Radiation stress
Definition of Radiation stress:
Radiation stress is the flux of momentum carried by ocean waves.
This is the common definition for Radiation stress, other definitions can be discussed in the article

Notes
The fact that waves can exert a net force on the water body directed along the wave propagation direction was first recognised by LonguetHiggins and Stewart (1962^{[1]}, 1964^{[2]}). The radiation stress is the momentum transferred through the water body per unit time (the flux of momentum) by wave orbital motion. For shorenormal waves, the radiation stress induced by the dynamic pressure of crossshore wave orbital motion is usually indicated by the symbol [math]S_{XX}[/math]. The radiation stress is called a stress because for obliquely incident waves, crossshore momentum is transferred by both crossshore wave orbital motion and longshore wave orbital motion and longshore momentum is transferred by both longshore wave orbital motion and crossshore wave orbital motion. For obliquely incident waves, a crossshore gradient in the wave orbital motion, for example due to wave breaking, will exert a stress on the water mass in crossshore direction as well as in longshore direction. A gradient in the stress in crossshore direction generates a water level setup at the coast and a gradient in the stress in longshore direction generates a longshore current. Forcing by radiation stress gradients related to wave breaking is commonly an order of magnitude greater than forcing due to wind stress or other wave nonlinearities.
Analytical expressions for the radiation stresses based on linear wave theory are given in Shallowwater wave theory#Radiation Stress (Momentum Flux) and Shallowwater wave theory#Radiation Stress Components for Oblique Waves. These expressions underestimate the radiation stresses when waves become skewed and asymmetric while propagating into shallow water  see Shallowwater wave theory#Finite amplitude waves. Calculations with 5th order nonlinear Stokes theory give a 17% higher value of [math]S_{XX}[/math] in the case of very steep waves^{[3]}.
For irrotational periodic (regular) gravity waves an exact expression of [math]S_{XX}[/math] is given by^{[4]}
[math]S_{XX} = 4 E_k  3 E_p + \rho h \bigl\langle u_b^2 \bigr\rangle , \quad E_k = ½ \rho \Bigl\langle \int_{h}^{\eta} (u^2+w^2)dz \Bigr\rangle , \quad E_p = ½ \rho g \bigl\langle \eta^2 \bigr\rangle , [/math]
where [math]u=[/math] crossshore wave orbital velocity, where [math]u_b=[/math] crossshore wave orbital velocity at the bottom, [math]w=[/math] vertical wave orbital velocity, [math]\eta=[/math] wave surface elevation, [math]h=[/math] water depth, [math]\rho=[/math] water density, [math]g=[/math] gravitational acceleration, and where [math] \bigl\langle … \bigr\rangle [/math] represents the average over the wave period.
References
 ↑ LonguetHiggins, M.S. and Stewart, R.W. 1962. Radiation stress and mass transport in gravity waves, with application to 'surf beats'. Journal of Fluid Mechanics 13: 481–504
 ↑ LonguetHiggins, M.S. and Stewart, R.W. 1964. Radiation stresses in water waves; a physical discussion, with applications. Deep Sea Research 11: 529–562
 ↑ Gao, X., Ma, X., Li, P., Yuan, F., Wu, Y. and Dong, G. 2023. Nonlinear analytical solution for radiation stress of higherorder Stokes waves on a flat bottom. Ocean Engineering 286 (2023) 115622
 ↑ LonguetHiggins, M.S. 1975. Integral properties of periodic gravity waves of finite amplitude. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 342 (1629): 157–174