Scaling of far-field wake angle of non-axisymmetric pressure disturbance

It has been recently emphasized that the angle of maximum wave amplitude $\alpha$ in the wake of a disturbance of finite size can be significantly narrower than the maximum value $\alpha_K = \sin^{-1}(1/3) \simeq 19.47^\mathrm{o}$ predicted by the classical analysis of Kelvin. For axisymmetric disturbance, simple argument based on the Cauchy-Poisson initial-value problem suggests that the wake angle decreases following a Mach-like law at large velocity, $\alpha \simeq Fr_L^{-1}$, where $Fr_L=U/\sqrt{gL}$ is the Froude number based on the disturbance velocity $U$, its size $L$, and gravity $g$. In this paper we extend this analysis to the case of non-axisymmetric disturbances, relevant to real ships. We find that, for intermediate Froude numbers, the wake angle follows an intermediate scaling law $\alpha \simeq Fr_L^{-2}$, in agreement with the recent prediction of Noblesse \textit{et al.} [Eur. J. Mech. B/Fluids {\bf 46}, 164 (2014)]. We show that beyond a critical Froude number, which scales as $A^{1/2}$ (where $A$ is the length-to-width aspect ratio of the disturbance), the asymptotic scaling $\alpha \simeq Fr_B^{-1}$ holds, where now $Fr_B = A^{1/2} Fr_L$ is the Froude number based on the disturbance width. We propose a simple model for this transition, and provide a regime diagram of the scaling of the wake angle as a function of parameters $(A,Fr_L)$.