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Linear Wave Theory Is a Solution of the Laplace Equation

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Linear Wave Theory Is a Solution of the Laplace Equation The University of the West Indies Organization of American States PROFESSIONAL DEVELOPMENT PROGRAMME: COASTAL INFRASTRUCTURE DESIGN, CONSTRUCTION AND MAINTENANCE A COURSE IN COASTAL DEFENSE SYSTEMS I CHAPTER 5 COASTAL PROCESSES: WAVES By PATRICK HOLMES, PhD Professor, Department of Civil and Environmental Engineering Imperial College, England Organized by Department of Civil Engineering, The University of the West Indies, in conjunction with Old Dominion University, Norfolk, VA, USA and Coastal Engineering Research Centre, US Army, Corps of Engineers, Vicksburg, MS, USA. St. Lucia, West Indies, July 18-21, 2001 Coastal Defense Systems 1 5-1 CDCM Professional Training Programme, 2001 5. Coastal Processes: WAVES P.Holmes, Imperial College, London 5.1 Introduction 5.2 Linear (Uniform) Waves. 5.3 Random Waves. 5.4 Waves in Shallow Water. 5.1 Introduction. Wave action is obviously a major factor in coastal engineering design. Much is known about wave mechanics when the wave height and period (or length) are known. In shallow water the properties of waves change; they change height and their direction of travel, which must be included in design calculations. However, waves generated by winds blowing over the ocean surface are not of the same height and period, they are random waves for which probability/statistical models have to be used. This section of the notes discusses these three topics: linear waves, random waves and waves in shallow water. 5.2 Linear (Uniform) Wave Theory. Figure 1 gives the general definitions for two-dimensional, linear water wave theory for which the following notation is needed: x,y are Cartesian co-ordinates with y = 0 at the still water level (positive upwards) η(x,t) = the free water surface; t = time u,v, = velocity components in the x,y directions, respectively φ(x,y,t) = the two-dimensional velocity potential ρ = the fluid density; g = gravitational acceleration a = wave amplitude = H/2; H = wave height k = wave number = 2π/L; L = wave length σ = wave frequency = 2π/T; T = wave period d = mean water depth; C = wave celerity = L/T Linear wave theory is a solution of the Laplace equation: δ2φ/δx2 + δ2φ/δy2 = 0 5.1 The particular flow in any condition is determined by the boundary conditions, in this case specific boundary conditions at the free surface of the fluid and at the bottom. (The details can be found in any good text on wave theories) Coastal Defense Systems 1 5-2 CDCM Professional Training Programme, 2001 y Direction of Propagation Wave Length Still W ater Level x Wave Height depth Particle Orbital M otion Bottom Figure 1. Definitions for Linear Water Wave Theory The solution for the potential function satisfying the Laplace equation (1) subject to the boundary conditions is: φ = (ga/σ). cosh k(y + d). sin (kx - σt) 5.2 cosh kd = (gHT/4π). cosh [(2π/L)( y + d)]. sin 2π(x/L – t/T) 5.3 cosh (2πd/L) η = a cos(kx - σt) = (H/2) cos 2π(x/L – t/T) 5.4 1/2 σ = (gk tanh kd) 5.5 or L = (gT2/2π) tanh (2πd/L) 5.6 or C = ((gL/2π) tanh (2πd/L))1/2 5.7 In “deep” water, d/L > 0.5, tanh (2πd/L) ≈ 1.0 Coastal Defense Systems 1 5-3 CDCM Professional Training Programme, 2001 2 2 ∴ Lo = gT /2π = 1.56T 5.8 where subscript “o” denotes deep water. In “shallow” water, d/L < 0.04, tanh (2πd/L) ≈ 2πd/L ∴ L = T (gd)1/2; C = √gd 5.9 For all depths the wave length, L can be found by iteration from: d/LO = d/L tanh (2πd/L) 5.10 Particle Velocities. From the derivation of the Laplace equation, for irrotational flow, u = δφ/δx and v = δφ/δy 5.11 so that from equation 2 the horizontal and vertical velocities of flow are given by: u = (πH/T). cosh [2π(y + d)/L]. cos 2π(x/L – t/T) 5.12 sinh 2πd/L v = (πH/T). sinh [2π(y + d)/L]. sin 2π(x/L – t/T) 5.13 sinh 2πd/L In “deep” water these simplify to: u = (πH/T). exp(2πy/LO). cos 2π(x/L – t/T) 5.14 v = (πH/T). exp(2πy/LO). sin 2π(x/L – t/T) 5.15 Note that y is measured positive upwards from the still water level. Pressure. In wave motion the pressure distribution in the vertical is no longer hydrostatic and is given by: p/ρg = cosh 2π[(y + d)/L] . η - y 5.16 cosh 2πd/L The cosh2π[(y + d)/L] / cosh 2πd/L term is known as the “pressure response factor” which tends to zero as y increases negatively, important when using pressure transducers for wave recording. Coastal Defense Systems 1 5-4 CDCM Professional Training Programme, 2001 Energy. The total energy per wave per unit width of crest, E, is: E = ρgH2L/8 5.17 Note that wave energy is proportional to the SQUARE of the wave height. Group Velocity. Group velocity, CG, is defined as the velocity at which wave energy is transmitted. Physically this can be seen if a group of, say, five waves is generated in a laboratory channel. The leading wave will disappear but a new wave will be created at the rear of the group, so there will always be five waves. Thus the group travels at a slower speed than the individual waves within it. CG = nC, n = ½ [1 + (4πd/L)/( sinh 4πd/L)] 5.18 Power. The mean power transmitted per unit width of crest, P, is given by: 2 P = CG ρgH /8 5.19 Non-Linear Wave Theories. As noted above the boundary conditions used to obtain a solution for wave motion were linearised, that is, applied at y = 0 not on the free water surface, y = η, hence the term Linear (or Airy) Wave Theory. For a non-linear solution the free surface boundary conditions have to be applied at that free surface, η. But η is unknown! Therefore solutions have been developed, notably by Stokes, in series form for which the coefficients of the series can be derived. Thus, the free-surface is given by: η = a cos(θ) + b cos(2θ) + c cos (3θ) + d cos(4θ) + e cos(5θ) ……5.20 To obtain a solution to, say, third order, terms greater than order three are ignored. Taken to first order the solution is, of course, a linear wave. Another wave theory applicable in shallow water is Cnoidal Wave Theory. Solutions are given in terms of elliptic integrals of the first kind; the solution at one limit is identical with linear wave theory and at the other is identical to Solitary Wave Theory. As the name implies, the latter is a single wave with no trough and the mass of water moving entirely in the x direction. More recently, numerical solutions for wave motion have been established and in offshore engineering it is common to see numerical solutions up to 18th or 25th order being used to obtain velocity and acceleration information for the derivation of wave loads on offshore structures. Figure 2, based on the U.S. Army corps of Engineers “Shore Protection Manual” 1984, indicates the preferred theory for given wave parameters, but note that the figure is illustrative only. Coastal Defense Systems 1 5-5 CDCM Professional Training Programme, 2001 Shallow water Transitional Water Deep Water HO/LO = 0.14 Stokes’ 4th Order 0.01 BREAKING Stokes’ 3rd Order 2 H/d = 0.78 H/gT H = HB/4 NON-BREAKING Stokes’ 2nd Order Stream 0.001 Function L2H/d3 ≈ 26 Cnoidal Theory Linear Theory 0.0001 0.001 0.01 0.1 d/gT2 Figure 2. Parameter Space for Wave Theories - illustrative only. Coastal Defense Systems 1 5-6 CDCM Professional Training Programme, 2001 5.2. RANDOM WAVES 5.2.1. Introduction. Waves generated by winds blowing over the ocean have a range of wave periods, T (s), or wave frequency, ω (radians/s) and are clearly not of the same height, H Surface elevation (η (t)) Time (t) Figure 1 A Typical Wave Record (m). These variations arise initially from the random nature of pressure pulses acting on the water surface and the subsequent, complex growth of the perturbations in that free surface. Based on linear wave theory it is clear that waves with longer periods travel at a higher celerity that those with a shorter period. Thus, as the waves travel away from the area in which they are generated the longer period waves travel faster and the water surface elevation is more periodic, usually being termed "swell". Within or close to the generation area the sea surface elevation is a complex three-dimensional process with short-crested waves travelling in a range of directions relative to the wind direction. A record of such waves will look like Figure 1. The surface elevation is measured relative to still water level and a second record taken at some later time would have a similar appearance, i.e., its general properties would not change - it is a stationary process. Any linear derivatives of the surface elevation, for example, the water particle velocity or acceleration would have similar appearances. Coastal Defense Systems 1 5-7 CDCM Professional Training Programme, 2001 For design purposes it is necessary to quantify this in some way. This can be done in two ways: in terms of the probability properties of the record or in terms of its frequency content, specifically the distribution of energy as a function of frequency, called the wave spectrum. 5.2.1 Wind Generation of Waves and Wave Prediction.
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