Analysis Of Narrow V-shaped Wakes Of High-speed Vessels

It has long been observerd that the far-field waves created by a ship that steadily advances at high speed in calm water of large depth appear to be contained within a wedge -ψ_max≤ψ≤ψ_max where ψ_max can be significantly smaller than the cusp angle ψ^K≈19^?28^′.Several alternative theories have been put forward to explain this phenomenon.The two-point wavemaker model based on wave-interference effects shows that the longitudinal interference of waves created by the bow and the stern of a monohull ship or the lateral interference of waves cre-ated by the twin hulls of a catamaran can lead to largest waves at angles that are significantly shaller than ψ^K.In this paper,the wave-interference effects on the far-field waves created by a monohull ship of length L or by a catamaran with identical twin hulls of length L at a lateral separation distance S,that advances at constant speed V along a straight path in calm water of large depth are considered.The hulls are represented via a continuous distribution of sources around hull sur-face.Systematic computations are performed for a wide range of Froude numbers （?）,hull spacings s≡S/L,and seven simple mathematically-defined hulls that correspond to a broad range of main hull-shape parameters（beam/length,draft/length,beam/draft,waterline entrance angle）.Here,g denotes the acceler-ation of gravity.The dominant ray angles,where the largest waves created by the monohull ships or the catamarans are found,are determined numerically via a realistic yet practical method.This method is based on the numerical determina-tion of the peaks of the amplitude function in the Fourier-Kochin representation of far-field ship waves,evaluated by means of the Hogner approximation,and the stationary-phase approximation.The main general conclusion of the study is that,although the amplitudes of the waves created by a ship are strongly influenced by the hull geometry（as well known）,the dominant ray angles associated with the largest waves created by a ship（due to interference effects）only weakly depend on the hull geometry.Thus,the ray angles associated with the dominant waves are mostly a kinematic feature of the waves created by a ship.An important practical consequence of this finding is that the wake angles that correspond to the domi-nant waves created by a ship can be explicitly determined in terms of the Froude number F for monohull ships,or in terms of the Froude number F_sand the hull spacing s for catamarans,via simple analytical relations.These relations,obtained via systematic numerical computations,are given here.The numerical determined dominant ray angles take into account the longitudinal interference effects of the waves created by the bow and the stern as well as the lateral interfence effects of the waves created by the port and starboard side of the monohull ship and by the twin hulls of the catamran.Thus,the analytical relations obtained here provide more accurate predictions of the dominant ray angles ψ_maxof high-speed mono-hull ships and catamarans than that derived by the simple two-point wavemaker model.The obtained dominant ray angles,which provide new upper bounds for the apparent wake angles of high-speed ships,supplement Kelvin’s upper bound ψ^K.The results of this study deepen the understanding of narrow V-shaped ship wakes,which can be used to gather useful information about ships from satellite images of ship wakes.