# CMS-Wave:Governing Equations

## Wave-action balance equation with diffraction

Taking into account the effect of an ambient horizontal current or wave behavior, CMS-Wave is based on the steady wave-action balance equation (Mase 2001)

 $\frac{\partial (c_x N) }{\partial x} + \frac{\partial (c_y N) }{\partial y} + \frac{\partial (c_{\theta} N) }{\partial \theta} = \frac{\kappa}{2 \sigma} \biggl[ (c c_g \cos ^2 \theta N_y)_y - \frac{c c_g}{2} \cos ^2 \theta N_{yy} \biggr] - \epsilon_b N - S$ (1)

where $N = E(\sigma,\theta)/\sigma$ is the wave-action density to be solved and is a function of frequency σ and direction θ. E(σ,θ) is spectral wave density representing the wave energy per unit water-surface area per frequency interval. In the presence of an ambient current, the wave-action density is conserved, whereas the spectral wave density is not (Bretherton and Garrett 1968; Whitham 1974). Both wave diffraction and energy dissipation are included in the governing equation. Implementation of the numerical scheme is described elsewhere in the literature (Mase 2001; Mase et al. 2005a). C and Cg are wave celerity and group velocity, respectively; x and y are the horizontal coordinates; Cx, Cy, and Cθ are the characteristic velocity with respect to x, y, and, θ respectively; Ny and Nyy denote the first and second derivatives of N with respect to y, respectively; κ is an empirical parameter representing the intensity of diffraction effect; εb is the parameterization of wave breaking energy dissipation; S denotes additional source Sin and sink Sds (e.g., wind forcing, bottom friction loss, etc.) and nonlinear wave-wave interaction term.

Symbol Description
$\sigma$ Wave frequency
$N$ Wave action
$E$ Spectral wave density
$c$ Wave celerity
$c_g$ Wave group velocity

# References

Mase, H., K. Oki, T. S. Hedges, and H. J. Li. 2005. Extended energy-balance-equation wave model for multidirectional random wave transformation. Ocean Engineering 32(8–9):961–985.---- Documentation Portal