depth. Bottom friction and dissipation from the surrounding land boundaries (i.e. entrance

losses at the mouth of a harbor) may also be empirically incorporated into MSE models.

A simplified version of the MSE is known as the ` arabolic approximation' (PA), which

p

usually greatly reduces the excessive computational demands of MSE model at the

expense of further assumptions and simplifications which may render the numerical

predictions inaccurate and inappropriate for many coastal and ocean engineering problems

(Panchang et al. 1998).

The only purpose of adapting the PA is to convert the MSE to a set of simpler

equations that describe a wave propagating in a prescribed direction while still taking both

refraction and diffraction in the lateral direction into account. The greatest advantage of

PA is its numerical efficiency, it can be solved rather easily by numerical means and thus

could be used for predicting wave transformation over a relatively large coastal region.

When reflection is of major interest, as it is in harbors, the MSE should be used since the

PA ignores reflection. One must also be reminded that the PA assumes that the length

scale of the wave amplitude variation in the direction of wave propagation (x direction) is

much longer than that in the transverse direction (y direction). The PA is derived on the

assumption that percentage changes of depth within a typical wavelength are small

compared to the wave slope. For details about PA models, see Booij (1981), Liu (1983),

Kirby (1983), Liu and Tsay (1984), and Kirby and Dalrymple (1984). The PA has been

verified extensively by laboratory studies and field applications (Berkhoff et al. 1982), Liu

and Tsay (1984), Kirby and Dalrymple (1984), Vincent and Briggs (1989), (Demirbilek

1994, Demirbilek et al. 1996a and 1996b), and Panchang et al (1998).

The mild-slope wave equation (also known as the "combined refraction-

diffraction" equation), first suggested by Eckart (1952) and later re-derived by Berkhoff

(1972, 1976) and others, is now well-accepted as the method for estimating coastal wave

conditions. It can be used to model a wide spectrum of waves, since it passes, in the limit,

to the deep and shallow water equations. Although the equation was developed in the

mid-seventies, computational difficulties precluded the development of a model for the

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