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WAVES AND WAVE-INDUCED HYDRODYNAMICS models can be used across the surf zone. Some examples are Scha¨ffer, Madsen, and Deigaard (1993) and Kennedy et al. (2000). If the initial wave ﬁeld is expanded in terms of slowly varying (in x) Fourier modes, Boussinesq equations yield a set of coupled evolution equations that predict the amplitude and phase of the Fourier modes with distance (Freilich andGuza 1984; Liu, Yoon, and Kirby 1985; and Kaihatu and Kirby 1998). Field applications of the spectral Boussinesq theory show that the model predictions agree very well with normally incident ocean waves (Freilich and Guza 1984). Elgar and Guza (1985) have shown that the model is also able to predict the skewness of the shoaling wave ﬁeld, which is important for sediment transport considerations. The KdV equation (from Korteweg and deVries 1895) results from the Boussi- nesq theory bymaking the assumption that thewaves can travel in one direction only. A large body of work exists on the mathematics of this equation and its derivatives such as the Kadomtsev and Petviashvili (1970) or K–P, equation, for KdV waves propagating at an angle to the horizontal coordinate system. In the surf zone and on the beach face, the simpler nonlinear shallow water equations (also fromAiry) canprovide goodestimates of thewaveformandvelocities because these equations lead to the formationof bores,which characterize thebroken waves: ∂u ∂t + u ∂u ∂x = −g ∂η ∂x (5.13) ∂η ∂t + ∂(h + η)u ∂x = 0 (5.14) Hibberd and Peregrine (1979) and Packwood (1980) were the early developers of this approach, andKobayashi and colleagues have produced several workingmodels (Kobayashi, De Silva, and Watson 1989; Kobayashi and Wurjanto 1992). 5.2.2 WAVE REFRACTION AND SHOALING As waves propagate toward shore, the wave length decreases as the depth decreases, which is a consequence of the dispersion relationship (Eq. (5.3)). The wave period is ﬁxed; the wavelength and hence the wave speed decrease as the wave encounters shallower water. For a long crested wave traveling over irregular bottom depths, the change in wave speed along the wave crest implies that the wave changes direction locally, or it refracts, much in the same way that light refracts as it passes through media with different indices of refraction.∗ The result is that the wave direction turns toward regions of shallowwater and away from regions of deepwater. This can create regions of wave focusing on headlands and shoals. The simplest representation of wave refraction is the refraction of waves propa- gating obliquely over straight and parallel offshore bathymetry. In this case, Snell’s law, developed for optics, is valid. This law relates the wave direction, measured by an angle to the x-axis (drawn normal to the bottom contour), and the wave speed C ∗ The classic physical example is a pencil standing in a water glass. When viewed from the side, the part of the pencil above water appears to be oriented in a different direction than the part below water. P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 5.2 WATER WAVE MECHANICS 95 in one water depth to that in deep water: sin θ C = sin θ0 C0 = constant, (5.15) where the subscript 0 denotes deep water. Wave refraction diagrams for realistic bathymetry provide a picture of howwaves propagate from theoffshore to the shoreline of interest. Thesediagrams canbedrawn by hand, if it is assumed that a depth contour is locally straight and that Snell’s law can be applied there. Typically at the offshore end of a bathymetric chart, wave rays of a given direction are drawn (where the ray is a vector locally parallel to the wave direction; following a ray is the same as following a given section of wave crest). Then each ray is calculated, contour by contour, to the shore line, with each depth change causing a change inwave direction according to Snell’s law.Nowmost of these calculations are done with more elaborate computer models or more sophisticated numerical wave models such as a mild-slope, parabolic, or Boussinesq wave model. Another effect of the change in wavelength in shallow water is that the wave height increases. This is a consequence of a conservation of energy argument and the decrease in group velocity (Eq. (5.8)) in shallow water in concert with the decrease in C (note that n, however, goes from one-half in deep water to unity in shallow water but that this increase is dominated by the decrease in C). This increase in wave height is referred to as shoaling. A convenient formula that expresses both the effects of wave shoaling and re- fraction is H = H0KsKr, (5.16) where H0 is the deep water wave height, Ks is the shoaling coefﬁcient, Ks = √ Cg0 Cg , and Kr is the refraction coefﬁcient, which for straight and parallel shoreline contours can be expressed in terms of the wave angles as follows: Kr = √ cos θ0 cos θ Given the deepwater wave height H0, the group velocity Cg0 , and the wave angle θ0, the wave height at another depth can be calculated (when it is used in tandem with Snell’s law above). Wave diffraction occurs when abrupt changes in wave height occur such as when waves encounter a surface-piercing object like an offshore breakwater. Behind the structure, no waves exist and, by analogy to light, a shadow exists in the wave ﬁeld. The crest-wise changes inwave height then lead to changes inwave direction, causing the waves to turn into the shadow zone. The process is illustrated in Figure 5.2, which shows the diffraction of waves from the tip of a breakwater. Note that the wave ﬁeld looks as if there is a point source of waves at the end of the structure. In fact, diffraction can be explained by a superposition of point wave sources along the crest (Huygen’s principle). P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 96 WAVES AND WAVE-INDUCED HYDRODYNAMICS Figure 5.2 Diffraction of waves at a breakwater (from Dean and Dalrymple 1991). 5.2.3 WAVE PROPAGATION MODELS Historically, wave models used to predict the wave height and direction over large areas were developed for a wave train with a single frequency, which is referred to as a monochromatic wave train in analogy to light. Monochromatic models for wave propagation can be classiﬁed by the phenomena that are included in themodel. Refraction models can be ray-tracing models (e.g., Noda 1974), or grid models (e.g., Dalrymple 1988).Refraction–diffractionmodels aremore elaborate, involving either ﬁnite element methods (Berkhoff 1972) or mathematical simpliﬁcations (such as in parabolic models, e.g., REF/DIF by Kirby and Dalrymple 1983). Spectral models entail bringing the full directional and spectral description of the waves from offshore to onshore. These models have not evolved as far as monochro- matic models and are the subject of intense research. Examples of such work are Brink-Kjaer (1984); Booij, Holthuijsen, and Herbers (1985); Booij and Holthuijsen (1987); and Mathiesen (1984). Recent models often include the interactions of wave ﬁelds with currents and bathymetry, the input of wave energy by the wind, and wave breaking. For example, Holthuijsen, Booij, and Ris (1993) introduced the SWAN model, which predicts directional spectra, signiﬁcant wave height, mean period, average wave direction, radiation stresses, and bottom motions over the model domain. The model includes nonlinear wave interactions, current blocking, refraction and shoaling, and white capping and depth-induced breaking. 5.2.4 WAVE BREAKING In deep water, waves break because of excessive energy input, mostly from the wind. The limiting wave height is taken as H0/L0 ≈ 0.17, where L0 is the deep water wavelength. In shallow water, waves continue to shoal until they become so large that they become unstable and break. Empirically, Battjes (1974) has shown that the breaking wave characteristics can be correlated to the surf similarity parameter ζ , which is P1: FCH/SPH P2: