Lagrangian solution for irrotational progressive water waves propagating on a uniform current: Part 1. Fifth-order analysis

•A proof of particle label in wave motion is same as it is in the still.•The particle motion period is always larger than the progressive wave period.•The 3D particle trajectories are described.•Ucosθ=−cw: the particle trajectory is like a 3D turned prolate helix.•The streakline for a fixed position is same as the particle trajectory.

A three-dimensional (3D) Lagrangian solution up to the fifth-order is found for the Boundary Value Problem (BVP) of irrotational, progressive water waves propagating in the presence of uniform current in water of constant depth. The BVP is formulated in the Lagrangian framework which uses Lagrangian label (a,b,c) and time t as the independent variables. The (a,b,c) of a particle in wave motion are the coordinates of the particle position in the still water. In the solution, wave–current interaction is embedded in the Lagrangian velocity potential in such way that pressure is not affected by current in the wave–current field. In this study, we present motion properties of particles such as the particle motion period, drift velocity, the Lagrangian mean level, and the 3D particle trajectory and streakline, aspects that are hardly described by the Eulerian solution. It is found that the particle motion period is always larger than the progressive wave period. As the wave–current becomes steady motion, the stable wavy track of the particle motion may exist in the wave–current field. When currents vanish, the present solution deduces to the same solution of the progressive wave propagation by Chen et al. (2010).