## Modeling Principles

There are two main classes of wave models:

Phase-resolving models
In these models the sea surface is resolved, i.e. the surface is covered with a grid which is fine compared with the wave length, and the gridded values of the vertical displacement $\zeta \left(x,y,t\right)$ are computed
Phase-averaged models
In these models the statistics of the sea surface is computed, i.e. on points of a grid the action or energy spectra are computed.

### Phase-resolving Models

Phase-resolving models can be used only for regions of limited size because the computational grid has to be fine enough to resolve all relevant wave lengths. The actual limitation depends on the type of model used.
There are time-dependent models in this class, and stationary models in which the wave motion is purely periodical.

The most widely used time-dependent models are based on the Bousinesq equation. This equation is the result of an integration over depth of the original 3-dimensional wave equation. The integration over depth limits the Bousinesq equation to relatively shallow water, i.e. the depth must not be large compared with the wave length. Recent research has relaxed this limit to a certain extent.
The limit does not apply for the 3-dimensional model SWASH developed recently by Stelling and Zijlema (see the SWASH site and the paper "SWASH: An operational public domain code for simulating wave fields and rapidly varied flows in coastal waters" by Marcel Zijlema, Guus Stelling and Pieter Smit, Coastal Engineering 58 (2011) 992–1012). Because this model resolves also the vertical, it requires more computer time than the Bousinesq equation, but due to a sophisticated treatment of the vertical the difference is not prohibitive any more.

Stationary models assume a harmonic wave motion, i.e. $\zeta \left(x,y,t\right) = Re\left[ Ζ\left(x,y\right)exp\left(i\omega t\right)\right]$, where $Ζ\left(x,y\right)$ is a complex variable. The absolute value of this variable is the amplitude of the wave, and the argument is related to the phase.
The partial differential equation governing Z was derived by Berkhoff (1972). This so-called mild-slope equation allows waves in all directions. The solution method requires the solution of a large system of linear equations.
The parabolic approximation of the mild-slope-equation allows waves in a limited set of directions. The computation progresses from a seaward boundary on which an incident wave field is defined, towards the shore. This procedure makes the model suitable for regions of the order of a few hundred wave lengths, and it has been applied to regions extending from the coast to 10 or 20 m depth. The fact that reflected waves cannot be taken into account makes this method unsuitable for harbours.

Phase-resolving models have originally been developed for regular wave motion. Irregular sea states can be computed; in the case of the mild-slope-equation one has to make computations for a range of wave frequencies, and add the results taking into account the distribution of energy over the frequencies.
In the case of the Bousinesq equation and the 3-d model the same can be done, or one can extend the computation over a longer time span, using an irregular incident wave field.

All phase-resolving models mentioned here account for refraction and diffraction, including the effect of currents. Mild dissipation effects are accounted for as well. The Bousinesq equation is non-linear, and it is used (mainly in research environments) to study non-linear effects in shallow water. The mild-slope-equation on the other hand is linear.

### Phase-averaged Models

Phase-averaged models are based on the energy or action balance equation.

#### Approximate diffraction

An approximation of diffraction for use in spectral models was developed by Herman et al(2002). In this approximation an additional $c$θ$is used which is calculated from derivatives of the wave amplitude. The most simple approximation uses an amplitude based on the total energy of the spectrum. A second approximation uses a different wave amplitude for each spectral frequency, i.e. based on integrals of the energy density over direction. The figure shows the results obtained by SWAN with and without diffraction for an experiment by Yu et al. In this experiment waves are transmitted through a gap in a breakwater. The spectrum of the incident waves is a single peak, so in this case there is no difference betweeen the two approximations.$