The steady-state solution of wave-current interaction based on the third-order Stokes wave theory

Abstract

This manuscript reports on the interaction of a current-free monochromatic surface wave field with a wave-free uniform current field. The existing reasonable theories of wave–current interactions are primarily based on weak current assumptions and derived from linear theory, resulting in calculation bias in the analysis of nonlinear wave–current interactions. Moreover, experimental data on high-order wave–current interactions still need to be collected. Thus, steady-state solutions named the third-order wave–current theory based on the third-order wave dispersion relationship and the principle of wave–current energy conservation were derived. The wave–current interaction experiment was set up to cover 164 sets of experimental conditions, including 33 types of periodic waves from the second to the fifth order and six different current velocities. The effects of water depth, current velocity, wave period, and height on the wave height and wavelength in the wave–current interaction field were investigated. A comparison of the mean relative error (MRE) and the determination coefficient (R2) of the wavelength with the experimental data revealed that the third-order wave–current theory outperformed the traditional linear theory, with an optimal reduction of 75% and an enhancement of 25%, respectively. Additionally, the third-order wave–current theory reduces the MRE by 25%–40% in the wave height calculation, with R2 consistently outperforming the linear theory. The third-order wave–current theory can significantly improve the calculation accuracy of the theoretical method in solving nonlinear wave–current interactions.