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because the wavelength appears in the argument of the hyperbolic tangent. Thismeans that iterative numericalmethods, such as Newton–Raphson methods, or approximate means are used to solve for wavelength; see, for example, Eckart (1951), Nielsen (1983), or Newman (1990). One convenient approximate method is due to Fenton and McKee (1989), which has a maximum error less than 1.7 percent, which is well within most design criteria. Their relationship is L = L0 { tanh [( 2π √ (h/g)/T )3/2]}2/3 (5.4) EXAMPLE A wave train is observed to have a wave period of 5 s, and the water depth is 3.05 m. What is the wavelength L? 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 91 First, we calculate the deep water wavelength L0 = gT 2/2π = 9.81 (5)2/ 6.283 = 39.03 m. The exact solution of Eq. (5.3) is 25.10 m. The approxima- tion of Eq. (5.4) gives 25.48 m, which is in error by 1.49 percent. The waveform discussed above is a progressive wave, propagating in the positive x direction with speed C . Near vertical walls, where the incident waves are reﬂected from the wall, there is a superposition of waves, which can be illustrated by simply adding the waveform for a wave traveling in the positive x direction to one going in the opposite direction: η(x, t) = H 2 cos(kx + σ t) The resulting standing wave is η(x, t) = H cos kx cos σ t, which has an amplitude equal to twice the amplitude of the incident wave. By sub- tracting, instead of adding, we have a standing wave with different phases involving sines rather than cosines. Examining the equation for the standing wave, we ﬁnd that the water surface displacement is always a maximum at values of kx equal to nπ , where n = 0, 1, 2, . . . . These points are referred to as antinodalpositions.Nodes (zero water surface displacement) occur at values of kx = (2n − 1)π/2, for n = 1, 2, 3, . . . , or x = L/4, 3L/4, . . . . Under the progressivewave,Eq. (5.1), thewater particlesmove in elliptical orbits, which can be decomposed into the horizontal and vertical velocity components u and w as follows: u(x, z, t) = Hσ 2 cosh k(h + z) sinh kh cos(kx − σ t) (5.5) w(x, z, t) = Hσ 2 sinh k(h + z) sinh kh sin(kx − σ t) (5.6) At thehorizontal seabed, the vertical velocity is zero, and thehorizontal velocity is u(x,−h, t) = ub cos(kx − σ t), where ub = Hσ2 sinh kh = Aσ The parameter A is half the orbital excursion of the water particle over the wave period and has been shown to be related to the size of sand ripples formed on the bottom. The associated pressure within the wave is given by p(x, z, t) = −ρgz + ρg H 2 cosh k(h + z) cosh kh cos(kx − σ t) = −ρgz + ρg ( cosh k(h + z) cosh kh ) η(x, t) (5.7) P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 92 WAVES AND WAVE-INDUCED HYDRODYNAMICS Table 5.1 Asymptotic Forms of the Hyperbolic Functions Large kh Small kh Function ( >π) (<π/10) cosh kh 12 e kh 1 sinh kh 12 e kh kh tanh kh 1 kh in which the ﬁrst term on the right-hand side is the hydrostatic pressure component, and the second is the dynamic (wave-induced) pressure component. The dynamic pressure is largest under thewave crest, for thewater column is higher at that location and at a minimum under the wave trough. Finally, the propagation speed of the wave energy turns out to be different than the propagation speed of the waveform owing to the dispersive nature of the waves. This is expressed as the group velocity Cg, which is deﬁned as Cg = nC, where n = 1 2 ( 1+ 2kh sinh 2kh ) (5.8) The rate at which the waves carry energy, or the energy ﬂux per unit width, is deﬁned as F = ECg. Both Cg and n are functions of the water depth; n is 0.5 in deep water and unity in shallow water. To simplify equations involving hyperbolic functions in shallow and deep water, asymptotic representations of the functions can be introduced for relative water depths of h/L < 1/20 and h/L > 1/2, respectively. These asymptotes are shown in Table 5.1. 5.2.1 OTHER WAVE THEORIES TheAiry theory is called the linear theory because nonlinear terms in such equations as the Bernoulli equation were omitted. The measure of the nonlinearity is generally the wave steepness ka, and the properties of the linear theory involve ka to the ﬁrst power. For this text, the linear wave theory is sufﬁcient; however, the tremendous body of work on nonlinear wave theories is summarized here. Higher order theories (including terms of order (ka)n , where n is the order of the theory) have been developed for periodic and nonperiodic waves. For periodic waves, for example, the Stokes theory (Stokes 1847, Fenton 1985, ﬁfth order), and the numerical StreamFunctionwave theory (Dean 1965,Dalrymple 1974) show that, because of nonlinearity, larger waves travel faster and wave properties are usually more pronounced at the crest than at the trough of the wave, causing, for instance, the wave crests to be more peaked than linear waves. The Stream Function theory, 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 93 which allows any order, is solved numerically, and the code is available from the authors. For variable water depths, including deepwater, approximations have beenmade to the governing equations to permit solutions over complicated bathymetries. A major step in this effort was the development of themild-slope equation by Berkhoff (1972), which is valid for linear waves. Modiﬁcations for nonlinear waves and the effects ofmean currentsweremade byBooij (1981) andKirby andDalrymple (1984). In shallow water, the ratio of the water depth to the wavelength is very small (kh < π/10 or, equivalently, as before, h/L < 1/20). Taking advantage of this, wave theories have been developed for both periodic and nonperiodic waves. The earliest was the solitarywave theory ofRussell (1844),whonoticed thesewaves being created by horse-drawn barges in canals. The wave form is η(x, t) = H sech2 (√ 3H 4h3 (x − Ct) ) (5.9) This unusual wave decreases monotonically in height from its crest position in both directions, approaching the still water level asymptotically. The speed of the wave is C = √ gh ( 1+ H 2h ) (5.10) In the very shallow water of the surf zone on a mildly sloping beach, waves behave almost as solitary waves. Munk (1949) discusses this thoroughly. For periodic shallow water waves, the analytic cnoidal wave theory is sometimes used (Korteweg and deVries 1895). This theory has, as a long wave limit, the solitary wave, and as a short wave limit, the linear wave theory. Amore general theory that incorporates all of the shallow water wave theories is derived from theBoussinesq equations (seeDingemans 1997 for adetailedderivation of the various forms of the Boussinesq theory). For variable depth and propagation in the x direction, equations for the depth averaged velocity u and the free surface elevation can be written as (Peregrine 1967) ∂u ∂t + u ∂u ∂x = −g ∂η ∂x + h 2 ∂2 ∂x2 ( h ∂u ∂t ) − h 2 6 ∂2 ∂x2 ( ∂u ∂t ) (5.11) ∂η ∂t + ∂(h + η)u ∂x = 0 (5.12) For constant depth, the solitary wave and the cnoidal waves are solutions to these equations. For variable depth and a two-dimensional problem, numerical so- lutions by several techniques are available. Some of the more well-known include that of Abbott, Petersen, and Skovgaard (1978), which is based on ﬁnite-difference methods. Several recent developments have extended the use of the Boussinesq equations in coastal engineering. One of these is the modiﬁcation of the equations to permit their use in deeper water than theoretically justiﬁed. These extensions include those of Madsen, Murray, and Sørenson (1991), Nwogu (1993), and Wei et al. (1995). The second effort has been to include the effects of wave breaking so that the Boussinesq P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 94