(2) All cases are important, but the first and third are relatively complex and require a numerical model

for reasonable treatment. The second case, swell propagating across a shallow region, is a classic building

block that has served as a basis for many coastal engineering studies. Often the swell is approximated by a

monochromatic wave, and simple refraction and shoaling methods are used to make nearshore-wave

estimates. Since the process of refraction and shoaling is important in coastal engineering, the next section

is devoted to deriving some simple approaches to illustrate the need for more complex approaches.

(3) Often it is necessary for engineers to make a steady-state assumption: i.e., wave properties along the

outer boundary of the region of interest and other external forcing are assumed not to vary with time. This

is appropriate if the rate of variation of the wave field in time is very slow compared to the time required for

the waves to pass from the outer boundary to the shore. If this is not the case, then a time-dependent model

is required. Cases (a) and (c) would more typically require a time-dependent model. Time-dependent models

are not discussed here due to their complexity. Examples are described by Resio (1981), Jensen et al. (1987),

WAMDI (1988), Young (1988), SWAMP Group (1985), SWIM Group (1985), and Demirbilek and Webster

(1992a,b).

In order to understand wave refraction and shoaling, consider the case of a steady-state, monochromatic (and

thereby long-crested) wave propagating across a region in which there is a straight shoreline with all depth

contours evenly spaced and parallel to the shoreline (Figure II-3-3). In addition, no current is present. If a

wave crest initially has some angle of approach to the shore other than 0 deg, part of the wave (point A) will

be in shallower water than another part (point B). Because the depth at *A, h*A , is less than the depth at *B, h*B,

the speed of the wave at A will be slower than that at B because

tanh *k h*A <

tanh *k h*B ' *C*B

(II-3-2)

ω

ω

The speed differential along the wave crest causes the crest to turn more parallel to shore. The propagation

problem becomes one of plotting the direction of wave approach and calculating its height as the wave

propagates from deep to shallow water. For the case of monochromatic waves, wave period remains constant

(Part II-1). In the case of an irregular wave train, the transformation process may affect waves at each

frequency differently; consequently, the peak period of the wave field may shift.

(1) The wave-propagation problem can often be readily visualized by construction of wave rays. If a

point on a wave crest is selected and a wave crest orthogonal is drawn, the path traced out by the orthogonal

as the wave crest propagates onshore is called a ray. Hence, a group of wave rays map the path of travel of

the wave crest. For simple bathymetry, a group of rays can be constructed by hand to show the wave

transformation, although it is a tedious procedure. Graphical computer programs also exist to automate this

process (Harrison and Wilson 1964, Dobson 1967, Noda et al. 1974), but to a large degree such approaches

II-3-6

Estimation of Nearshore Waves

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