Abstract
The classic problem of ship waves of infinitely small amplitudes is studied within the recently developed approach of reference solutions [V. G. Gnevyshev and S. I. Badulin, Moscow Univ. Phys. Bull. 72, 415 (2017)]. The evolution of narrow-banded Gaussian wavetrains of a finite volume is considered as an alternative to the conventional inherently point-wise tracking of wavetrains within the asymptotic methods of the stationary phase or the steepest descent. The approach allows for avoiding general problems of the methods: occurrence of singularities of wave fields. The non-singular solutions for a stationary ship wake are presented in an analytical form in terms of two key dimensionless quantities: the Froude number and the aspect ratio of the initial (boundary) domain of wave generation. Systems of transverse and diverging ship waves as well as amplitude and phase effects of their transformation can be analyzed separately within this approach for small Froude numbers. The essential role of dispersion of three-dimensional water waves is emphasized and detailed.
Highlights
This paper presents asymptotic solutions to the classic problem of stationary deep water ship waves first formulated by Lord Kelvin
The reference solution approach for linear water waves is presented. It allows for transparent physical analysis of the classic problem of stationary ship waves
Explicit formulas are obtained for wave variables without explicit reference to dynamical equations
Summary
This paper presents asymptotic solutions to the classic problem of stationary deep water ship waves first formulated by Lord Kelvin. The Kelvin analysis of wave kinematics predicts universal wake patterns in the form of arms of chevron accompanied by a system of transverse arcs inside a gusset of total opening at approximately 39○ (twice the Kelvin angle, θK ≈ 19.47○). The Kelvin analysis of wave kinematics predicts universal wake patterns in the form of arms of chevron accompanied by a system of transverse arcs inside a gusset of total opening at approximately 39○ (twice the Kelvin angle, θK ≈ 19.47○) This analysis was later extended by studies of amplitude distributions based on asymptotical approaches of the stationary phase (see the work of Thomson7) and the steepest descent.. The novel approach recently sketched by the authors, the so-called reference solution method, follows the idea of Lord Kelvin According to this approach, the reference wavetrains are Gaussian, which allow one to develop an analytic approach to many problems of wave propagation in multidimensional dispersive. Solutions will be presented in an explicit form where particular terms have transparent physical meaning In this way, the effects of amplitude and phase transformation can be analyzed separately.
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