(2) Wave breaking. The model tests whether the local wave height has exceeded a fixed threshold, which

is set at *h /d *= 0.78. For local wave heights exceeding this value, a breaking wave energy flux decay model

is started in order to remove energy from the wave train. The model used is described in Dally, Dean, and

Dalrymple (1985). The reader is referred to Kirby and Dalrymple (1986a) for further details.

(3) Wave damping mechanisms.

(a) In addition to the strong wave breaking mechanism described above, REF/DIF 1 also provides the

user with three selectable bottom damping mechanisms. These are: laminar bottom boundary layer damping,

sand-bed percolation damping, and turbulent bottom boundary layer damping.

(b) At present, no laboratory or field data sets clearly point to the need for including bottom damping

effects in model simulations. Laboratory experiments usually include too short a propagation distance for

damping effects to accumulate significantly. In the field, damping due to bottom effects may be balanced

or overshadowed by wave growth resulting from wind-wave interaction, and so one should not be considered

in the absence of the other. At present, it is recommended that these user-selectable damping mechanisms

not be included in model simulations.

(4) Wave nonlinearity.

(a) Wave nonlinearity has a strong effect on the phase speed of waves and thus can significantly modify

both refraction and diffraction effects. For example, waves shoaling on a plane beach refract more slowly

than predicted by linear theory, since the increase in wave height with decreasing water depth speeds up the

waves, in opposition to the direct, linear-theory effect decreasing depth, which slows them. Diffraction

effects are typically enhanced. Phase speed is greater in a high-amplitude, illuminated area than in a low-

amplitude, shadowed area; this causes refractive bending of waves into the shadow area, causing an increase

in wave height in the shadow zone relative to the predictions of linear theory.

(b) REF/DIF 1, designed to predict the propagation of a monochromatic wave in intermediate water

depth, includes the effects of nonlinearity as predicted by third-order Stokes wave theory (Kirby and

Dalrymple 1983). Since the model is often used to predict wave-height distributions into the surf zone and

up to dry land boundaries, the model must also be corrected to avoid the singularities arising from the

invalidity of Stokes theory in shallow water. In order to provide a smooth correction to the model results in

the shallow-water limit, Kirby and Dalrymple (1986b) provided an algorithm that gives a smooth patch

between Stokes theory and an empirical modification to linear theory developed by Hedges (1976). The

approximate theory does not cause any degradation in solution accuracy in comparison to the Stokes theory

alone for intermediate depth experiments; see Kirby and Dalrymple (1986b) for relevant documentation.

(5) Numerical noise filter. Higher-order forms of the parabolic approximation have the undesirable effect

of allowing high-wave number noise (i.e., noise with rapid lateral variation) to propagate rapidly across the

computational grid. This effect has been described in detail by Kirby (1986a), and is usually found in

association with the start of surf zones around complicated planforms such as island shores. The resulting

noise component may be damped by the application of various types of smoothing filters. The three-point

moving average filter described by Kirby (1986a) has been found to be heavy-handed in practical

applications, and has been replaced in present versions of the REF/DIF 1 model by a damping filter included

in the governing differential equation, whose effect is centered around the lateral wave number, which spread

rapidly in the undamped model. A full description of the damping method and a range of tests may be found

in Kirby (1993).

(6) Examples of REF/DIF1 results laboratory verification. REF/DIF 1 (and the parabolic approximation

model in general) are capable of providing a detailed picture of the water surface in the region of study if the

Estimation of Nearshore Waves

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