and

sin2 θ M2C

sin θ cos θ M2C

M2C

% cos2 θ

&2

(II-3-22)

M*x *M*y*

M*y * 2

(3) These equations are solved for a set of rays for each wave component of interest (typically

combinations of periods and directions). Since this analysis is linear, often a unit wave height is applied for

the offshore wave height, which yields a series of refraction and shoaling coefficients at sites of interest. Then

the wave transformation for any non-unit initial wave height is obtained by multiplication. This is permissible

as long as wave breaking does not occur along a wave ray.

(1) Estimating wave propagation patterns with wave rays is intuitively and visually satisfying, and often

very useful. The engineer obtains a good picture of how a wave propagates to a site. However, the procedure

has several drawbacks when applied to even mildly irregular bathymetry. One problem is ray

convergence/crossing; another is bathymetry inadequacy on ray paths.

(2) An example calculation from Noda et al. (1974) illustrates the basic problem. Bathymetry is highly

regular, but has undulatory contours (Figure II-3-7). From the ray pattern, convergence and divergence of

adjacent rays are apparent as the waves sweep over the undulations in bathymetry. However, in shallow

water near the shore, the rays are sufficiently perturbed by the bathymetry that several converge, with the ray

spacing going to zero (in some ray programs the rays actually are computed to cross). Remembering the

conservation of wave energy argument used to define the refraction coefficient, the flux across an orthogonal

between the rays remains constant. As the spacing between rays approaches zero, the energy flux becomes

infinite. Practically, if strong wave convergence occurs, breaking either due to depth constraints (Part II-4)

or steepness constraints (Part II-1) naturally limits the wave height. However, situations which generate

strong gradients or discontinuities in wave height along a wave crest give rise to *diffraction *effects, which

can reduce the wave height and keep it below the breaking value.

(3) The second problem with ray theory is the sensitivity of the wave ray calculations. In most locations,

the bathymetry is not well-known. Discretizations of the bathymetry can produce sharper local gradients in

the computational depth field than may exist locally or, conversely, may reduce local gradients. Most wave

ray calculation schemes calculate each wave ray uncoupled from all others. Ray paths are very sensitive to

gradients in bathymetry. The smoothing algorithms that are used to numerically compute the required

derivatives can alter the ray field significantly. Since the ray calculations are uncoupled, adjacent rays may

take radically different paths due to how the bathymetry was discretized or smoothed. Also, if the ray

calculations were started at slightly different spatial locations, the resulting patterns may be significantly

different for the same reason. *In the cases where ray patterns are unstable with respect to perturbations of*

(4) Wave propagation discussion has centered on the concept of waves traveling from deep water to

shallow. At some locations, the bathymetry is such that waves propagating from offshore towards a beach

may initially propagate from deeper to shallow water, then propagate across a zone where the water becomes

deeper again. In the region where the wave is propagating at an angle to the progressively shallower depths,

the process of refraction previously described occurs: the waves turn more shore-normal. Once the depth

gradient reverses, the wave turns in the opposite direction (because of the reversed depth gradient, the part

Estimation of Nearshore Waves

II-3-15

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