at its own – higher – phase velocity. The amplitude of this bound long wave varies with the square of the wave height, and is only significant in shallow water; Wave–current interaction – in varying mean-flow fields, the energy exchanges between the waves and the mean flow, as well as the mean-flow forcing, can be modeled by means of the radiation stress. Definitions and values derived from linear wave theory One-dimensional wave propagation For uni-directional wave propagation – say in the x-coordinate direction – the component of the radiation stress tensor of dynamical importance is Sxx. It is defined as: where p(x,z,t) is the fluid pressure, is the horizontal x-component of the oscillatory part of the flow velocity vector, z is the vertical coordinate, t is time, z = −h(x) is the bed elevation of the fluid layer, and z = η(x,t) is the surface elevation. Further ρ is the fluid density and g is the acceleration by gravity, while an overbar denotes phase averaging. The last term on the right-hand side, ½ρg(h+)2, is the integral of the hydrostatic pressure over the still-water depth. To lowest (second) order, the radiation stress Sxx for traveling periodic waves can be determined from the properties of surface gravity waves according to Airy wave theory: where cp is the phase speed and cg is the group speed of the waves. Further E is the mean depth-integrated wave energy density (the sum of the kinetic and potential energy) per unit of horizontal area. From the results of Airy wave theory, to second order, the mean energy density E equals: with a the wave amplitude and H = 2a the wave height. Note this equation is for periodic waves: in random waves the root-mean-square wave height Hrms should be used with Hrms = Hm0 / , where Hm0 is the significant wave height. Then E = ρgHm02. Two-dimensional wave propagation For wave propagation in two horizontal dimensions the radiation stress is a second-order tensor with components: With, in a Cartesian coordinate system (x,y,z): where and are the horizontal x- and y-components of the oscillatory part of the flow velocity vector. To second order – in wave amplitude a – the components of the radiation stress tensor for progressive periodic waves are: where kx and ky are the x- and y-components of the wavenumber vector k, with length k = |k| = and the vector k perpendicular