Waves and wave induced hydrodynamics

P1: FCH/SPH CB373-05 P2: FCH/SPH CB373 QC: FCH/SBA June 27, 2001 11:28 T1: FCH Char Count= 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= 88 ρg H , P1: FCH/SPH CB373-05 P2: FCH/SPH CB373 QC: FCH/SBA June 27, 2001 11:28 T1: FCH Char Count= 89 5.2 WATER WAVE MECHANICS where the water 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 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 and Dalrymple 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 in Figure 5.1, which shows a wave assumed to be propagating in the positive 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 H cos k(x − Ct) = cos(kx − σ t) 2 (5.1) Because the wave motion is assumed to be periodic (repeating identically every wavelength) in the wave direction (+x), the factor k, the wave number, is used to ensure 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 CB373-05 P2: FCH/SPH CB373 QC: FCH/SBA June 27, 2001 11:28 T1: FCH Char Count= 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 = gk kh (5.2) The “dispersion” meant by this relationship is the frequency dispersion of the waves: longer period waves travel faster than shorter period waves, and thus if one started out 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 frequency and wave number L= g T kh = L kh, 2π (5.3) defining the deep water wavelength L , 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 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 because the wavelength appears in the argument of the hyperbolic tangent This means that iterative numerical methods, 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 = L 2π (h/g)/T 3/2 2/3 (5.4) EXAMPLE A wave train is observed to have a wave period of s, and the water depth is 3.05 m What is the wavelength L? P1: FCH/SPH CB373-05 P2: FCH/SPH CB373 QC: FCH/SBA June 27, 2001 11:28 T1: FCH Char Count= 91 5.2 WATER WAVE MECHANICS First, we calculate the deep water wavelength L = gT /2π = 9.81 (5)2 / 6.283 = 39.03 m The exact solution of Eq (5.3) is 25.10 m The approximation 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 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 subtracting, 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 antinodal positions 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 progressive wave, Eq (5.1), the water particles move in elliptical orbits, which can be decomposed into the horizontal and vertical velocity components u and w as follows: u(x, z, t) = H σ cosh k(h + z) cos(kx − σ t) sinh kh (5.5) w(x, z, t) = H σ sinh k(h + z) sin(kx − σ t) sinh kh (5.6) At the horizontal seabed, the vertical velocity is zero, and the horizontal velocity is u(x, −h, t) = u b cos(kx − σ t), where ub = Hσ = Aσ sinh kh 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 = −ρgz + ρg H cosh k(h + z) cos(kx − σ t) cosh kh cosh k(h + z) η(x, t) cosh kh (5.7) P1: FCH/SPH CB373-05 P2: FCH/SPH CB373 QC: FCH/SBA June 27, 2001 11:28 T1: FCH Char Count= 92 WAVES AND WAVE-INDUCED HYDRODYNAMICS Table 5.1 Asymptotic Forms of the Hyperbolic Functions Large kh ( >π) Small kh ( 1/2, respectively These asymptotes are shown in Table 5.1 5.2.1 OTHER WAVE THEORIES The Airy 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 Stream Function wave 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 CB373-05 P2: FCH/SPH CB373 QC: FCH/SBA June 27, 2001 11:28 T1: FCH Char Count= 93 5.2 WATER WAVE MECHANICS which allows any order, is solved numerically, and the code is available from the authors For variable water depths, including deep water, approximations have been made to the governing equations to permit solutions over complicated bathymetries A major step in this effort was the development of the mild-slope equation by Berkhoff (1972), which is valid for linear waves Modifications for nonlinear waves and the effects of mean currents were made by Booij (1981) and Kirby and Dalrymple (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 solitary wave theory of Russell (1844), who noticed these waves being created by horse-drawn barges in canals The wave form is η(x, t) = H sech2 3H (x − Ct) 4h (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 A more general theory that incorporates all of the shallow water wave theories is derived from the Boussinesq equations (see Dingemans 1997 for a detailed derivation 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 ∂η h ∂ ∂u +u = −g + ∂t ∂x ∂x ∂x2 ∂η ∂(h + η)u + =0 ∂t ∂x h ∂u ∂t − h2 ∂ ∂x2 ∂u ∂t (5.11) (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 solutions 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 CB373-05 P2: FCH/SPH CB373 QC: FCH/SBA June 27, 2001 11:28 T1: FCH Char Count= 94 WAVES AND WAVE-INDUCED HYDRODYNAMICS models can be used across the surf zone Some examples are Schaffer, ă 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 and Guza 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 Boussinesq theory by making the assumption that the waves 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 from Airy) can provide good estimates of the waveform and velocities because these equations lead to the formation of bores, which characterize the broken waves: ∂u ∂η ∂u +u = −g ∂t ∂x ∂x ∂η ∂(h + η)u + =0 ∂t ∂x (5.13) (5.14) Hibberd and Peregrine (1979) and Packwood (1980) were the early developers of this approach, and Kobayashi and colleagues have produced several working models (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 shallow water and away from regions of deep water This can create regions of wave focusing on headlands and shoals The simplest representation of wave refraction is the refraction of waves propagating 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 CB373-05 P2: FCH/SPH CB373 QC: FCH/SBA June 27, 2001 11:28 T1: FCH Char Count= 95 5.2 WATER WAVE MECHANICS in one water depth to that in deep water: sin θ0 sin θ = = constant, C C0 (5.15) where the subscript denotes deep water Wave refraction diagrams for realistic bathymetry provide a picture of how waves propagate from the offshore to the shoreline of interest These diagrams can be drawn 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 in wave direction according to Snell’s law Now most 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 refraction is H = H0 K s K r , (5.16) where H0 is the deep water wave height, K s is the shoaling coefficient, Ks = Cg0 , Cg and K r 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 in wave height then lead to changes in wave 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 CB373-05 P2: FCH/SPH CB373 QC: FCH/SBA June 27, 2001 11:28 T1: FCH Char Count= 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 the model Refraction models can be ray-tracing models (e.g., Noda 1974), or grid models (e.g., Dalrymple 1988) Refraction–diffraction models are more 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 monochromatic 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 /L ≈ 0.17, where L 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 CB373-05 P2: FCH/SPH CB373 QC: FCH/SBA June 27, 2001 11:28 T1: FCH Char Count= 97 5.2 WATER WAVE MECHANICS Table 5.2 Breaking Wave Characteristics and the Surf Similarity Parameter ζ→ ≈0.1 Breaker Type Spilling κ N r 0.8 6–7 10−3 0.5 1.0 2.0 1.0 1–2 1.1