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 $Bo_D = D / \lambda_c$ 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 $Bo_D$. For small $Bo_D$ (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 $Bo_D$ (weak capillary effects), we recover the law $\alpha \sim \sqrt{gD}/U$ found for ship wakes at large velocity. 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. This model, complemented by numerical simulations of the surface elevation induced by a moving Gaussian pressure disturbance, is in good agreement with experimental measurements.