Mach-like capillary-gravity wakes

We determine experimentally the angle $\alpha$ of maximum wave amplitude in the far-field wake behind a vertical surface-piercing cylinder translated at constant velocity $U$ for Bond numbers $\mathrm{Bo}_D = D / \lambda_c$ ranging between 0.1 and 4.2, where $D$ is the cylinder diameter and $\lambda_c$ the capillary length. In all cases the wake angle is found to follow a Mach-like law at large velocity, $\alpha \sim U^{-1}$, but with different prefactors depending on the value of $\mathrm{Bo}_D$. For small $\mathrm{Bo}_D$ (large capillary effects), the wake angle approximately follows the law $\alpha \simeq c_{\rm g,min} / U$, where $c_{\rm g,min}$ is the minimum group velocity of capillary-gravity waves. For larger $\mathrm{Bo}_D$ (weak capillary effects), we recover a law $\alpha \sim \sqrt{gD}/U$ similar to that found for ship wakes at large velocity [Rabaud and Moisy, Phys. Rev. Lett. {\bf 110}, 214503 (2013)]. Using the general property of dispersive waves that the characteristic wavelength of the wavepacket emitted by a disturbance is of order of the disturbance size, we propose a simple model that describes the transition between these two Mach-like regimes as the Bond number is varied. We show that the new capillary law $\alpha \simeq c_{\rm g,min} / U$ originates from the presence of a capillary cusp angle (distinct from the usual gravity cusp angle), along which the energy radiated by the disturbance accumulates for Bond numbers of order of unity. This model, complemented by numerical simulations of the surface elevation induced by a moving Gaussian pressure disturbance, is in qualitative agreement with experimental measurements.