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 spatial gradients in wave energy can induce a net force on the water body was first recognised by Longuet-Higgins and Stewart (1962[1], 1964[2]). The radiation stresses are the elements of a stress tensor representing the phase-averaged momentum transferred through the water body (the flux of momentum) by the horizontal wave orbital motion and the non-hydrostatic component of the pressure. The radiation stresses are given by (see Shallow-water wave theory#Radiation Stress (Momentum Flux))

$S_{XX} =\Bigl\langle \int _{-h}^{\eta } (p+\rho u^{2} )dz \Bigr\rangle - \int _{-h}^{0} p_0 dz , \qquad S_{YY} =\Bigl\langle \int _{-h}^{\eta }(p+\rho v^{2} )dz \Bigr\rangle -\int _{-h}^{0}p_0 dz , \qquad S_{XY} =\Bigl\langle \int _{-h}^{\eta } \rho u v dz \Bigr\rangle , \qquad (1)$

where $z=$ vertical coordinate, $h=$ water depth, $\eta =$ wave surface elevation, $\rho =$ water density, $p =$ pressure, $p_0 =$ hydrostatic pressure, $u=$ horizontal orbital velocity in $x$-direction, $v=$ horizontal orbital velocity in $y$-direction, and where $\bigl\langle … \bigr\rangle$ represents the average over the wave period.

For obliquely incident waves, cross-shore momentum is transferred by both cross-shore wave orbital motion and longshore wave orbital motion and longshore momentum is transferred by both longshore wave orbital motion and cross-shore wave orbital motion. For obliquely incident waves, a cross-shore gradient in the wave orbital motion, for example due to wave breaking, will exert a stress on the water mass in cross-shore direction as well as in longshore direction. A gradient in the stress in cross-shore direction generates a water level set-up 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.

Approximate analytical expressions of the radiation stresses can be obtained from linear wave theory. Assuming a uniform wave field propagating with an angle $\theta$ to the cross-shore $x$-axis the expressions are

$S_{XX} \approx \Big(n(1+\cos^2 \theta) - \large\frac{1}{2}\normalsize \Big) E , \qquad S_{YY} \approx \Big(n(1+\sin^2 \theta) - \large\frac{1}{2}\normalsize \Big) E , \qquad S_{XY} \approx nE \sin \theta \cos \theta , \qquad (2)$

where $E = \large\frac{1}{8}\normalsize \rho g H^2 =$ wave energy, $H=$ wave height, $k=$ wave number, $g=$ gravitational acceleration, and $n = \large\frac{1}{2}\normalsize + \Large\frac{kh}{\sinh (2kh)}\normalsize =$ ratio of group celerity and wave celerity.

These expressions underestimate the radiation stresses when waves become skewed and asymmetric while propagating into the shoaling zone - see Shallow-water wave theory#Finite amplitude waves. For example, calculations with 5th order nonlinear Stokes theory give a 17% higher value of $S_{XX}$ in the case of very steep waves[3]. On the other hand, in the surf zone the expressions (2) with sinusoidal waves overestimate the radiation stresses[4].

For irrotational periodic (regular) gravity waves an exact expression of $S_{XX}$ is given by[5]

$S_{XX} = 4 E_k - 3 E_p + \rho h \bigl\langle u_b^2 \bigr\rangle , \quad E_k = \large\frac{1}{2}\normalsize \rho \Bigl\langle \int_{-h}^{\eta} (u^2+w^2)dz \Bigr\rangle , \quad E_p = \large\frac{1}{2}\normalsize \rho g \bigl\langle \eta^2 \bigr\rangle ,$

where $u=$ cross-shore wave orbital velocity, where $u_b=$ cross-shore wave orbital velocity at the bottom, $w=$ vertical wave orbital velocity.

## References

1. Longuet-Higgins, 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
2. Longuet-Higgins, M.S. and Stewart, R.W. 1964. Radiation stresses in water waves; a physical discussion, with applications. Deep Sea Research 11: 529–562
3. Gao, X., Ma, X., Li, P., Yuan, F., Wu, Y. and Dong, G. 2023. Nonlinear analytical solution for radiation stress of higher-order Stokes waves on a flat bottom. Ocean Engineering 286 (2023) 115622
4. Madsen, P.A., Sorensen, O.R. and Schäffer, H.A. 1997. Surf zone dynamics simulated by a Boussinesq type model. Part I. Model description and cross-shore motion of regular waves, Coastal Engineering 32: 255-287
5. Longuet-Higgins, 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