Waves Primer expandedV2a · 2013. 2. 4. · wave form propagates), and the orbital velocity (i.e....

Wave Energy Coastal Waves Primer

(R. Budd)

NIWA Internal Project Report:

October 2004

NIWA Project: IRL05301

All rights reserved. This publication may not be reproduced or copied in any form without the permission of the client. Such permission is to be given only in accordance with the terms of the client's contract with NIWA. This copyright extends to all forms of copying and any storage of material in any kind of information retrieval system.

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, Greta Point, Wellington Private Bag 14901, Kilbirnie, Wellington, New Zealand Phone +64-4-386 0300, Fax +64-4-386 0574 www.niwa.co.nz

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

Wave Energy Coastal Waves Primer 1

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

fetch (harbour)

10 3 s 14 m 4.7 m/s .71 Deep

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.


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)


where ω is the radian frequency (ie ω = 2πf = 2π/T), g is gravitational acceleration

(9.81m/s2), 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, cg

(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.


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 Ep =

½ ρ g ς2. For a theoretical sinusoidal wave of amplitude H/2 averaged over a wave

period this becomes Ep = 1/16 ρgH2 .

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 ρgH2

regardless of depth, and the power transported per unit crest length is:

P = 1/2 ρga2Cg.

More generally, when energy is distributed across a spectrum (S) of wave periods or

frequencies (f) :

P = ρg ∫Cg(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 Cg = 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 h1 to h2,

E1Cg1 = E2Cg2 .

Substituting for E, the wave height must increase to compensate for the decrease of

group velocity:

H2 = H1√(Cg1/Cg2).


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.


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.

(2) from the spectrum. e.g. the dominant period is obtained from the peak of the

spectrum, Tp. The moments of the spectrum can also be used to give the theoretical

Tz. The significant wave height, Hs, is calculated from the area under the spectrum,

m0. i.e. Hs =4√(m0).

The spectrum in the lower panel shows an approximate f-4 fall off in wave energy from

the peak. In this case of ‘young’ waves, the peak frequency is 0.43 Hz (peak period

2.3s) and significant wave height, Hs = 0.30 m.


Figure 4: Open ocean example, from the Bay of Plenty, is more typical than the earlier harbour

spectrum in that it shows mixed wind-sea and swell; the swell frequency is 0.08

Hz (period 12 sec) and the dominant sea frequency is 0.12 Hz (period ~8sec).

5. WaveSim

This Matlab tool was developed to display movies of real (and artificial) wave fields.

The initial distribution took Waverider data from waveriders deployed at

Mokohinau Island and Mangawhai.

Figure 5: WAVESIM3 still frame showing wave orbits and water surface elevation.


6. Wave Loading on Structures

Forces on submerged wave-affected structures are well described elsewhere (e.g.

Grosenbaugh, 2002) with an emphasis on gas platforms and pipeline/cable scenarios.

The Figure below summarizes the relevant forces on an ideal structure, including those

due to buoyancy, acceleration/inertia and drag (Fb, Fa, Fd). These forces act in

response to the local water motion (u) which is characterized as a function of time (t)

and space (x). Implicit in this is time variation where some forces (Fb) are constant as

long as the structure is submerged, some are related to water velocity (Fd) and some

are related to water acceleration (Fa). The forces acting on the object (and to a lesser

extent on the mooring line) are transferred along the mooring as both a force (and if it

has any compressive strength a bending moment). To maintain equilibrium, there

must be a restorative force (Fr) and bending moment (Mr) acting at the base of the

mooring.

Figure 6: Dynamics showing hydrodynamics and forces on a submerged body.

With respect to wave energy converter modelling we will need to determine

(1) the basic shape of the structure

(2) the flexibility of all mooring elements

(3) the coefficients of drag and added mass for all elements.


Most of these properties can be derived from the literature, modelling of the

combined response will need to be verified with laboratory/field validation.

7. Wave energy absorption

Considering wave energy from the perspective of simply absorbing it we can look to a

number of successful floating breakwater approaches that can remove around 80% of

energy that could be considered wind-wave (e.g. Seymour and Hanes 1979).

Figure 7: Sketch of floating breakwater http://209.196.135.250/floating_breakwater.htm

The structural similarities between a floating breakwater and a NZ-style mussel farm

are clear.

Figure 8: Wave attenuation by a mussel farm. Energy Transmission Ratio ETR for the wave

attenuation scaling analysis as a function of frequency (x-axis) and initial wave

height (given in legend). From Plew et al. (2005).


8. Measuring waves

We can classify wave measurements in two categories: a) in-situ and b) remote

sensing, each with advantages and disadvantages.

Table 3: Wave measurement summary.

Device Sensor Advantage Disadvantages

a) In-situ

Wave-rider buoy Accelerometer (magnetometer)

Suitable for longterm deployments

Expensive to purchase and maintain

Pressure sensors (e.g. DOBIE)

Pressure Robust Inexpensive Suitable for longterm deployments

Limited to shallow depths

Wave Staff Resistance or Capacitance

Accurate to high frequencies

Limited to shallow depths Less robust

ADV Acoustic Also measures current Moderately expensive ($40k)

ADP Acoustic 3 independent measures of wave height. Measures current profile Robust at sea bed

Moderately expensive

b) Remote sensing Radar Doppler effect Measures over a range

of distances Requires suitable site Requires operator

Satellite Electromagnetic backscatter or altimeter

Covers vast areas of ocean

Poor spatial resolution Very poor temporal resolution Not highly accurate

9. Modeling Waves

Wind generated waves can now be modeled well in open-ocean deep water. The most

common community-developed model is WAM and is typically driven by

meteorological analysis data. This is used in NIWA’s wave hindcasts (See below).

In shallower coastal water, the SWAN model is proving efficient except in particularly

complicated situations e.g. variable bathymetry. Both of these models are spectral

models in that they provide a statistical spectrum of wave energy. Wave growth is


modeled as a consequence of the imbalance between: energy advection and the

‘source terms’: wind-input, dissipation due to breaking, dissipation by bottom friction,

and non-linear wave-wave interactions.

Figure 9: New Zealand regional WAM wave model for a particular time. Windfields input from

ECMWF reanalysis.

Figure 10: Mean Hs derived from NIWA wave hindcast averaging 20 years of the above results

– see (Gorman et al 2003).


To model an individual wave, various versions of a Boussinesq model are being

developed. The current challenge is to incorporate breaking effects in shallow water,

with the associated current generation.

For engineering applications, empirical relations are often used to predict wave

heights and periods. Both wave height and period grow with fetch (distance over

which the wind-forcing is acting) and duration (over which wind-forcing has been

acting). The most common reference for this type of wave prediction is the US Army

Corps Shore Protection Manual. Complications occur when: bathymetry is irregular,

wind field is influenced by complex orography, or currents are strong.

10. Wave Resources

A wave hindcast (Gorman et al. 2003; Gorman 2003) provided a wave climate

offshore of a number of selected sites from 1977 to 1997. The hindcast model was

driven by European Centre for Medium Range Weather Forecasting re-analysis

windfields, and run on a 1.125x1.125 degree grid for the SW Pacific and Southern

Ocean region. Directional spectra, saved at grid cells around the coast, have been

interpolated to points on the 50m isobath and filtered to account for limited fetch to

the coastline (Gorman et al. 2003). The hindcast generates 3-hourly estimates of

significant wave height Hs (~2a) (Fig. 11), the frequency of the peak of the wave

spectrum fp as well as a range of parameters relating to the energy flux. The Hs is the

average of the highest 1/3 of the waves which is equivalent to the square root of the

summed variance of the wave spectrum. It can be expected that the largest wave

height within each 3 hr sample period will be about 2Hs.

Figure 11: Raw hindcast data for 4 coastal-ocean sites spread around New Zealand. This

represents around 58,000 data points per site.


The analysis provided a suite of figures to demonstrate the resource availability - these

include

� A site map showing the approximate location

� A distribution of Hs occurrence statistics

� A distribution of peak period occurrence statistics

� A distribution of percentage occurrence of total omni-

directional wave energy flux occurrence statistics

� Monthly averages of Hs and peak Hs. The 20 year peak Hs

gives an indicator for survival modelling.

� Event-duration scatter diagram showing the number of events

exceeding a particular Hs for a given period.

Figure 12: Monthly averages of Hs at four sites around New Zealand.


Figure 13: Significant wave height occurrence statistics for a particular site.

Figure 14: Event-duration occurrences for a site for three values of Hs (2, 3 & 4 m) where the

symbols show the number of events exceeding Hs continuously for that duration.


11. Bibliography/Links

11.1 Links

Banks Peninsula Wave Rider

� NIWA http://www.niwa.co.nz/services/waves

� Ecan http://www.ecan.govt.nz/Coast/Wave-Buoy/wave-buoy.html

WAM http://www.ecmwf.int/products/data/technical/wam/representations.html

Ocean Engineering University of New Hampshire: http://www.unh.edu/oe/

European Centre for Medium-Range Weather Forecasts http://www.ecmwf.int/

SWAN wave model http://fluidmechanics.tudelft.nl/swan/default.htm

11.2 Literature

Barnett P.S. and E.P.M. Brown (1987). The potential for wave generation off New

Zealand. 8th Australasian Conference on Coastal and Ocean Engineering,

87/11. IE Aust, 30 Nov – 4 Dec 1987, Launceston, Tasmania.

Barstow, S., and R. Deo (1993). A wave energy resource climatology for the South

Pacific. Proc. European Wave Energy Symposium, Edinburgh, Scotland, July

'93.

Brown, E.P.M. (1988). An estimate of New Zealand’s wave power resource for

electricity generation. Proc. 1st Symposium of the N.Z Ocean Wave Society,

pp 17-27.

Brown, E.P.M. (1990). Wave Power investigations in New Zealand. IPENZ

Transactions, 1990, pp173-183.

Brown, E.P.M. (1990). Comparison of Ocean Wave Power calculation methods. An

estimate of New Zealand’s wave power resource for electricity generation.

Proc. 2nd Symposium of the N.Z Ocean Wave Society, pp 49-63

Dean, R.G. and R.A. Dalrymple 1984, Wave Mechanics for Engineers and Scientists,

Englewood Cliffs: Prentice-Hall, Inc.

Falnes, J. 2002 Ocean Waves and Oscillating Systems, CUP.

Gorman, R.M., Bryan, K.R. and Laing, A.K. (2003). Wave hindcast for the New

Zealand region - deep water wave climate. New Zealand Journal of Marine

and Freshwater Research 37(3): 589-612.


Gorman, R.M. and Laing, A.K. (2001). Bringing wave hindcasts to the New Zealand

coast. Journal of Coastal Research Special Issue 34: 30-37.

Gorman, R.M. (2003) The treatment of discontinuities in computing the nonlinear

energy transfer for finite-depth gravity wave spectra. Journal of Atmospheric

and Oceanographic Technology, 20, 206-216.

Grosenbaugh, M., S. Anderson, R. Trask, J. Gobat, W. Paul, B. Butman and R.

Weller, “Design and performance of a horizontal mooring for upper-ocean

research,” J. Atmos. Oceanic Technol., 19, pp.1376-1389, 2002.

Hornstra, M.W. (1983). Wave Power – a New Zealand study. M.E. Thesis, Dept of

Mech. Eng., University of Auckland. 232pp.

Komen, G.J., et al., 1994. Dynamics and modelling of ocean waves. Cambridge,

University Press

Laing, A.K. 1993. Estimates of wave height data for New Zealand waters from

numerical modelling. New Zealand Journal of Marine and Freshwater

Research 27: 157-175.

Pickrill, R.A., and J.S.Mitchell, (1979). Ocean wave characteristics around New

Zealand. New Zealand journal of marine and freshwater research, 13(4):504-

520.

Plew. D.; Stevens. C.; Spigel, R.;Hartstein, D. 2005. Hydrodynamic implications of

large offshore mussel farms. . IEEE Journal of Oceanic Engineering, Special

Issue on Open Ocean Aquaculture Engineering 30:95-108.

Reid, S.J. and B. Collen (1983). Analysis of wave and wind reports from ships in the

Tasman Sea and New Zealand waters. N.Z Met Service, Misc. Pub 182.

Seymour R.J. and D. M. Hanes, 1979 Performance analysis of tethered float

breakwaters, J. Waterway, Port, Coastal & Ocean Eng., 105, pp. 265-280.

Smith, M.J.; Stevens, C.L.; Gorman, R.M.; McGregor, J.A.; Neilson, C.G. (2001)

Wind-wave development across a large shallow intertidal estuary: a case study

of Manukau Harbour, New Zealand. NZ Journal of Freshwater and Marine

Research, 35: 985-1000.

Stevens, C.L.; Hurd, C.L.; Smith, M.J. (2001). Water motion relative to subtidal kelp

fronds, Limnology and Oceanography, 46(3): 668-678.

WAMDI group, 1988. The WAM model - a third generation ocean wave prediction

model. J. Phys. Oceanogr. 18, pp. 1775-1810


12. Brief Glossary:

ECMWF - European Centre for Medium-Range Weather Forecasts

Frequency (f) – number of crests (or troughs) passing a fixed point per second. The

inverse of wave period.

Period (T) – time interval between two successive crests passing a fixed point

Phase velocity (c) - speed that the waveform travels (L/T).

Significant wave height (Hs) – the most commonly used measure of wave height. It

represents the height of approximately the highest third of waves. Hs = 4.0√(mo).

Note that in a real sea in a three hour period you would expect to see a wave 2Hs in

height.

Swell - waves whose source region is very distant. As a result they have long periods

(e.g. 8-12 sec)

Wavelength (L) – the distance between successive crests

Wind-waves – generated by the local wind (e.g. 5 s period)

Young waves - waves which are generated when either a) waves have not had time to

develop to the point where the phase velocity approaches the wind speed or b) waves

have not had the distance over which to develop to the point where the phase velocity

approaches the wind speed

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