input wave conditions must be known, and proceeds onshore until dry land is

encountered (taking into account wave setup). A major difference from

NMLong is the additional iteration procedures that are required to solve the

governing equations. In comparison to NMLong, NMLong-CW requires the

following iterative steps:

longshore current is not known (which could be a current resulting

from waves, wind, and an external current). Thus, after the

longshore current has been calculated by means of the longshore

momentum equation, the wave calculations have to be redone,

followed by a new current calculation. This iteration between the

waves and current continues until convergence is achieved at

predefined level of accuracy.

Equation 6 is solved via a Newton-Raphson technique, as opposed to

in NMLong where the dispersion relation (without a current) is

solved explicitly employing a Pad approximation.

(Equation 11), proceeding from one grid point to next, both the wave

angle and wavelength are unknown at the new grid point. These

quantities are coupled through Equation 6 (in NMLong the dispersion

relation can be solved independently of Snell's law, and vice versa).

Overall, these iterative requirements might make NMLong-CW considerably

more time demanding to run than NMLong, depending on the computer

capabilities.

In calculating for random waves, Monte-Carlo simulation is employed by

simulating a large number of individual waves belonging to a certain

probability density function (pdf), typically taken to be a Rayleigh

distribution in deep water. Computations are performed with the governing

equations for each individual wave, and the statistical wave properties are

derived from the series of waves obtained at respective cross-shore locations.

The number of waves selected should be large enough to yield statistically

stable values on the mean wave properties when averaging for all the waves.

The advantage of a Monte-Carlo simulation technique is that no inference for

the shape of the pdf in the nearshore is necessary; the shape is obtained in the

simulations. The disadvantage of the method is that possible wave-wave

interaction and associated energy transfer are neglected. For random waves,

the wave forcing terms (radiation stresses and roller momentum fluxes) are

determined as averages for the selected number of waves in the Monte-Carlo

simulation before they are used in the momentum equations.

The wave energy balance equation employed to calculate the roller

properties across-shore (Equation 32) is discretized according to,

2*P*D,*i*∆*x *+ *m*R,*i*+1 cos αi +1Cr2,*i*+1 - *g*β D mR,*i*+1∆*x*

(33)

cos αiCr2,*i *+ *g*β D∆*x*

where

∆x = grid cell length

23

Chapter 3 Wave Model