to the wave crests. The phase and group speeds, cp and cg respectively, are the lengths of the phase and group velocity vectors: cp = |cp| and cg = |cg|. Dynamical significance The radiation stress tensor is an important quantity in the description of the phase-averaged dynamical interaction between waves and mean flows. Here, the depth-integrated dynamical conservation equations are given, but – in order to model three-dimensional mean flows forced by or interacting with surface waves – a three-dimensional description of the radiation stress over the fluid layer is needed. Mass transport velocity Propagating waves induce a – relatively small – mean mass transport in the wave propagation direction, also called the wave (pseudo) momentum. To lowest order, the wave momentum Mw is, per unit of horizontal area: which is exact for progressive waves of permanent form in irrotational flow. Above, cp is the phase speed relative to the mean flow: with σ the intrinsic angular frequency, as seen by an observer moving with the mean horizontal flow-velocity while ω is the apparent angular frequency of an observer at rest (with respect to 'Earth'). The difference k⋅ is the Doppler shift. The mean horizontal momentum M, also per unit of horizontal area, is the mean value of the integral of momentum over depth: with v(x,y,z,t) the total flow velocity at any point below the free surface z = η(x,y,t). The mean horizontal momentum M is also the mean of the depth-integrated horizontal mass flux, and consists of two contributions: one by the mean current and the other (Mw) is due to the waves. Now the mass transport velocity is defined as: Observe that first the depth-integrated horizontal momentum is averaged, before the division by the mean water depth (h+) is made. Mass and momentum conservation Vector notation The equation of mean mass conservation is, in vector notation: with including the contribution of the wave momentum Mw. The equation for the conservation of horizontal mean momentum is: where ⊗ denotes the tensor product of with itself, and τw is the mean wind shear stress at the free surface, while τb is the bed shear stress. Further I is the identity tensor, with components given by the Kronecker delta δij. Note that the right hand side of the momentum equation provides the non-conservative contributions of the bed slope ∇h, as well the forcing by the wind and the bed friction. In