Wave Patterns of Gravity–Capillary Waves from Moving Localized Sources

Vladimir Gnevyshev,Sergei Badulin

doi:10.3390/fluids5040219

Abstract

We study wave patterns of gravity–capillary waves from moving localized sources within the classic setup of the problem of ship wakes. The focus is on the co-existence of two wave systems with opposite signatures of group velocity relative to the localized source. It leads to the problem of choice of signs for phase functions of the gravity (“slow”) and capillary (“fast”) branches of the dispersion relation: the question generally ignored when constructing phase patterns of the solutions. We detail characteristic angles of the wake patterns: (i) angle of demarcation of gravity and capillary waves—“the phase Mach” cone, (ii) angle of the minimal group velocity of gravity–capillary waves—“the group Mach” cone, (iii, iv) angles of cusps of isophases that appear after a threshold current speed. The outer cusp cone is naturally associated with the classic cone of Kelvin for pure gravity waves. The inner one results from the effect of capillarity and tends to the “group Mach” pattern at high speeds of current. Amplitudes of the wave patterns are estimated within the recently proposed approach of reference functions for the problem of propagation of packets of linear dispersive waves. The effect of shape is discussed for elliptic reference sources.

Highlights

The problem of wavefield behind a moving source is well-known in various domains of modern physics and, in particular, in the dynamics of the atmosphere and ocean [1]
The main focus of the paper is an intermediate range of gravity–capillary waves where both effects of gravity and capillarity compete on equal terms
The dimensionless speed U∗, the Mach number related to the minimum of phase speed, determines characteristic wave patterns

Summary

Introduction

The problem of wavefield behind a moving source is well-known in various domains of modern physics and, in particular, in the dynamics of the atmosphere and ocean [1]. The brightest and deepest for understanding rich physics is the example of ship waves, first considered in the works of Lord. At the turn of the 20th century, Kelvin and his followers created the basis for many works developing the classical results (e.g., [6,7]). We assume a motionless source and the flow velocity U = (U, 0, 0) directed along the x −axis in the positive direction. We restrict ourselves to the case of two-dimensional waves, which include surface water waves, Rossby waves, and internal gravity waves in a depth-limited ocean (atmosphere).

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Conclusion