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referred to as the zero-, ﬁrst-, second-, and third-mode edgewaves, is shown inFigure 5.8 and compared P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 108 WAVES AND WAVE-INDUCED HYDRODYNAMICS Figure 5.8 Free surface elevations corresponding to the ﬁrst four edge wave modes and a fully reﬂected wave as a function of dimensionless distance offshore. with the normally incident standing wave solution (Eq. (5.48)). Notice that the mode number n corresponds to the number of zero crossings of the water surface elevation. This wave motion must satisfy the following edge wave dispersion relationship: σ 2 = gλ(2n + 1)m, which relates the alongshore wave number λ to the wave frequency and the beach slope. Ursell (1952) has shown that a more accurate representation of the dispersion relationship is σ 2 = gλ sin(2n + 1)m, which is equivalent to the previous one for small beach slope m. The waveform described by (Eq. (5.51)) is a standing edge wave, which will not propagate along a beach. Propagating waveforms may be found by adding together two standing waves as before. The second standing wave might be proportional to sin λy sin σ t , in which case, we obtain η(x, y, t) = Ae−λx Ln(2λx) cos(λy − σ t), (5.56) a wave that propagates in the positive y direction. By subtracting, instead of adding, we can obtain a wave propagating in the opposite direction. Theedgewave solutionabove is valid only for planarbeaches; additionalmeans to solve Eq. (5.46) are required for other beach proﬁles. For those that can be described by an exponentially increasing depth, h = h0(1− e−r x), where h0 is the offshore constant depth, Ball (1967) developed analytical solutions for the wave motion. For arbitrary beach proﬁles, Holman and Bowen (1979) and Kirby, Dalrymple, and Liu (1981) provide numerical methods. The effects of a longshore current on edge waves has been investigated byHowd, Bowen, and Holman (1992), who showed that the inﬂuence of the current is of the same magnitude as a variable beach proﬁle. Furthermore, the equations govern- ing edge waves on a uniform longshore current can be cast into the same govern- ing equation as before through a transformation of the depth. After introducing a P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 5.4 LOW-FREQUENCY MOTIONS AT THE SHORELINE 109 current-modiﬁed effective depth deﬁned as h′(x) = h(x)( 1− V (x)C )2 , Eq. (5.46) still can be used for the solution. The celerity C is the speed of the edge wave C = σ/λ. The generation of edge waves in nature has been a question of great interest, particularly in conjunction with the study of nearshore circulation. Guza and Davis (1974) developed a model showing that edge waves can be generated through a nonlinear resonant mechanism with the incident wave train. An incident wave train with a frequency σ can generate two edge waves with frequencies σ/2, which are called subharmonic edge waves. This mechanism has been veriﬁed in the laboratory∗ (Guza and Inman 1975). It should be pointed out that, from linear analyses, edge waves on a beach with straight and parallel contours cannot be caused by an incident wave train from off- shore. This can be proven with Snell’s law: sin θ C = sin θ0 C0 If we consider a wave train generated in shallow water but directed obliquely off- shore at angle θ to the beach normal, there are two possibilities: the wave propagates offshore to deep water (so-called leaky modes), or, as the wave encounters deeper water, thewave angle becomes larger until ﬁnally thewave propagates directly along- shore. From Snell’s law, the wave angle θ becomes 90◦. Part of the crest will be in shallowwater and part in deeper water, which will turn the wave back onshore. Then, if a signiﬁcant portion of the wave is reﬂected back offshore, this process repeats. This means that the wave is trapped against the shoreline and it is an edge wave. In other words, if a wave generated in shallow water has a propagation angle between 0o and θc, the wave propagates offshore. For angles greater than θc, waves become trapped as edge waves. The only way edge waves can be generated by waves incident from offshore is if the bathymetry is sufﬁciently irregular so that waves can approach the shallow water with large angles of incidence. For man-made structures, two examples are the trapping of waves by the ends of breakwaters (Dalrymple, Kirby, and Seli 1986) and the reﬂection of waves from groins and jetties such that they have the correct wave angle as they propagate onto the straight and parallel beach contours. Gallagher (1971) proposed another mechanism for edge wave generation based on the concept that the incident wave ﬁeld, which is composed of many waves and many frequencies, can have wave groups occurring at a period that satisﬁes the edge wave dispersion relationship. Bowen andGuza (1978) examined the limit case of two ∗ A very graphic laboratory experiment in a wave basin with reﬂecting sidewalls to illustrate this mecha- nism is to generate normally incident waves onto a sloping beach. The wave period should be chosen so that the width of the wave basin (length of the beach) is nπ/λ, where n ≥ 1. For n = 2, one edge wave ﬁts along the beach. Within several minutes of starting the wave generator, the presence of the edge waves will be obvious, for the run-up on the beach is distorted strongly by their presence. P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 110 WAVES AND WAVE-INDUCED HYDRODYNAMICS waves with slightly different frequencies in a laboratory which formed wave groups. Edge waves were observed as predicted by this mechanism. Lippmann, Holman, and Bowen (1997) provide a review of the mechanisms for the generation of edge waves and show that edge wave generation by spatially and temporally varying radiation stresses of the incident waves can account for most of the low-frequency motion in the surf zone. They use the linear wave equation (Eq. 5.46) with radiation stress driving terms. In the ﬁeld, there are now numerous studies that show very large edge wave mo- tions in the surf zone. In a major study, Huntley, Guza, and Thornton (1981) showed, using a longshore array of current meters, that low-frequency energy in the surf zone had wave lengths associated with the edge wave dispersion relationship given above. Figure 5.9 shows the ﬁeld data and the associated dispersion relationships. This low- frequency energy in some cases can have more energy content locally within the shallow portions of the surf zone than the incident wave ﬁeld, which has important ramiﬁcations (as yet not properly elucidated) for coastal processes. Figure 5.9 Edge wave dispersion relationships and wave num- bers forTorreyPinesBeach (Huntley,Guza, andThornton1981, copyright by the American Geophysical Union). P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 5.5 NEARSHORE CIRCULATION AND RIP CURRENTS 111 Amanifestation of the low-frequency wave motion is the slowly varying location of the limit of wave uprush on the beach face. An observer standing at this limit of wave uprush may soon ﬁnd it necessary to move to other locations up and down the beach face as the surf zone slowly rises and falls owing to the edge wave or standing wave motions within the surf zone over the course of minutes. 5.4.3 SHEAR WAVES In 1986, a ﬁeld experiment atDuck,NorthCarolina, revealed a surprising behavior of the longshore current (Oltman-Shay, Howd, and Birkemeier 1989). The very strong longshore current that were caused by large waves began to oscillate with a low frequency. The alongshore wavelengths of this motion, detected by current meters distributedalong the shoreline,weremuch less thanpredictedbyedgewaveor gravity wave theories. Bowen and Holman (1989) provided a theory for a wave motion that depended on the cross-shore shear in the longshore