Wave Energy Coastal Waves Primer

Wave Energy Coastal Waves Primer (R. Budd) NIWA Internal Project Report: October 2004 NIWA Project: IRL05301 Wave Energy Coastal Waves Primer Murray Smith Craig Stevens Richard Gorman Prepared for FRST Wave Energy Project Group NIWA Internal Report: October 2004 NIWA Project: IRL05301 National Institute of Water & Atmospheric Research Ltd 301 Evans Bay Parade, GretaPoint, Wellington PrivateBag 14901, Kilbirnie, Wellington, New Zealand Phone+64-4-386 0300, Fax +64-4-386 0574 www.niwa.co.nz All rights reserved. This publication may not be reproduced or copied in any form without the permission of theclient. Such permission is to begiven only in accordancewith the terms of theclient's contract with NIWA. This copyright extends to all forms of copying and any storageof material in any kind of information retrieval system. Contents 1. Introduction 1 2. Linear theory Kinematics: 1 2.1 Wave classification 1 2.2 Velocity 2 3. Energy 4 4. Real waves 5 5. WaveSim 7 6. Wave Loading on Structures 8 7. Wave energy absorption 9 8. Measuring waves 10 9. Modeling Waves 10 10. Wave Resources 12 11. Bibliography/Links 15 11.1 Links 15 11.2 Literature 15 12. Brief Glossary: 17 1. Introduction Any wave energy technology development requires a solid understanding of wave theory and practice. Here we summarise some important points relevant to the project. 2. Linear theory Kinematics: 2.1 Wave classification It is convenient to classify waves as deep, intermediate (transitional) or shallow. We do this since deep and shallow waves exhibit quite different properties. This is shown for example a) in the way the water motion (and energy) is attenuated with depth b) the way the wave speed depends on wave period (or frequency). Table 1: Wave classification, based on the ratio of depth (h) to wavelength (L), or kh where the wavenumber, k, is 2π/L: Shallow h/L < 1/20 kh < π/10 Intermediate 1/20 < h/L < ½ π /10 < kh < π Deep ½ < h/L < ∞ π < kh < ∞ Table 2: Representative values of key wave parameters for typical coastal waves that we are likely to encounter, propagating in water depths, h. These are period, T, wavelength, L, and phase velocity, C: h (m) T (s) L C h/L Classification Young waves, small 10 3 s 14 m 4.7 m/s .71 Deep fetch (harbour) Young waves, coastal 5 5 s 30 m 6.1 m/s .17 Intermediate Swell on coastal shore 3 12 s 64 m 5.3 m/s .05 Shallow Swell on open ocean 300 12 s 225 m 18.7 m/s 1.3 Deep Note that ‘young’ waves generated in a shallow harbour can still be categorized as deep-water waves, since their short wavelengths do not ‘feel’ the bottom. WaveEnergy Coastal Waves Primer 1 2.2 Velocity We need to also distinguish between the phase velocity of the wave (i.e. the speed the wave form propagates), and the orbital velocity (i.e. the velocity that water particles or tracers travel). In deep water the orbital velocity ideally traces out circular motion, while in shallow water this becomes increasingly elliptical with a decreasing amount of vertical excursion. Ideally for small amplitude waves there is no net transport of water, only of energy. In reality, for finite amplitude waves, unclosed particle orbits do result in a small net migration (Stokes drift). Figure 1: Wave orbital velocities are elliptical in shallow water, circular in deep water. (From Dean & Dalrymple, 1984) The schematic in Figure 1 illustrates the shallow-water elliptical motion in contrast to deep-water circular motion. For deep-water waves, the velocity (and pressure) is attenuated exponentially with depth. Thus most of the wave energy is confined to surface layers. In contrast for shallow water waves, the horizontal excursion is the same at all depths beneath the wave, but the total vertical excursion increases linearly from zero at the bed, to H (the wave height) at the surface. One of the other important properties of water waves is that they are dispersive, i.e. the speed that they travel depends on their wavelength (or period), unlike non- dispersive sound waves. This fact is often overlooked since we are accustomed to viewing shallow-water waves breaking on the beach which, as we will see, appear to be non-dispersive. The relationship between frequency and wavelength is given by the general dispersion relationship: ω2 = gk tanh (kh) ……. (1) WaveEnergy Coastal Waves Primer 2 where ω is the radian frequency (ie ω = 2πf = 2π/T), g is gravitational acceleration (9.81m/s 2), k is the wavenumber (2π/L), h is water depth, T the wave period, and L the wavelength. This transcendental equation is usually solved iteratively (e.g. Newton method) or using a lookup table. The dispersion equation becomes simple for the two extremes of deep and shallow water: Deep water: kh → ∞, tanh (kh) → 1, ω2 = gk i.e. frequency independent of depth Shallow water: kh → 0, tanh(kh) → kh, ω2 = gk 2h. Of immediate interest is the phase velocity, c (speed of crests) and group velocity, c g (speed of energy propagation). Deep: c = ω/k = g/ ω cg = ∂ω /∂k = c/2 Shallow: c = √gh cg = ∂ω /∂k = c. Waves slow down with decreasing depth, and in the ‘shallow’ water extreme, crests travel at the same speed regardless of frequency (or wavelength). This is what we observe on a surf beach, and contrasts to deep water where fast (long wavelength) waves are continually overtaking and moving through slower (shorter) ones. Note that the frequency is always invariant (although it can be Doppler shifted by currents); it is the phase velocity and wavelength that decrease in shallower water. A practical consequence of the slowing phase velocity in shallow water is that wave crests refract to become parallel to the bathymetry. For a wave approaching the shore obliquely, those parts entering shallow water first are slowed, while deeper parts continue to catch up, thus swinging the wave crest around parallel to the depth contours and shoreline. The distribution of wave height is often treated as a random Gaussian process. However this belies the fact that there is often an underlying group structure, which is evident when time-space data is available. This structure is the ‘wave sets’ familiar to surfers. The practical implication of this is that the space-time occurrence of wave breaking in deep water is not random, but structured, and in certain cases (narrow band wave spectra) is predictable. It is wave breaking that provides the strongest impact forces on structures. WaveEnergy Coastal Waves Primer 3 3. Energy The energy of water waves is equally divided between a) potential energy and b) kinetic energy. The potential energy relates to the raising of water from the trough to a crest. The potential energy per unit area of a water surface raised by ς is given by E p = ½ ρ g ς 2. For a theoretical sinusoidal wave of amplitude H/2 averaged over a wave period this becomes Ep = 1/16 ρgH 2 . Kinetic energy can be shown to have an equal value integrated over depth. However it can be seen in Figure 1 that the depth distribution of kinetic energy is quite different between deep and shallow water waves. The total average potential plus kinetic energy (after integrating over depth) per unit surface area is: E = 1/8 ρgH 2 regardless of depth, and the power transported per unit crest length is: 2 P = 1/2 ρga Cg. More generally, when energy is distributed across a spectrum (S) of wave periods or frequencies (f) : P = ρg ∫C g(f) S(f) df (This may depend on the distances one is integrating over). The rate that energy is transported by the waves, the energy flux, F, is: F = E C g = E (ω/k) ½ (1 + 2kh/sinh (2kh)). On large scales energy will be lost to bottom friction, but if the depth changes rapidly with respect to this scale, we can treat E as constant. One consequence of the conservation of energy is that as waves enter shallow water from depth h 1 to h 2, E1Cg1 = E 2Cg2 . Substituting for E, the wave height must increase to compensate for the decrease of group velocity: H2 = H 1√(C g1/C g2). WaveEnergy Coastal Waves Primer 4 This shoaling equation tells us for example that a 10 s period wave approaching the shore from deep water will increase in amplitude by 11% by the time it reaches 5m depth, and 54% at 2m depth. Refraction is normally taken into account as well. 4. Real waves In the real world, waves are not monochromatic sine waves (see Fig. 2 below), but due to the complex forcing mechanisms, occur as a continuous wave spectrum. In addition the spectrum of waves has a directional spread so a fully resolved measurement will also specify the direction from which a wave component is travelling. Figure 2: Water surface velocities towards and away from the observer recorded using microwave radar from RV Tangaroa in 2004 showing the complex structure in time and space. Data slices are shown above and to the right of image panel. WaveEnergy Coastal Waves Primer 5 Figure 3: The wave example above comes from Wellington Harbour. The top panel shows a typical water elevation times series. Typically, it is not a simple sinusoid but shows a combination of frequencies. The bottom panel shows the resulting wave spectrum. In order to define the wave parameters such as period and height, there are generally 2 approaches: (1) from the timeseries e.g. the average time between zero crossings, Tz.

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