
Modeling Principles
There are two main classes of wave models:
 Phaseresolving 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
$\zeta (x,y,t)$
are computed
 Phaseaveraged 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.
Phaseresolving Models
Phaseresolving 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 timedependent models in this class, and stationary models
in which the wave motion is purely periodical.
The most widely used timedependent models are based on the Bousinesq equation.
This equation is the result of an integration over depth of the original 3dimensional
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 3dimensional 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.
$\zeta (x,y,t)\; =\; Re[\; {\rm Z}(x,y)exp(i\omega t)]$,
where ${\rm Z}(x,y)$ 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 socalled mildslope equation allows waves in all directions.
The solution method requires the solution of a large system of linear equations.
The parabolic approximation of the mildslopeequation 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.
Phaseresolving models have originally been developed for regular wave motion. Irregular sea states
can be computed; in the case of the mildslopeequation 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 3d model the same can be done, or one can
extend the computation over a longer time span, using an irregular incident wave field.
All phaseresolving 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 nonlinear, and it is used (mainly in research environments)
to study nonlinear effects in shallow water.
The mildslopeequation on the other hand is linear.
Phaseaveraged Models
Phaseaveraged 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}$is\; used\; which\; is\; calculated\; from\; derivatives\; of\; the\; wave\; amplitude.\; The\; most\; simple\; approximation\; uses\; an\; amplitude\; based\; on\; the\; total\; energy\; of\; the\; spectrum.\; A\; second\; approximation\; uses\; a\; different\; wave\; amplitude\; for\; each\; spectral\; frequency,\; i.e.\; based\; on\; integrals\; of\; the\; energy\; density\; over\; direction.\; The\; figure\; shows\; the\; results\; obtained\; by\; SWAN\; with\; and\; without\; diffraction\; for\; an\; experiment\; by\; Yu\; et\; al.\; In\; this\; experiment\; waves\; are\; transmitted\; through\; a\; gap\; in\; a\; breakwater.\; The\; spectrum\; of\; the\; incident\; waves\; is\; a\; single\; peak,\; so\; in\; this\; case\; there\; is\; no\; difference\; betweeen\; the\; two\; approximations.$

