EM 1110-2-1100 (Part II)
30 Apr 02
(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).
II-3-3. Refraction and Shoaling
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, hA , is less than the depth at B, hB,
the speed of the wave at A will be slower than that at B because
g
g
tanh k hA <
tanh k hB ' CB
CA '
(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.
a. Wave rays.
(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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