\kappa }^{-1}$ Left: Color or shaded diagram of the surface deformation $\zeta (\mathbf{r})$ computed numerically from Eq. (2). This image represents a square region of size $400{\kappa }^{-1}$ around the center of rotation, red or gray color corresponds to maximal $\zeta (\mathbf{r})$ values, while green or light gray corresponds to minimal values of $\zeta (\mathbf{r})$. The cross indicates the center of the trajectory and the moving object is located in the region of highest deformation. Right: Photography of the wave crests generated on a water surface by a needle rotating at a velocity. On both pictures, the black curve represents the Archimedean spiral of radius given by Eq. (5).

Figure 3

Inset: typical time dependence of the laser deflection angle (arbitrary units) during a rotation period $T_{\mathrm{rot}}=2\pi /\Omega$, the fast oscillation frequency is given by $⟨\omega ⟩=2\pi /T$. Main figure: Dependence of the ratio $⟨\omega ⟩/\Omega$ on $\kappa \mathcal{R}$ for different needle velocities. The dashed curves represent experimental results, while the continuous curve display the numerical results of our model. Red or gray, green or light gray, and blue or dark gray curves (diamonds, squares, and circles, respectively) correspond to $V/c_{\min }=0.69$, 0.76, and 0.84. The black curve correspond to the analytical estimate $⟨\omega ⟩/\Omega =\kappa \mathcal{R}$.