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P1: FCH/FYX P2: FCH/FYX QC: FCH/UKS T1: FCH CB373-FM CB373 July 18, 2001 11:7 Char Count= 0 COASTAL PROCESSES with Engineering Applications ROBERT G. DEAN University of Florida ROBERT A. DALRYMPLE University of Delaware P1: FCH/FYX P2: FCH/FYX QC: FCH/UKS T1: FCH CB373-FM CB373 July 18, 2001 11:7 Char Count= 0 The Pitt Building, Trumpington Street, Cambridge, United Kingdom The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcón 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org First published in printed format ISBN 0-521-49535-0 hardback ISBN 0-511-03791-0 eBook Cambridge University Press 2004 2001 (Adobe Reader) © P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 CHAPTER FIVE Waves and Wave-Induced Hydrodynamics Freak waves are waves with heights that far exceed the average. Freak waves in storms have been blamed for damages to shipping and coastal structures and for loss of life. The reasons for these waves likely include wave reﬂection from shorelines or currents, wave focusing by refraction, and wave–current interaction. Lighthouses, built to warn navigators of shallow water and the presence of land, are often the target of such waves. For example, at the Unst Light in Scotland, a door 70 m above sea level was stove in by waves, and at the Flannan Light, a mystery has grown up about the disappearance of three lighthouse keepers during a storm in 1900 presumably by a freak wave, for a lifeboat, ﬁxed at 70 m above the water, had been torn from its mounts. A poem by Wilfred W. Gibson has fueled the legend of a supernatural event. The (unofﬁcial) world’s record for a water wave appears to be the earthquake and landslide-created wave in Lituya Bay, Alaska (Miller 1960). A wave created by a large landslide into the north arm of the bay sheared off trees on the side of a mountain over 525 m above sea level! 5.1 INTRODUCTION Waves are the prime movers for the littoral processes at the shoreline. For the most part, they are generated by the action of the wind over water but also by moving objects such as passing boats and ships. These waves transport the energy imparted to them over vast distances, for dissipative effects, such as viscosity, play only a small role. Waves are almost always present at coastal sites owing to the vastness of the ocean’s surface area, which serves as a generating site for waves, and the relative smallness of the surf zone, that thin ribbon of area around the ocean basins where the waves break and the wind-derived energy is dissipated. Energies dissipated within the surf zone can be quite large. The energy of a wave is related to the square of its height. Often it is measured in terms of energy per unit water surface area, but it could also be the energy per unit wave length, or energy ﬂux per unit width of crest. The ﬁrst deﬁnition will serve to be more useful for our purposes. If we deﬁne the wave height as H , the energy per unit surface area is E = 1 8 ρgH 2, 88 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 89 where thewater density ρ and the acceleration of gravity g are important parameters. (Gravity is, in fact, the restoring mechanism for the waves, for it is constantly trying to smooth the water surface into a ﬂat plane, serving a role similar to that of the tension within the head of a drum.) For a 1-m high wave, the energy per unit area is approximately 1250 N-m/m2, or 1250 J/m2. If this wave has a 6-s period (that is, there are 10 waves per minute), then about 4000 watts enter the surf zone per meter of shoreline.∗ For waves that are 2 m in height, the rate at which the energy enters the surf zone goes up by a factor of four. Now think about the power in a 6-m high wave driven by 200 km/hr winds. That plenty of wave energy is available to make major changes in the shoreline should not be in doubt. The dynamics and kinematics of water waves are discussed in several textbooks, including the authors’ highly recommended text (Dean andDalrymple 1991),Wiegel (1964), and for the more advanced reader Mei (1983). Here, a brief overview is provided. The generation of waves by wind is a topic unto itself and is not discussed here. A difﬁcult, but excellent, text on the subject is Phillips’ (1980) The Dynamics of the Upper Ocean. 5.2 WATER WAVE MECHANICS The shape, velocity, and the associated water motions of a single water wave train are very complex; they are even more so in a realistic sea state when numerous waves of different sizes, frequencies, and propagation directions are present. The simplest theory to describe the wave motion is the Airy wave theory, often referred to as the linear wave theory, because of its simplifying assumptions. This theory is predicated on the following conditions: an incompressible ﬂuid (a good assumption), irrotational ﬂuid motion (implying that there is no viscosity in the water, which would seem to be a bad assumption, but works out fairly well), an impermeable ﬂat bottom (not too true in nature), and very small amplitude waves (not a good assumption, but again it seems to work pretty well unless the waves become large). The simplest form of a wave is given by the linear wave theory (Airy 1845), illustrated inFigure 5.1,which showsawaveassumed tobepropagating in thepositive x direction (left to right in the ﬁgure). The equation governing the displacement of the water surface η(x, t) from the mean water level is η(x, t) = H 2 cos k(x − Ct) = H 2 cos(kx − σ t) (5.1) Because the wave motion is assumed to be periodic (repeating identically every wavelength) in the wave direction (+x), the factor k, thewave number, is used to en- sure that the cosine (a periodic mathematical function) repeats itself over a distance L , the wavelength. This forces the deﬁnition of k to be k = 2π/L . For periodicity in time, which requires that the wave repeat itself every T seconds, we have σ = 2π/T , where T denotes the wave period. We refer to σ as the angular frequency of the waves. Finally, C is the speed at which the wave form travels, C = L/T = σ/k. The C is referred to as the wave celerity, from the Latin. As a requirement of the Airy ∗ If we could capture 100 percent of this energy, we could power 40 100-watt lightbulbs. P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 90 WAVES AND WAVE-INDUCED HYDRODYNAMICS Figure 5.1 Schematic of a water wave (from Dean and Dalrymple 1991). wave theory, the wavelength and period of the wave are related to the water depth by the dispersion relationship σ 2 = gk tanh kh (5.2) The “dispersion” meant by this relationship is the frequency dispersion of the waves: longerperiodwaves travel faster than shorterperiodwaves, and thus if one startedout with a packet of waves of different frequencies and then allowed them to propagate (which is easily done by throwing a rock into a pond, for example), then at some distance away, the longer period waves would arrive ﬁrst and the shorter ones last. This frequency dispersion applies also to waves arriving at a shoreline from a distant storm. The dispersion relationship can be rewritten using the deﬁnitions of angular fre- quency and wave number L = g 2π T 2 tanh kh = L0 tanh kh, (5.3) deﬁning the deep water wavelength L0, which is dependent only on the square of the wave period. This equation shows that the wavelength monotonically decreases as the water depth decreases owing to the behavior of the tanh kh function, which increases linearly with small kh but then becomes asymptotic to unity in deep water (taken arbitrarily as kh > π or, equivalently, h > L/2). The dispersion relationship is difﬁcult to solve