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Virtual wave stress in deep-water crossed surface waves

Waves excited on the surface of deep water decay in time and/or space due to the fluid viscosity, and the momentum associated with the wave motion is transferred from the waves to Eulerian slow currents by the action of the virtual wave stress. Here, based on the conservation of the total momentum, we found the virtual wave stress produced by calm gravity waves under the assumption that the slow Eulerian currents are weak in the sense that the Froude number is small and the scattering of surface waves by slow flow inhomogeneities can be neglected. In particular, we calculated the virtual wave stress generated by a propagating wave and two orthogonal standing waves. The obtained results possess Euler invariance, are consistent with previously known ones, and generalize them to the case of the excitation of almost monochromatic waves propagating in arbitrary directions.