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ρu(x, z) dzdt, (5.21) where the time interval between t1 and t2 is a long time (many wave periods for irregular waves; one wave period for periodic waves). If we integrate over the depth from the bottom only to themeanwater surface, z = 0, rather than the instantaneous water surface η, we obtain M equal to zero, as predicted by linear theory. If we 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 101 continue the integration up to η, the mass transport becomes M = E C , (5.22) which shows that there is a nonlinear transport of water in the wave direction due to the larger forward transport of water under the wave crest because the total depth is greater when compared with the backward transport under the trough. From this formula, the mass transport is larger for more energetic waves. Thismass transporthasmomentumassociatedwith it,whichmeans that forceswill be generatedwhenever thismomentum changesmagnitude or direction byNewton’s second law. To determine this momentum, we integrate themomentum ﬂux from the bottom to the surface as follows: M = 1 t2 − t1 ∫ t2 t1 ∫ η −h (ρu)u dzdt (5.23) This quantity has as a ﬁrst approximationM = MCg = En, which indicates that the ﬂux of momentum is described by the mass transport times the group velocity. Offshore of the breaker line, there is a depression of the mean water level from the still water level due to the waves, which is called setdown and is denoted as η. This quantity, originally elucidated by Longuet-Higgins and Stewart (1963), is (e.g., Dean and Dalrymple 1991) η = − H 2k 8 sinh 2kh , (5.24) where, again, H is the wave height. Because the wave height increases as waves shoal, the setdown increases as well, reaching a maximum at the breakerline that is approximately 5 percent of the breaking water depth. Longuet-Higgins and Stewart (1963) introduced the concept of wave momentum ﬂux, designating the sum of the momentum ﬂux and the mean pressure as the radia- tion stress, based on the analog with light, which develops a radiation pressure when shining on an object. The quantities are related in the following way: M+ 1 2 ρgh2 = Sxx + 12ρg(h + η) 2, (5.25) where Sxx is the radiation stress representing the ﬂux in the x direction of the x component of momentum and η is the mean water level elevation. The formula for Sxx is Sxx = E ( 2n − 1 2 ) (5.26) There is an equivalent term for y-momentum carried in the y direction and the mean pressure, which is Syy = E ( n − 1 2 ) (5.27) These expressions apply for waves traveling in the x direction. If the waves were traveling in the θ direction, where θ is an angle the wave direction makes with the P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 102 WAVES AND WAVE-INDUCED HYDRODYNAMICS x-axis, then we would have the following radiation stresses: Sxx = E [ n(cos2 θ + 1)− 1 2 ] (5.28) Syy = E [ n(sin2 θ + 1)− 1 2 ] (5.29) Sxy = Syx = En sin 2θ2 (5.30) The last equation is for the ﬂux of x momentum in the y direction, or vice versa, and it arises owing to the obliquity of the waves to the coordinate axis. This equation can be rewritten in the following form by introducing the wave celerity in numerator and denominator: Sxy = EnC cos θ ( sin θ C ) The ﬁrst part of this expression is recognized as the shoreward ﬂux of wave energy, which, on a beach characterized by straight and parallel contours, is constant until breaking begins in the surf zone, and the second term, in parentheses, is Snell’s law, which is also constant. Therefore, for this idealized beach, Sxy is constant from offshore to the breaker line. 5.2.6 WAVE SETUP The wavemomentum ﬂuxes are proportional to the wave energy. If the waves break, then themomentum ﬂux decreases. This change inmomentum ﬂuxmust be balanced by forces; therefore, wave breaking induces forces in the surf zone that act in thewave direction.Wewill nowexamineeachof these forces by resolving them into anonshore and alongshore direction. The onshore-directed momentum ﬂux is Sxx . As the wave propagates into the surf zone, the momentum ﬂux is equal to its value at the breaker line. At the limit of wave uprush, the value is zero. This gradient in momentum ﬂux is balanced by a slope in water level within the surf zone ∂η/∂x . Balancing the forces, the following differential equation results: ∂η ∂x = − 1 ρg(h + η) ∂Sxx ∂x (5.31) Integrating the mean water level slope, we get the wave set up η, as noted earlier. Us- ing the shallow water asymptote and the spilling breaker assumption in the radiation stress terms, Sxx becomes Sxx = 316ρgκ 2(h + η)2, where κ is the breaking index. As noted earlier, the value of κ is about 0.8 for spilling breakers. For a monotonic beach proﬁle, we can solve Eq. (5.31) for η as follows: η = ηb −K(h − Hb), (5.32) P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 5.3 CROSS-SHORE AND LONGSHORE CURRENTS 103 where ηb is the mean water level at the breaker line and K = (3κ2/8)/(1+ 3κ2/8). Experiments have been carried out to verify this model in the laboratory (with good results; see Bowen, Inman, and Simmons 1968) and in the ﬁeld. (This is the same wave setup that was introduced in the last chapter as a component of storm surge.) 5.3 CROSS-SHORE AND LONGSHORE CURRENTS The wave-induced mass transport M must engender a return ﬂow (which on a long- shore uniform beach is the undertow), for there can be no net onshore ﬂow of water because of the presence of the beach. The amount of seaward mass ﬂux is therefore equal to M . This ﬂow is not distributed uniformly over the depth but has a distinct proﬁle caused by the variation in wave-induced stress over the depth. In the alongshore direction, the greatest change in radiation stress occurs owing to the Sxy term, which is affected, of course, by breaking. To balance this change in momentumﬂux, a longshorewater level slope is possiblewhere the shoreline is short; bounded at the ends by headlands, inlets, or man-made structures; or when there are rip currents. For an inﬁnitely long uniform shoreline with uniform wave conditions, this water-level slope cannot exist, for it leads to inﬁnite or negative water depths in the surf zone. Some other mechanism for developing a longshore balancing force is required.Mean currents ﬂowing along the shorelinewill develop bottom stresses and canbalance the gradients in the radiation stress terms.The resulting longshore current is then directly engendered by the obliquely incident waves and the process of wave breaking. By balancing the frictional forces and the gradients in the radiation stress, Bowen (1969a), Longuet-Higgins (1970a and b), and Thornton (1970) developed equations for generating the longshore current and, in fact, since these early models, numerous studies have been made of the current ﬁeld. The steady-state equation of motion in the alongshore direction is −ρg(h + η)∂η ∂y − [ ∂Sxy ∂x + ∂Syy ∂y ] + [τs − τb]+ ∂[(h + η)τxy] ∂x = 0, (5.33) in which the last term represents lateral shear stress coupling. The y derivatives can be neglected for the case of a long straight beach because these terms would lead to inﬁnitemagnitudes of η and Syy if they existed. The remaining terms (after neglecting the surface shear stress for the case of no wind and lateral shear stress coupling) can be written as ∂Sxy ∂x = −τb (5.34) Determining a simple expression for the bottom friction term that results from the mean current that is coexistent with the oscillatory wave ﬁeld is difﬁcult. Longuet- Higgins (1970) developed what Liu and Dalrymple (1978) call the small (incident wave) angle model deﬁned by τb = ρ f4π umV (5.35) The f is the empirical Darcy–Weisbach friction factor, used in pipe ﬂow calculations, which is known to be a function of the ﬂowReynolds number and the sand roughness, P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27,