On the Lagrangian Period in Steep Periodic Waves

We show that, for periodic steep waves at finite water depth, the excess Lagrangian period, \(\Delta T_\mathrm{{L}}\), made nondimensional by the Eulerian period, \(T_0\), is equal to the mean Lagrangian drift velocity, \(u_\mathrm{{L}}\), made nondimensional by the wave speed, c. This result is exact within second-order theory. Measurements in wave tank obtain that the two quantities are approximately equal. The vertical mean of \(\Delta T_\mathrm{{L}}\) and \(u_\mathrm{{L}}\) are zero in the experiments. The vertically averaged Lagrangian period thus equals the Eulerian period. The experimental and theoretical vertical derivative of \(\Delta T_\mathrm{{L}}\) are the same in the middle of the water column but strongly differ above the bottom and below the wave surface. These differences are due to the secondary streaming effect in the viscous boundary layers in the experiments, at the bottom and free surface. Fully nonlinear calculations complement the investigation.