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We present the kinematic model of the ship wake in the presence of horizontal subsurface current linearly varying with the depth of water. An extension of the Whitham-Lighthill theory for calm water is developed. It has been established that the structure of ship waves under the action of a shear flow can radically differ from the classical Kelvin ship wake model. Co propagating ship and shear current lead to increasing the total wedge angle up to full one 180 degrees and decreases for the counter shear current. At relatively large unidirectional values of the shear current, cusp waves in the vicinity of the wedge boundary are represented by transverse waves and, conversely, by diverging waves directed almost perpendicular to the ship track for the opposite shear current. The presence of a shear flow crossing the direction of the ship's movement gives a strong asymmetry of the wake. An increase in the perpendicular shear flow leads to an increase in the difference between the angles of the wake arms. The limiting value of the shear current corresponds to one or both arms angles equal to 90 degrees. Transverse and divergent edge waves for this limiting case coincide.

International audience; In this Brief Research Report, we have shown, within the framework of the vor-NLS equation in deep water, that the Peregrine breather travelling at the free surface of a shear current of slowly varying vorticity may transform into grey solitons. Contribution to the field Based on numerical simulations of the nonlinear Schrödinger equation in infinite depth we conjecture the existence of grey solitons at the sea surface of slowly varying vortical current. This occurrence of dark solitons is due to the transformation of the Peregrine breather traveling at the free surface of the varying vortical current. Ethics statements Studies involving animal subjects Generated Statement: No animal studies are presented in this manuscript. Studies involving human subjects Generated Statement: No human studies are presented in this manuscript. Inclusion of identifiable human data Generated Statement: No potentially identifiable human images or data is presented in this study.

Asymptotic solutions up to third-order which describe irrotational finite amplitude standing waves are derived in Lagrangian coordinates. The analytical Lagrangian solution that is uniformly valid for large times satisfies the irrotational condition and the pressure p = 0 at the free surface, which is in contrast with the Eulerian solution existing under a residual pressure at the free surface due to Taylor's series expansion. In the third-order Lagrangian approximation, the explicit parametric equation and the Lagrangian wave frequency of water particles could be obtained. In particular, the Lagrangian mean level of a particle motion that is a function of vertical label is found as a part of the solution which is different from that in an Eulerian description. The dynamic properties of nonlinear standing waves in water of a finite depth, including particle trajectory, surface profile and wave pressure are investigated. It is also shown that the Lagrangian solution is superior to an Eulerian solution of the same order for describing the wave shape and the kinematics above the mean water level.