Coastal Engineering

f7

February 1989

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TU

Delft

Coastal Engineering

Ir. E.T.J.M. van der Velden

COASTAL ENGINEERING

VOLUME 11

Ir. E.T.J.M. van der Velden

Delft University of Technology Department of Civil Engineering Delft

The Netherlands f7

February 1989 Morfologie van kusten en estuaria

Technische Universiteit Delft Faculteit der Civiele Techniek

TABLE OF CONTENTS page 1 Introduction 1.1 Purpose 1.2 Coasta1 changes 1.3 Abstract 1.4 Literature 2 Waves 2.1 Introduction 2.2 Wave characteristics 2.2.1 Regu1arwaves 2.2.2 Irregu1arwaves

2.2.3 Regu1arwaves and a current 2.3 Wave measurements

2.4 Refraction,diffraction,ref1ectionand breaking 2.4.1 Breaking 2.4.2 Refraction 2.4.3 Diffraction 2.5 Literature 1 1 2 4 5 7 7 8 10 15 19 20 21 22 26 28 31 3 Sediment transport 33 3.1 Introduction 33

3.2 Initiationof motion or critica1shear stress 40

3.3 Transportby a currents 44

3.3.1 Velocitydistributionwith a currenta10ne 44 3.3.2 Concentrationdistributionwith a current a10ne 52 3.3.3 Transportformu1asfor currentsa10ne 63

3.4 Transportby waves 71

3.4.1 Velocitydistributionfor waves a10ne 71 3.4.2 Concentrationdistributionfor waves a10ne 77

3.4.3 Transportby waves a10ne 80

3.5 Transportby waves and currents 83

3.5.1 Appoach of Bijker 83

3.5.2 Commentson the Bijker formu1a 94

3.6 Measurementtechniques 96 3.7 Literature 102 4 Littora1 transport 106 4.1 Introduction 106 4.2 Definitions 107 4.3 Longshorecurrent forces 109 4.3.1 Radiationstress 109 4.3.2 Tida1 forces 131 4.3.3 Turbulentforces 133 4.3.4 Bottom frictionforces 135 4.3.5 Longshorecurrentcomputations 140 4.3.6 Examp1e 144 4.3.7 Additiona1drivingforces 148 4.4 Sedimentsources 149 4.4.1 C1assification 149 4.4.2 Sedimenttransportbudget 153 4.5 Sand transportmechanism 153 4.6 Literature 156

5 Longshore sediment transport 158

5.1 Introduction 158

5.2 CERC formu1a 160

5.3 Bijker transport formu1a 168

5.4 Adapted river transport formu1as 177

5.5 Literature 178

6 Onshore-offshore sediment transport (cross-shore transport) 180

6.1 Introduction 180

6.2 OSTRAN 183

6.3 CROSTRAN 185

6.4 Equilibrium beach profile 187

6.5 Literature 192

7 Coasta1 changes 7.1 Introduction

7.2 Single 1ine theory

7.3 Multiple line theories (two-1ine approach) 7.4 Literature 194 194 196 220 221 8 222 222 222 224 227 230 234 Dune 8.1 8.2 8.3 8.4 8.5 8.6

erosion and storm surges Introduction

Dune erosion mechanism

Principles of a sca1e-series

Main resu1ts of dune erosion research Dune design

Literature

9 Sedimentation in channe1s and trenches 9.1 Introduction

9.2 Current pattern across a channe1 9.3 Wave pattern above a channe1 9.4 Bed load and suspended load 9.5 Sedimentation ca1cu1ations 9.6 Literature 235 235 235 239 241 241 259 10 Coasta1 engineering in practice

10.1 Introduction 10.2 Erosion prob1ems 10.3 Accretion prob1ems 10.4 Wave prob1ems 10.5 Structure planning 10.6 Shore protection works 10.7 Literature 260 260 261 265 266 267 267 273 List of symbo1s Subject index 274 281

LIST OF FIGURES Fig.l.l.1 Fig.l.2.1 Fig. 2 .1.1 Fig.2.2.1 Fig.2.2.2 Fig.2.2.3 Fig.2.2.4 Fig.2.2.5 Fig.2.2.6 Fig.2.2.7 Fig.2.2.8 Fig.2.2.9 Fig.2.2.10 Fig.2.4.1 Fig.2.4.2 Fig.2.4.3 Fig.2.4.4 Fig.2.4.5 Fig.2.4.6a Fig.2.4.6b Fig. 3 .1.1 Fig. 3.1. 2 Fig. 3.1. 3 Fig. 3 .1.4 Fig.3.2.1 Fig.3.2.2 Fig.3.2.3 Fig.3.2.4 Fig.3.3.1 Fig.3.3.2 Fig.3.3.3 Fig.3.3.4 Fig.3.3.5 Fig.3.3.6 Fig.3.3.7 Fig.3.3.8 Fig.3.3.9 Fig.3.3.10

Examp1es of coasta1 morpho1ogy prob1ems. Transport mass ba1ance.

Estimated re1ative ocean surface wave energy, a c1assification of surface waves by wave band, primary

disturbing force, and primary restoring force (adopted from Shore Protection Manua1 1984).

Water-surface time-history record.

Definition sketch of progressive, sinusoida1, surface waves.

Regions of va1idity for various wave theories.

Simp1ified hyperbo1ic functions for progressive waves. Orbita1 motion under a sha110w water wave and a deep water wave (Shore Protection Manua1 [1984]).

Loca1 f1uid ve10cities and acce1erations (Shore protection Manua1 [1984]).

Uniform probabi1ity distribution. Zero-downcrossing wave period. Energy spectrum with peak-period. Wave height.

Maximum crest ang1e.

Breaker types as a function of

e

(Battjes [1974]) Wave refraction over straight parallel contours.

Refraction diagram of Long Branch, New Jersey (from Pierson [1950]).

Diffraction of an incident wave train (a) and diffraction of the ref1ected wave train (b).

Computed wave height pattern (Berkhoff [1981]). Computed wave height pattern (Berkhoff [1981]).

Sediment transport principle sketch.

Sediment concentrations as a function of time (99 individua1 records)

Bed load transport. Suspended transport. Forces on a grain. Drag force, FD. Lift force, FL. Shie1ds curve. Force ba1ance.

Shear stress as a function of height above the bed. Velocity distribution for a uniform stationary current.

Vertica1 transport of horizontal momentum.

Distribution of velocity, shear stress and fluid diffusion coefficient.

Variation of concentration with height above the bed. Mass balance of sediment.

Forces on a sphere in "clear" stationary water. D50 distribution over the water depth.

Coleman distribution of sediment diffusion coefficient.

Fig.3.3.11 Fig.3.3.12 Fig.3.3.13 Fig.3.3.14 Fig.3.3.15 Fig.3.3.17 Fig.3.3.18 Fig.3.4.1 Fig.3.4.2 Fig.3.4.3. Fig.3.4.4 Fig.3.4.5 Fig.3.4.6 Fig.3.4.7 Fig.3.4.8 Fig.3.4.9 Fig.3.4.10 Fig.3.5.1 Fig.3.5.2 Fig.3.5.3 Fig.3.5.4 Fig.3.6.1 Fig.3.6.2 Fig.3.6.3 Fig.3.6.4 Fig.3.6.5 Fig.3.6.6 Fig.4.2.1 Fig.4.2.2 Fig.4.3.1 Fig.4.3.2 Fig.4.3.3 Fig.4.3.4 Fig.4.3.5 Fig.4.3.6 Fig.4.3.7 Fig.4.3.8 Fig.4.3.9 Fig.4.3.l0 Fig.4.3.11 Fig.4.3.12 Fig.4. 3.13 Fig.4.3.l4

Bhattacharya distribution of the sediment diffusion

coefficient.

Rouse/Einstein distribution of the sediment diffusion coefficient.

Van Rijn f1uid diffusion coefficient and concentration

profile.

Bottom concentration versus reference concentration.

Typical concentration, velocity and transport

profiles.

Concentration in the bottom layer.

Computation of the mean velocity in the bottom layer. Water particle movement in waves.

Variation of velocity with height.

Velocity profiles near the bottom at various phases in one wave cyc1e.

Wave friction parameters (p, fw)'

Concentration distribution according to Bosman.

Sediment concentrations versus time above ripp1e crest. Bottom velocity versus time.

Water ba1ance under waves.

Wave-induced current, measured in a flume with no net mass

transport (Nie1sen [1985]).

Vortex formations.

Plan view and specific velocity components at an e1evation

Zt above the bottom.

Shear stress component at an elevation Zt above the

bottom.

Components of the mean shear stress.

Suspended sediment transport parameters.

Samples drawn from two different homogeneous

concentrations. Isokynetic suction. Tranverse suction.

Determination of concentration suction samples.

Hydrocyc1one.

Transmission and scattering.

Definition sketch. Definition axis system.

Radiation stress (caused by waves).

Wave-induced changes in horizontal momentum caused by

pressure and velocity f1uctuations.

Principal radiation stresses.

SXX' Syy and wave height as a function of water depth.

Radiation stresses for ob1ique approaching waves.

Mohr Circ1e ana1ysis.

Definition sketch - radiation stress component, Syy'

Laboratory measurements of h' on a slope 1:12;

T - 1.14s; Ho - 6.45 10-2 m; Hb - 8.55 10-2 m. The 1ine

'theoretical curve' represents Eq.4.3.l9. Source: Bowen et

al [1968].

Average water level with waves.

Equilibrium of force in the entire breaker zone.

Computed wave set-up.

Current caused by diffracted waves. Circu1ation current in breaker zone.

Fig.4.3.l5 Fig.4.3.l6 Fig.4.3.l7 Fig.4.3.l8 Fig.4.3.l9 Fig.4.3.20 Fig.4.3.2l Fig.4.3.22 Fig.4.3.23 Fig.4.3.24 Fig.4.5.l Fig 4.5.2 Fig.5.2.l Fig.5.3.l Fig.5.3.2 Fig.6.1. 1 Fig.6.l.2 Fig. 6.1. 3 Fig. 6 .1.4 Fig.6.2.l Fig.6.2.2 Fig.6.2.3 Fig.6.3.l Fig.6.4.l Fig.6.4.2 Fig.6.4.3 Fig.6.4.4 Fig.6.4.5 Fig.7.l.l Fig.7.l.2 Fig.7.1. 3 Fig.7.2.l Fig.7.2.2 Fig.7.2.3 Fig.7.2.4 Fig.7.2.5 Fig.7.2.6 Fig.7.2.7 Fig.7.2.8 Fig.7.2.9 Fig.7.2.l0 Fig.7.2.11 Fig.7.2.l2 Fig.7.2.l3 Fig.7.2.l4

Refraction theory principles. Computed radiation shear stress. Tidal current along the shore.

Effect of turbulence on the velocity profile.

Influence of the y component of the bottom shear stress on the current direction.

Simplified velocity distribution in the breaker zone. Effect of turbulence on the longshore current distribution. Contributions irregular wave field to longshore current. Longshore velocity distribution (regular wave field). Comparison of calculated velocity profiles.

Bed regime boundaries for waves and for a current. Eddy formation near ripples.

Results of the calculations.

Examples of velocity profiles and related sand transport profiles.

Sensitivity of Bijker formula.

Onshore-offshore sediment transport. Summer and winter beaches.

Bar movement offshore resulting from three storms (Birkemeier [1984]).

Recovery of an offshore bar. Undertow under breaking waves.

Sediment concentration profile (Bosman [1982]).

Measured and computed (b - 0.25) profile development on a small-scale laboratory beach.

Measured and computed profile development. Equilibrium beach profile (Bruun [1954]). Development of the equilibrium beach profile.

Beach profile dependence on wave height, grain diameter and water level changes.

Schematized beach profile for cross-shore transport at point A.

Schematised D-profile.

Single line theory. 2- and 3-line theories.

Principles of grid computations. Beach profile schematization. Continuity equation relationship. Shore plan showing 6yj6x and ~'.

Accretion of the shore near a breakwater. Shoreline accretion parameters

Accretion geometry.

Shoreline development at the lee side of the breakwater. Profile at the point at which sand starts to pass round the end of the breakwater.

Validity zones of models.

Re-distribution of increasedjdecreased beach slope. 'Outbreaking' water.

Tidal current passing a breakwater. Harbor entrance plan.

Fig.7.2.15 Fig.7.3.1 Fig. 8 .1.1 Fig.8.2.1 Fig.8.2.2 Fig.8.3.1 Fig.8.4.1 Fig.8.5.1 Fig.8.5.2 Fig.8.5.3 Fig.8.5.4 Fig.9.2.1 Fig.9.3.1 Fig.9.5.1 Fig.9.5.2 Fig.9.5.3 Fig.9.5.4 Fig.10.2.1 Fig.10.2.2 Fig.10.2.3 Fig.10.3.1 Fig.10.6.1 Fig.10.6.2 Fig.10.6.3 Fig.10.6.4

Beach accretion 1ines.

Shore plan and profile (two-1ine schematization).

Areas be10w mean sea level in the Netherlands.

Schematic cross section of dunes in the Netherlands. Erosion as a function of time.

Principle of a scale series.

Examp1e of a calculated eros ion profile. Computed retreat distances in severe storms.

Predicted and measured water level during the January-February 1953 surge at Flushing.

Position of the Netherlands on the North Sea. Frequency of exceedance curve of Hook of Holland.

Current pattern across a channe1. Attraction and def1ection of waves. Sedimentation mechanism.

Continuity equation re1ationships.

Simp1ifications of sedimentation mechanism of a current crossing a channe1 made by Bijker [1980].

Total sediment transport in the channe1.

Temporary cross shore sediment transport.

Position of "the coastline" of a stab1e coast with storm events.

Erosion due to a longshore sediment transport gradient. Entrance channel depth maintenance.

Beach nourishment a10ng the Dutch coast, in the period 1952-1985 (Roelse [1985]).

Location and number of groins a10ng the Dutch coast (Vellinga [1986]).

Dikes along the Dutch coast.

Princip1es of dune foot revetment and locations a10ng the Dutch coast.

LIST OF TABLES Table 2.2.1 Table 3.3.1 Table 3.5.1 Table 3.6.1 Tab1e 4.3.1 Table 4.3.2 Tab1e 4.3.3 Tab1e 4.3.4 Table 4.4.1 Tab1e 5.1.1 Table 5.2.1 Tab1e 5.2.2 Table 5.3.1 Table 5.3.2 Table 5.3.3 Tab1e 7.1.1 Table 7.2.1 Table 7.2.2 Table 7.2.3 Table 9.2.1 Table 9.5.1 Table 9.5.2

Formulas for shal1ow, transitiona1 and deep water waves. Va1ues of Einstein integra1 factor

Q

.

Values of Einstein integral factor, Q, and va1ues of the ratio suspended load to bed load, Ss/Sb' according to Bijkers transport formu1a.

Transverse suction rate for different particle sizes.

Radiation stress va1ues.

Radiation shear stress as a function of distance from the shore, y.

Longshore current for points within the breaker zone. Longshore current determination, alternative methods compared, for a deep water wave height and period of 2 m and 7 s.

Size c1assification of sediment particles.

General description of longshore sediment transport predictors (after F1eming et al. (1986])

CERC-formu1a coefficients.

Table 5.2.3 Sediment transport as a function of ~o·CERC-formu1a derivatives using H i .

Computed sediment transport va1ues ïo~ a series of points within the breaker zone.

Sediment transport rates computed with different

longshore current profiles (for exp1anation see text). Tota1 sand transport for different longshore current profiles.

Numerical bottom change mode1s (TOW (1980]). Shoreline accretion parameters.

Corrections to breakwater tip transport computations Stip/S.

Coast1ine computations.

Resu1t of flow 1ine computation. Channe1 sedimentation parameters. Tota1 sediment transport.

1 INTRODUCTION

1.1 PURPOSE

In general "Coastal Engineering" concerns all engineering problems in coastal areas. Because coastal engineering is so extensive, a

subdivision is made into three main areas according to the type of problems experienced. These three main categories are Harbors, Morphology and Offshore. One cannot say that Morphology is the most important of all three, but the other two categories, Harbors and Offshore do have to take into account the morphology; harbors can be harassed by siltation; offshore constructions, like pipelines, can be harassed by scour. This course discusses only coastal morphology.

'Coastal morphology' means the physical shape and structure of the coast. In other words: coastal morphology is the study of the

interaction between waves and currents, and the coast, which results in sediment movement and eventually in coastal changes. Of course hardly any sediment move ment takes place on rocky coasts. These areas do respond to wave and current interaction, but because of the large time scale of this response, the changes are of more concern to geology than to coastal engineering. Coasts frequently consist of sandy material which responds rapidly to the influence of waves and currents.

Problems originating from sediment transport generally concern either siltation or scour (erosion). Some examples of siltation and scour, see Fig.l.l.l, are:

Siltation: - upstream of breakwaters - harbor entrances

Scour: - downstream of breakwaters

- dune eros ion - under pipelines - near breakwater toe - around pile foundations Siltation and scour: - deformation of islands

Siltation problems can of ten be solved by dredging. This can be very costly and so it is of interest to know the amount of siltation. In the case of new coastal projects, the dredging costs have to be taken into account when deciding on the optimal design.

Scour problems of ten have a more serious nature particularly when they relate to safety in some way or another. Scour around a pipelirte can for example cause it to float and break. In the case of beach erosion the safety of the land behind the coast may be endangered particularly if this land is below sea level.

The main subject of this course is how to describe and predict coastal changes, and the type, mechanism and consequences of sediment

TOE EROSICl-I

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DUNE EROSION

DATUM

Fig.1.1.1 Examp1es of coasta1 morph010gy prob1ems.

1.2 COASTAL CHANGES

Coasta1 changes occur mostly as a resu1t of changes in sediment transport along the coast. If at one point, ray A (Fig.l.2.l), the sediment transport is for any reason larger (or smaller) than at

another point, ray B, accretion (or eros ion) will take place in between the two rays. This can be illustrated by considering the mass balance of the area between rays A and B, assuming no exchange takes place in a

cross shore direction, that is offshore or onshore.

Af ter calculating the sediment transport S at each location along the coast, that is, across each ray, the coastal changes can be predicted.

SA=SB SA>SB SA<SB stabie accretion erosion SEA t.S=O

- - - -

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area underconsideration

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LAND =

Fig.1.2.1 Transport mass ba1ance.

SEDIMENT TRANSPORT

First of all we want to know if any sediment transport wi11 take p1ace. Sediment transport on1y occurs provided there is sediment to be

transported. Furthermore the sediment has first to be moved. Sediment in water can on1y be moved if the water movement is strong enough to lift or ro11 the grains from and over the bottom. This point of initiation of movement is described by the critica1 velocity or the critica1 shear stress. If the critica1 shear stress (or critica1 velocity) is exceeded, the grains wi11 move, ro11 or be brought into suspension and thus sediment transport begins.

For examp1e, let us consider a sand bottom a10ng the coast. The critica1 velocity wil1 be about 0.20 mis (the determination of this critica1 velocity will be exp1ained in Chapter 3). An average current velocity of more than 0.20 mis for examp1e due to the tide wi11 cause sediment transport. Velocities of more than 0.20 mis (near the bottom) are caused in water depth of, for examp1e, 2.5 m even with waves as low as 0.25 m. It wi11 be clear that waves have an important inf1uence on the sediment transport a10ng the coast.

Sediment transport concerns the amount of sediment in mot ion and the rate, that is, the velocity at which this amount of sediment is transported. The amount of sediment is considered in terms of

concentration. Sediment transport is denoted in the most general way by: S c V where S transport c concentration V velocity 1.2.1

The above equation indicates the transport at one point. The total transport in one 'ray', SA (Fig.l.2.l), can be calculated by

integrating the specific transport (S - c V) from the bottom to the water surface and along the ray from the waterline to the region outside the breaker zone where transport no longer takes place. Note: In this lecture note the character V is used for the current velocity. In most cases the average velocity is meant (actually V). It will be clear from the topic under discussion which velocity parameter

is intended (the instantaneous velocity or the average velocity). However, if the velocity is unspecified the average is intended.

1.3 ABSTRACT

Because sediment transport is of ten directly related to the wave characteristics and waves greatly influence the transport, some wave theory topics are treated separately in CHAPTER 2.

CHAPTER 3 specifically treats the parameters included in sediment transport theories. These parameters are discussed separately. The relation between some parameters is mentioned.

CHAPTER 4 deals with more practical aspects and treats the reasons for transport and how it is initiated. This chapter only concerns coastal areas.

CHAPTERS 5 and 6 integrates all the aspects described in the earlier chapters into one transport formula for either longshore or cross shore sediment transport.

With these specific transport formulas for longshore and cross shore sediment transport, it should be possible to describe how a coast changes as a result of specific sea conditions; the case of coastal changes more or less beneath the water line is treated in CHAPTER 7; the case of coastal change above the water line, namely dunes, is treated in CHAPTER 8.

CHAPTER 9 considers the problems of entrance channels and trenches and discusses the related siltation and wave and current problems.

Having discussed the description and prediction of coastal changes in the earlier chapters, we finally review how we can influence these changes in CHAPTER 10. This chapter also discusses how we can improve the security of a coastline and prevent problems developing. Specific problems and solutions are mentioned with their particular advantages and disadvantages.

1.4 LITERATURE

Examples of literature about coastal engineering include:

- Journalof Waterway, Port, Coastal and Ocean Engineering, published quarterly by the American Society of Civil Engineers, New York, USA

- Shore and Beach, published half-yearly by the American Shore and Beach Preservation Association, Miami, Florida, USA

- Coastal Engineering in Japan, published annually by the Japan Society of Civil Engineers, Tokyo, Japan

- proceedings of the International Conference on Coastal

Engineering, published every two years by the American Society of Civil Engineers, New York, USA

- Coastal Engineering, published quarterly by Elsevier Scienee Publishers B.V., Amsterdam, the Netherlands

- Technical Reports CERC, US Army Corps of Engineers, Coastal Engineering Research Center (CERC), Washington, USA

- Teehnical Reports of Delft Hydraulics, Delft, the Netherlands These publieations generally provide the specific technical details of a problem and its solution.

Some general reference books of specific interest to coastal engineers are listed below. Each of these tells something but usually not

everything about a wide spectrum of coastal engineering topics.

- J.F.A. Sleath, Sea Bed Mechanies: Cambridge University: Wiley-Interseience Publication, New York, 1984

- K. Horikawa, Coastal Engineering, University of Tokyo Press,

1978

- W.H. Graf, Hydraulics of Sediment Transport, Me Graw-Hill, New York, 1971

- M.S. Yalin, Mechanics of Sediment Transport, Pergamon Press, Oxford, Great Britain, 1972

- R.M. Sorensen, Basic Coastal Engineering, Wiley-Interscienee Publication, New York, 1978

- J.B. Herbich, R.E. Schiller, R.K. Watanabe, W.A. Dunlap, Seafloor Scour, Marine Technology Society, New York, 1984 - Muir Wood, Coastal Hydraulics, MacMillan, Civil Engineering Hydraulics, London, Great Britain, 1969

A very practical handbook, which covers nearly all the topics in coastal engineering is the Shore Protection Manual [1984], Coastal

Engineering Research Center, Department of the Army, Vicksburg,

Mississippi.

Specific literature references are included at the end of each chapter. These references provide background information and information on

recent developments. These references are noted in the text by author and year of publication.

2.1 INTRODUCTION

If the sea was always quiet and smooth, coastal engineering would not be very interesting. But this is not the case and fluctuations in the sea level, long and short term and of ten very irregular, make the sea so powerfu1 and the coastal engineering so important and interesting. For many years people were afraid of the sea because they knew nothing about these f1uctuations which were unpredictable and of ten came complete1y by surprise. Today we know a lot more, but not all, about these fluctuations, where they originate and what their consequences are. Though nolonger "afraid" we still must exercise great caution in our dealings with the sea.

The purpose of this chapter is not to give a 'textbook' on wave theory but to give a short review of the wave theory which is necessary to complete this lecture note. Since waves or wave movements are used in the remainder of the lecture note for calculating and describing

sediment transport it is useful to first discuss the subject of waves and give the basic formulas and parameters. We assume that the reader is familiar with common parameters like H, c, k, wand T in wave theory. A list of symbols is given af ter chapter 10.

There are many different types of water fluctuations, for example: - wind waves / surface waves (capillary waves, ship waves) - astronomical tides

- tsunamis - seiches - wave set-up

- storm surges / wind set-up - climatological variations

Most of these fluctuations occur periodically, their periods varying fr om seconds to months. Fig.2.l.l shows a wave energy spectrum. In the figure periodic fluctuations are given with their range of periods (horizontally) and the relative quantity of energy (vertically). The figure also indicates the forces initiating the wave, the forces opposing the wave, and the way in which we designate the wave. Of greatest interest in coastal engineering (in Europe there are no tsunamis) are wind-generated waves, having periods from 1 s to about 20 s. This chapter discusses how wind-generated waves can be analysed and described and what happens if they re ach the coast. Chapter 3 discusses the consequences of wind-generated waves, or more generally surface waves, for sand transport along the coastline.

24h 12h 5 mln 30s 1s 0,1s

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trans- I I long I infra' gravity I ultra , capillary

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tidal "period iqrovity 'I waves I gravity I waves

waves "waves Iwaves Iwaves I

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PERIOD (sl (otter Kinsman 1965J

Fig.2.l.l Estimated relative ocean surface wave energy,

a classification of surface waves by wave band, primary disturbing force, and primary restoring force (adopted from Shore Protection Manual 1984).

2.2 WAVE CHARACTERISTICS

If we put a buoy in the sea and keep it in one place, we can record its level. Measuring the level of the water surface as a function of time we get a graph similar to that shown in Fig.2.2.l.

Fig.2.2.l presents a very irregular picture. In order to get a better idea about wave movements, wave fields are of ten schematized into regular waves or even a sinusoidal wave form. Fig.2.2.2 gives a definition sketch for the most elementary properties of such a sinusoidal progressive wave. It should be noted that the axis-system starts at the sea bottom. In most wave theories the axis-system starts at mean sea level (therefore for mean sea level z-O). However because this course is about sediment transport and in sediment transport the

x-axis (z-O) is on the sea bottom we will treat the wave theories with the x-axis (z-O) on the sea bottom. One has to bear in mind therefore

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-Fig.2.2.1 Water-surface time-history record.

In rea1ity waves are not regu1ar and are certain1y not sinusoida1. However it is often very usefu1 to work with the regu1ar monochromatic

(and in theoretica1 prob1ems a1so sinusoida1) waves because the resu1ts give a good impression of what happens, and areasonabie simi1arity to rea1ity. A lot of research has a1ready been done on regular waves and on the consequences of regu1ar waves (e.g., water velocity profiles, wave penetration in harbors, sand concentrations).

z

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Fig.2.2.2 Definition sketch of progressive, sinusoida1, surface waves.

Nowadays theories and computer programs have been extended to handle irregu1ar waves. In some cases equipment is avai1ab1e to produce known irregu1ar wave fie1ds (model research) or to ca1cu1ate the consequences of irregu1ar wave fie1ds (computers). However, if the sophisticated equipment needed for irregu1ar waves is not avai1ab1e usefu1 resu1ts can still be obtained with regu1ar monochromatic waves.

When using irregular waves it is first necessary to analyse and

characterise wave records such as that shown in Fig.2.2.1. This can be done probabilistica11y. The approach used for analysing and

characterising an irregular wave record is discussed below af ter a discussion on regular waves.

2.2.1 REGULAR WAVES

Even truely monochromatic regular waves are difficult to describe because of their non-sinusoidal profile. There are several theories, varying both in complexity and in accuracy, for describing these waves.

The simp lest and generally most useful theory is the Airy theory

(1845). Airy presented this wave theory in which he simplified the wave profile to a linear sinusoidal wave form. His theory provides equations for the most important properties of surface gravity waves, and

prediets these properties within useful limits in most practical conditions, even though real water waves are not sinusoidal.

As non-linear theories for periodic, regular waves we can mention:

- Small amplitude theories. The higher order Stokes theories which are based on waves with both a small wave height / wave length ratio

(H/À«l) and a small wave height / water depth ratio (HÀ2/h3«26). The first order Stokes (non-linear) theory is, in fact, the linear theory of Airy, discussed before.

- Shallow water theories (À/h»l, H/h<l). In this category we find the

cnoidal wave and the solitary wave theory (Boussinesq [1872]).

Rotational wave theories in which the distribution of vorticity is

taken into account. An example of this theory is the Gerstner [1802] or trochoidal wave theory.

- Edge waves which are waves that develop on a sloping bottom and propagate along the shore with the wave crests perpendicular to the

shore line.

- Numerical theories which give the most accurate solutions (also for

near-breaking waves). We can also ment ion the stream-function wave

theory of Dean [1965], a publication which gives tabulated results

(often used in offshore technology), Cokelet [1977] for an exact solution in the case of a steady wave train in water of constant depth, and the vocoidal theory of Swart and Loubser [1978]. The

latter suggest that their theory is avai1ab1e for a wider range of conditions than Dean's theory.

Fig.2.2.3 indicates the regions to which the various wave theories apply. The figure is based on similar plots by Le Mehaute [1976] and Dean (1970).

Linear sinusoidal waves.

In this theory equations are given for properties like:

- the partiele velocity (u, w) at any height in the water column

- the partiele acceleration (ax,az) at any height in the water column

- the particle displacement (ç,r) at any height in the water column

- the pressure (p) at any height in the water column

- the wave speed (c)

- the wave group speed (Cg)

- the wave length (À)

- the wave profile (~(x»

- the wave energy per wave length per unit crest length (Et)

- the energy per unit water surface area (E)

- the wave power (U)

Shallow_;.t..I-=-o( Ironsitionol ~I_ Deep

water water Iwatèr

The general equations for the above properties are found in Table 2.2.1

in the "Transitional Water Depth" column. The equations for

transitional water depth are not easy to solve because simplifications

can not be made for transitional water depths. The equations can be

simplified however both for deep and shallow water (see Table 2.2.1).

The simplifications are made in the hyperbolic function terms

(Fig. 2.2.4).

;: '" .,., ... ~ b() 1 3 I

~I~

I <"'(0 I f-4 I o o I o

~I~

I E-< o u I o oe u o -< I -c

..,

3 lil o u ~

_..,

3 ~_.., 3 r:: .... lil >< Ol lil o u N Ol

..,

3 r:: ....

..

~

_...

3 + ~ N ..c: bi.

_...

I 0.

..

o :.: 110

...

ee <,_.... I o

..,

""

""

o v I 3 1 '" I ""'IE-< I () u 110 U

11~

~I'"

I :J

...

3 Ol o U

~I~

~I'"

I :l:

...

3 r:: .... lil N

'"

..c: lil o U W 3 r:: .... lil lil o u ] ~ ~

_...

3 r:: .... lil

11~

=71'" I

'"

~

_...

3 lil o o

~I~

_:.:1'" I ~o Ol o U W 3

]I~

.;; .;; o 0 U U

~IN

+ ~ N

.

ë 110

...

I 0. lil o U

..

_:.: 110

...

.,

<, ... I

...

""

110 110 ., U <, ...

""

I I

""

~ U) Z o H U) Z

""

:>: H Cl I 3 I '" I ""'IE-< I U ~ I

...

₃ W 3 I :l: r:: .... lil

~I~

I 71 71 ... lil lil 71 ... lil Ol o U N Ol r:: .... lil lil o U N

I

cG ~ W 3 ë 110

...

I 0. "'" ';: bO

...

.,

<, ... I ":.: bO

...

.,

<, ... I

""

a () I bO U a <,'" El lil 'il" >< Ol ><

'"

..

_lil <, El El El "El <,_...., QJ ... ... .... o

'"

0. QJ :>

'"

:l: » '" ...

'"

QJ ...

_.,

U QJ :> Ol :> ~ ....

'"

..

...

..

u g-o ... 00

.,

;. Ol :l: .c '"00 r:: " ... " ~ :. »

..,

.... U o ...

_.,

;. ...

..

'"

...U '"r:: .... 0 '"N "'.... lil ... 0.0 ..c:

'"

..

u lil :l: ... Ol U .... u ... ~ r:: o ...._u lil

...

..

...

..

u u Ol_... .. Ol ...u u r:: ...0 U N

'"

... lil... 0.0 .a ... Q) u lil :l: u r::

..

El Q) u Ol ... 0. lil .... -e ... lil U .... U_...

..

;. ... .. lil ...u ur:: ....0 U N ... .... lil ... 0.0 ..c: ...

_..

u Ol :0 ... Ol o ...._u ...

..

;.

..

~ lil lil

..

... 0.

..

o Ol

....

_... ::l lil .g

'"

i:il ...

..

r::

..

..

~ ;I o .... .... .... U

..

0. '" o () o

""

'"

~ I o u r::0 o

""

I ... ~ o .... u .... lil ~ .... u ~ ... ... Ol ..c:

..

.... o .... lil

'"

]

~ u

""

I U r::

""

I bO U

""

I ~ lil El <, ...., ...

..

;I o 0.

..

~

For relatively deep water (h > >"0/2 so kh > 11" where k - 211"/>" is the wave number):

sinh kh- 1/2(e(kh)

-

e(-kh) _{1/2 e(kh) tor kh -> "" 2.2.1} cosh kh 1/2(e(kh) ₊ e(-kh ) _{1/2 e(kh) for kh->}

""

2.2.2

tanh kh 1 for kh->

_""

2.2.3

For relatively _{shallow water (h}

_<

_{),0/25 so kh}

_<

_1/2):

sinh kh kh for kh -> 0 2.2.4 cosh kh 1 for kh -> 0 2.2.5 tanh kh kh for kh -> 0 2.2.6 y 20

15--Fig.2.2.4 Simplified hyperbolic

functions for progressive waves.

There are different views about the criteria for deep and shallow water waves. Mathematicians, who are concerned with the accuracy of the

computations, take a wide range of transitional water depths between the deep water limit and the shallow water limit. Engineers, while still being concerned with accuracy are also concerned about the complexity of computations because of the time involved, and take a more practical, smaller range of transitional water depths.

Mathematicians

shallow water deep water

h/>'o hl>' kh _h/>'o h/>. kh

<1/25 <1/12 <1/2 >1/2 >1/2 >11"

<1/20 <1/11 <1/1.7 >1/4 >1/4 >11"/2

Engineers

Some usefu1 re1ationships to remember are:

w2 - (2~/T)2 - gk tanh kh k = 2~/>.

ko - 2~/>'o

dispersion re1ation wave number wave number deep water

2.2.7 2.2.8 2.2.9 n __C_g_ 0.5 [ 1 + __ 2_k_h__ ] C sinh 2kh 2.2.10

The water partiele displacement is shown for a sha110w water wave and for a deep water wave in Fig.2.2.5. In deep water the effect of the waves does not extend down to the bed; in sha110w water the water makes

an oscillating movement over the entire depth. Near the surface the water particles describe an el1iptical path, near the bottom the water partieles make an horizontal osci11ating movement (Fig.2.2.5).

J~l

SWL

"":-,-,-

-

-

~~-~

'-_- \ ~ '---z=h

Q

Circular Orbils I' 2A=H

lp

,

,

I' I' ,I f 1 I _ti: I 1 1 ElliplicalOrbils 1

I

j

A*B

f I

1_

u I ~ I I I I I I I I z z t= O' u ;t0

Shallow-waTer wave _{Deepwo Ier wave}

Fig.2.2.5 Orbita1 motion under a sha110w water wave and a deep water wave (Shore Protection Manual [1984]).

Fig.2.2.6 shows the relation between the direction of the velocity and the acceleration of water particles at certain phases in the wave period.

-

-+

-

-

--1- -======:>Wove Propoqction Velocity

00000

u =+ u=0 u=- u= 0 u =+ w= 0 w=+ w=O w=- w=O Acceleration

C)E)G)GC)

ox=O 0x=+ ox=O ox=- ox=O o;z=- oz=O O2=+ O2=0 0%=

-Fig.2.2.6 Local fluid velocities and accelerations (Shore protection Manual [1984]).

2.2.2

IRREGULAR WAVES

It is not possible to use the regular wave theories described above to analyse and describe wave records of the type shown in Fig.2.2.l

measured at sea or near the shore. The waves shown in Fig.2.2.1 are caused by the wind and the wind has a very turbulent character. In theory each wind velocity could make its own wave (period, wave height). All the wind velocity components could together then make a very irregular water surface profile.

It has been found to be a very useful method to consider wind waves as a superposition of a lot of sinusoidal waves with different amplitudes, frequencies, phases and directions, referred to as spectral components

(Battjes [1984]). Describing the variation of the water level in time as the sum of a lot of sinusoidal terms we get:

'7(t) 2.2.11

where the instantaneous surface elevation

the amplitude of the ith eosine component

the frequency of the ith cosine component in cycles per unit time

the phase of the ith cosine component

This expression is cal led the one-dimensional random phase model (it is one-dimensional because the elevation is only a function of time t). In this expression there is only one stochastic value, the phase ai' The values of 0i are stochastic independent values each with a uniform probability distribution function (see Fig.2.2.7)

-1'( 1'( ex

p(a)

Fig.2.2.7 Uniform.probability distribution.

The amplitude, ai' and the frequency, f., are related to each other.

Each frequency has his own specific amplitude, which of course depends on the particular wave record. So if the relation between ai and fi is known, the wave field is known. The problem is how to find this

particular relationship.

Aspectral analysis (Fourier-transformation) of the surface elevation in one point as a function of time (~(t» can be used to find a

spectral variance density function E(f) in which f is the frequency in cycles per unit time. E(f) is defined such that its integral, over all positive values of f, equals the variance of ~(t) where the variance of

the surface elevation -(o~(t»2. Because the variance is proportional to the average energy, the spectral variance density function is often called the wave ener~y spectrum. This energy spectrum therefore

indicates how the total energy of the wave field is distributed over the various frequencies. The total average energy of the wave field per unit surface area itself can be found by multiplying the area beneath the energy spectrum curve by (1/2)pg:

co E - (1/2)pg

J

E(f)df

o

2.2.12 where

E

f

the mean energy per unit surface area the frequency in cycles per unit time

Because the variance and the energy are also proportional to the square of the amplitude of the surface elevation, the spectrum can also be seen as the relation between the amplitude ai (actually ai2) and the frequency fi which we need to know for Eq.2.2.l2. With this energy spectrum the wavefield can be described and reproduced. Here we only want to know certain characteristics of the wave field. Most of the characteristics can be expressed in terms of moments of E(f) denoted by ~:

<Xl

~

-

J

f(n) E(f)df n - 0, I, 2, ... 2.2.13

o

As we can see mo is the area beneath the energy spectrum curve, which was equal to U~(t)2.

mo -

J

E(f)df - ",(t)'

o

2.2.14

Characteristic wave periods (Battjes [19771);

- The 'zero-downcrossing wave period', To.

The 'zero-downcrossing wave period' is the mean time interval

between consecutive zero-downcrossings (Fig.2.2.8).

T_o - 2.2.15

1C

,

Tl

S

1./ rr

~ f\ 1\ ~

r

1\

1

!

IJ.

i-'::I.

L

(\,

"

I

r\

~ j

IJ

V

\

~

f\

J

~)

r

~

\,

_J

!\J

v

~

i

V

-r- r_ V

-IQ

10

• =

zero - down

crossno

Fig.2.2.8 Zero-downcrossing wave period.

- The peak-period, Tp.

The peak-period is the period at which the spectrum has its

maximum energy (Fig.2.2.9).

2.2.16

- The average period of the one-third highest waves, T1/3.

Because this period is very often nearly the same as the visual estimate of the "characteristic" wave period of a wave field, it has been called the significant wave period (Tsig).

frequency

Fig.2.2.9 Energy spectrum with peak-period.

A wave height is defined as the difference between the minimum and maximum water level between two zero-downcrossings (Fig.2.2.10).

Tl

10

s ..

1./

rr

(\

-

r

I

r

11

~ I

IH

Jlo..

I

r:

/

~

A

_j

"

j

_V

,

ij

~

'-hJ

"\)

fJ

-,

J

I\J

"V

~

V

ll-

I- V

•

zero- downcrbssinos

Fig.2.2.10 Wave height.

Characteristic wave heights are: - The mean wave height, ~H'

~H - J21Cm~ 2.2.18

- The root mean square wave height, Hrms'

Hrms is a measure of the average wave energy. Hrms - J(LHi2)/N. If an irregu1ar wave field has to be reproduced by a wave field of monochromatic waves with the same average energy per unit surface area, the monochromatic waves shou1d have the wave height Hrms of the irregular wavefield.

- The average height of the one-third highest waves, Hl/3 or Hsig.

As was the case with the significant wave period Tl/3, this wave

height is also referred to as the significant wave height, Hsig·

2.2.20

2.2.3 REGULAR WAVES AND A CURRENT

Up to now we have discussed waves alone. Very often there will also be a current flowing in the direction of or at an angle to the wave

propagation. This current will influence some of the wave

characteristics, and, in their turn, the waves will influence certain

current characteristics.

Clearly, if the current is in the same direction as the wave

propagation direction, the wave height will decrease and the wave

length will increase. If the current is in the opposite direction, the

wave height will increase and the wave length will decrease.

The wave frequency will change because the wave celerity will change. The wave celerity, in the absence of current, is given by:

so c = wik 2.2.21 c wave celerity w wave frequency k wave number, 21f/À w c k where

For currents and waves c will change to c', given by:

c'-c+V// 2.2.22

where c' wave celerity in case of waves and a current

is the current velocity component in the direction of

the wave propagation.

V/I

The frequency will change to w'. given by:

w' - c'k - (c + V/I) k - w + kV// 2.2.23

where w' : wave frequency in the case of waves and a current

2.3 WAVE MEAVSUREMENTS

There are different types of wave measurements. Which type has to be used depends on what the measurement is needed for. For example, when designing a flexible structure. e.g. a rubble mound breakwater, the failure of which is gradual and repairable. it is common to use the significant wave height as a main criterion. In other cases, for example. when designing a fixed rigid structure. e.g. an offshore drilling tower, the total wave spectrum is needed as a main criterion. In the case of an expensive piece of equipment with a limited

seakeeping abili ty, the percentage of exc·eedance of a particular wave height on a day-by-day basis is needed for construction planning.

The most inexpensive method of wave measurements is visual obseryati.on. These measurements are made from ships and yield an estimate of Hsi ' Tsig and lIsig (llsig is the significant direction of wave propagatio~b. This method ~f obs~rvation is used all over the world. The World Meteorological Organisation (WHO) collects and coordinates these measurements.

If more detailed measurements are needed instrumental obseryations can be made:

- Buoys.

Waverider.

A waverider is a floating buoy which records the vertical

acceleration. From the records a reliable estimation of the vertical elevation of the water surface can be made for frequencies in the range of wind waves (0.05 Hz to 1 Hz).

Pitch-roll-buoy.

A pitch-roll-buoy is a development of the waverider. It measures accelerations of the water surface in three directions. The records of this buoy yield an estimate of the vertical water surface

elevation and the direction of wave propagation. - Gauges.

Continuous gauge.

This is basically a vertical wire, pair of wires or tube which pierces the water surface. These are components of an electric

circuit which measure the variance of the resistance of the wire. The resistance varies with the depth of immersion of the wire. The

results yields a reliable estimate of the vertical water surface elevation.

Step resistance gauge.

This is basically a vertical staff with electrodes spaced at discrete intervals piercing the water surface. Depending on the water surface elevation the electrodes make contact. The records of this gauge yield a stepwise estimate of the vertical water surface elevation. Pressure gauge.

This gauge stays completely under water and measures the underwater

can be determined. This system filters out the higher frequencies depending on the depth at which it is placed.

- Remote sensing

A new development is the remote sensing of the water surface by means of a laser or radar device. These methods are able to measure wave

frequencies and wave directions over a large area simultaneously.

2.4 REFRACTION. DIFfRACTION. REFLECTION AND BREAKING

Now that we can analyse wave fields or predict wave fields, we have to

know what happens when these waves approach a coast. Different

situations can occur:

Breaking:

If a wave approaches water which is gradually becoming shallower

(a sand beach), the wave will be affected by the bottom when the

water depth becomes less than half the wave length, assuming that

the wave crest is parallel to the depth contours. Nearing the

breaker line the wave celerity (c) and therefore the wave length

(À) decreases while the wave height (H) increases (approaching from deep water there will be initially a slight decrease in wave height). A steeper wave profile therefore develops. At a certain water depth the wave height (or in some cases the wave steepness) becomes so large that the wave will break and the wave energy will be dissipated.

Reflection:

If the bottom slope has a steep profile like a dam or a dike, the waves will partly break and partly reflect. The steeper the bottom profile, the greater the wave reflection. A vertical wall will reflect practically all the wave energy and a standing wave will develop in front of the wall.

Refraction:

If a wave approaches water which is gradually becoming shallower (a sand beach), and the wave crest makes an angle to the depth contours, it will refract. The part of the wave crest which is already in shallower water will have a less celerity and therefore the wave crest will bend, the angle between the wave crest and the depth contours diminishing. This phenomenon is called refraction. Again when the water depth becomes too small the wave will break.

Diffraction:

If a wave meets an obstacle (an offshore island, a breakwater), a part of the wave crest will be reflected seaward. The remainder will bend around the obstacle and thus penetrate into the zone in the lee of the obstacle. This phenomenon is called diffraction.

Both refraction and diffraction cause the wave crests to bend, however, for different reasons. According to simple refraction theory the energy

flux remains constant between wave orthogonals (lines perpendicular to

the wave crest extending in the direction of the wave propagation).

Because the part of the wave which is in shallower water travels more

slowly than the part in deeper water, the wave bends. In the case of

diffraction the energy flux leaks over a wave orthogonal and as a

result the energy is less in the bending part of the wave than the

initial wave energy. The celerity of the wave crest, and therefore the

wave length, stays the same (because the water depth is assumed to stay

constant), the wave height, in contrast to refracted waves, will

therefore decrease.

The different phenomena 'breaking', 'refraction' and 'diffraction' will

be discussed in this paragraph.

2.4.1 BREAKING

The celerity of waves (c) in shallow or transitional water depths is a

function of the water depth (h) (Table 2.2.1). Decreasing water depths

produce decreasing celerities. To find arelation between the wave

height (H) and the water depth (h) we have to examine the energy flux

balance. To find the energy which enters or leaves the balance area we

need the energy flux. Wave energy flux is the rate at which energy is

transmitted in the direction of wave propagation across a vertical

plane perpendicular to the direction of wave advance and extending

over the entire depth. The energy flux is also cal led the wave power.

The average energy flux, per unit wave crest width, is:

u

- Ec_g - Ene

where U wave power or energy

E wave energy per unit

Cg wave group velocity

c wave celerity

n ratio Cg to c

2.4.1 flux per unit wave crest width surface area

By assuming that this energy flux does not change as the wave progresses through water of varying depth, we find:

2.4.2

where 1 and 2 are subscripts indicating the location at which the parameters are evaluated

If we choose Location 2 in deep water where the wave properties are more easily evaluated, we find:

where wave height ratio c to c wave gr8up ve10city wave ce1erity

For deep water n - no - 1/2 and:

2.4.4

This can a1so be written as:

J

_c_o_1__

2.4.5

The parameter Ksh is ca11ed the shoa1ing factor. The shoa1ing factor can be found in various tab1es but can a1so be calculated from Eq.2.4.5 or from the more specified equation:

Ksh -

J

1 _ 2.4.6 nh kh (1 2kh) ta + sinh 2kh where k h wave number (-2~/À) water depth

As we see, Ksh is pure1y a function of kh and therefore h/À.

In shal10w water the shoaling factor can be reduced (using Eq.2.4.5 and cl-)gR and nI-I) to:

2.4.7a

With a bit of algebra this becomes:

[ 1Ào ]1/4

--

-8~ h 4rç 0.4466

j ___.::::

h 2.4.7b Nearing the breaker line therefore the wave height increases. This

increasing wave height (and decreasing wave length) gives an increasing

steepness. As we can imagine there will obviously be an upper limit for the wave height:

due to a maximum wave steepness (H/À)

- due to a maximum wave height water depth ratio (H/h)

The first criterion, wave steepness, is valid in both shallow and deep water. The wave steepness is defined as the ratio of the wave height to wave length (H/À). From theoretical considerations the limiting

steepness is found to be:

[ 27rh ]

(1/7) tanh -À- 2.4.8

In deep water Eq.2.4.8 reduces to

2.4.9 which occurs when the crest angle is abou~ 1200 (Fig.2.4.l)~

Fig.2.4.l Maximum crest angle. In shallow water Eq.2.4.8 becomes:

2.4.10 Therefore ~ax ~ 0.9 h from which an upper limit for the second

criterion (the wave height to water depth ratio) is found more or less automatically. The depth at which the wave breaks is calléd the breaker depth. The ratio breaker wave height to breaker depth is often called the breaker index, denoted by ~.

2.4.11 Solitary wave theory gives:

A more practical and of ten used value for this breaker index is 0.6.

All the above breaking wave relations have been derived for a

horizontal bottom. In reality the bottom will be sloping. Depending on

the beach slope:

- the wave willor will not break or

- different kinds of breaker types will occur.

Battjes [1974] found a parameter to indicate whether the wave will

break or not. This surf similarity parameter reads:

where

f - 211' --- 2.4.13

wave height

deep water wave 1ength

beach slope (beach slope is a1so denoted by m - tan a)

If f

>

1 breaking wi11 occur. This is quite simi1ar to Irribarren's

[1949] approach; in this case the parameter reads:

tan a

e ----

2.4.14

It

e

<

4/~ ~ 2.3 breaking occurs.

Waves break in a different way depending on the beach slope and the

wave steepness. Three main types of breakers can be differentiated:

'surging breakers'. 'p1unging breakers' and 'spi11ing breakers' . The

transition from surging to p1unging breakers is often referred to as a

'co11apsing breaker'. Fig.2.4.2 shows how the different breaker types

can be recognized. The typica1 va1ue of the Irribarren parameter.

e.

for the breaker type (Battjes [1974]) is a1so given in the figure.

SURGING

COLLAPSING

SPILLING

2.4.2 REFRACTION

Refraction occurs if, for some reason, one section of a wave crest has a bigger celerity than its neighbour. Refraction occurs therefore not only in shoaling water if the waves approach obliquely, but also in cases where there is a gradient in the current velocity e.g., in tidal entrances, in major ocean currents, in harbour ent rance channels. Refraction in the case of currents is discussed in Chapter 9, which deals with entrance channels and trenches. The present section

discusses only refraction caused by shoaling water when the wave crests make an angle with the depth contours.

The process of decreasing wave celerity in decreasing water depth can be considered as similar to the decreasing speed of light in media with

increasing density. Snel's Law of geometrical optics has therefore been

considered and shown to give a valid approximation when applied to water wave refraction problems (although in optics the light beam changes speed abruptly while in water wave refraction there is a gradual change in wave celerity).

We consider a long crested, monochromatic wave train approaching at an angle to the shore in a gradually shoaling area with bottom contours

that are essentially straight and parallel as shown in Fig.2.4.3.

hO,o= 0.5

===-1-

depth contours

!.!

-- -- ___L_ _

Fig.2.4.3 Wave refraction over straight parallel contours.

The direction of wave propagation is perpendicular to the wave crest i.e.,

in the direction of the orthogonals. Orthogonals are lines perpendicular to the wave crest extending in the direction of the wave propagation.

Orthogonals are sometimes cal led rays. We assume that the power

2.4.15

where U

b

the wave power per unit crest length the distance between orthogonals

Using Eq.2.4.l (U - Enc) we get:

2.4.16 where E the wave energy

the ratio of wave group velocity to wave celerity n

c wave celerity

Using Table 2.2.1 to find E [E - (1/8)pgH2] and no [no - 1/2]:

Hl

J

---

1 Co bo - KshKr 2.4.17

Ho 2nl cl bI where H the wave height

Ksh the shoaling coefficient (see Eq.2.4.5) Kr the refraction coefficient (Jbofbi)

To find the ratio bofbl we first make use of Snel's Law:

sin f/)o

, to find f/)l 2.4.18

Since the distance between given wave orthogonals, measured parallel to the depth contours remains constant, for parallel depth contours:

2.4.19 cos

The computation procedure indicated above is easily carried out for coasts with an simple bathymetry. In reality there will always be a much more complicated pattern of depth contours (Eq.2.4.l9 does not hold) and these "hand" calculations will be impossible. In this case

therefore two basic calculating techniques are available for refraction patterns: graphical and numerical. A precise description of the first method is given in the Shore Protection Manual, Volume I Chapter 2. Fundamentally all methods of refraction analyses are based on Snel's Law and conservation of wave energy flux.

A refraction diagram is given in Fig.2.4.4 as an example of the results of a refraction study. If the wave orthogonals converge there is an

'accumulation' of energy and relatively high wave heights can be

expected. In contrast if orthogonals diverge the energy is spread over a larger part of the wave crest so the wave height is reduced.

OflTHOGONAL - WAVE CRES T PA T TERN PERIOD 12 SECONDS AZIMUTH 112 5'

CREST IfHERVAL 45 WAVE

Fig.2.4.4 Refraction diagram of Long Branch, New Jersey (from Pierson (1950]).

2.4.3 DIFFRACTION

Diffraction occurs when there is a sharp variation in wave energy along a wave crest. When a wave train is passing an obstacle there are, in the first instance, no waves in the lee of the obstacle. There wil1 therefore be a gradient in the wave energy along the wave crest. The water away from the obstacle has more energy (all the initial wave energy) than the water behind the obstacle (in first instance zero, since there are no waves). Energy is now transported along the wave crest to the part behind the obstac1e and bending waves develope in the lee of the obstacle.

The degree of diffraction which occurs depends on the ratio of a

characteristic lateral dimension of the obstacle, e.g., the length of a detached breakwater, L, to the wavelength, À. When a thin pile is

standing in waves with a large wave length, L/À

«

1, clearly the diffraction will be nearly 100% implying that the wave field is approximately the same as if there is no pile. In the case of a detached breakwater, L/À

»

1, diffraction occurs around each breakwater head. There is a large zone in which diffracted waves

develop. Both the undisturbed waves passing the breakwater (Fig.2.4.5a) and also the reflected waves are diffracted (Fig.2.4.5b)

As with refraction and shoaling there is a diffraction coefficient which is defined as the ratio of the diffracted wave height to the incident wave height assuming that the latter is not disturbed by the obstacle.

where diffracted wave height

wave height of the incident wave which is not disturbed by the obstac1e.

a : INCIDENT WAVE TRAIN b:REFLECTED WAVETRAIN

Water wave diffraction is ana1ogous to the diffraction of light in the same way that water wave refraction is ana1ogous to the refraction of light. Using this relation, Wiegel [1964] ca1culated the diffraction coefficient at selected points in the vicinity of the obstac1e and

tabu1ated his results.

"

i

Graphica1 methods are also avai1ab1e for ca1culating diffraction coefficients. With the Cornu Spiral it is re1ative1y easy to find an approximation of the diffraction coefficient for one or two obstacles. The use of the Cornu Spiral is exp1ained in certain course

documentation and 1iterature on short wave theory (eg., Battjes [1986]). Another graphica1 approach invo1ves the wave diffraction

diagrams given in the Shore Protection Manua1 Volume I Chapter 2. These diagrams give the diffraction coefficient as a function of position (relative to a semi-infinite rigid impermeab1e breakwater and to a breakwater gap).

dittreetien zone

The disadvantage of the above methods is that they assume a constant water depth. In rea1ity there wi11 generally be a sloping bottom or an

uneven bed and the resu1ts wi11 therefore be inf1uenced by this bottom.

There are numerical mode1s which take into account the water depth and therefore can consider diffraction, refraction and ref1ection all

together. The LINGO software package (linear wave propagation) , which

computes wave propagation phenomena in area with constant or variab1e water depth is an examp1e of such a numerical model. A part of the

LINGO package, GOLDHA, has been deve10ped for wave penetration into

harbours (Berkhoff [1981]). Fig.2.4.6 shows the resu1ts of the

computation of wave penetration into a harbour. In Fig.2.4.6a the wa11s

AB and BC have ref1ection coefficients of 0.05 and 0.30; in Fig.2.4.6b

Fig.2.4.5 Diffraction of an incident wave train (a) and

0.60 and 1.00. It is obvious that, with higher reflection, the wave

height within the harbour increases. In the figure the lines are

representing places of equal wave height (in terms of percentage of

incident wave height).

A

B

I!J"'I. 2S'" sa•• 7S·,. - ••- 100.,. q ,,, I) tI ,'.o!»""J,'.Jl

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_{15 ..} 2S'" -··-100"'110 ---- 12~·'" o o .,

.

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,'" tO (J:0.) "' T. I,. I 8

Fig.2.4.6a Computed wave height pattern (Berkhoff [1981]).

Another example of a numerical wave propagation model is CREDIZ

(Dingemans et al [1984]).This model determines the combined effect of

an arbitrary bottom topography and current patterns on a monochromatic

wave field and includes the effect of energy dissipation due to

A

" ''.!!./'' ~:

-

'

-

.

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,

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Fig.2.4.6b Computed wave height pattern (Berkhoff [1981]).

LITERAIURE

Airy, G.B. (1845): On Tides and Waves: Encyc10pedia Metropo1itana. (1974): Surf Simi1arity: proceedings

Coasta1 Engineering, Copenhagen, Battjes, J.A.

Conference on pp 467-479.

14th International Vol.I, Chapter 26,

Battjes, J.A. (1977): Probabilistic Aspects of Ocean Waves: Delft University of Technology, Department of Civil Engineering, Laboratory of Fluid Mechanics, Report NO.77-2.

Battjes, J.A. (1984): Wind Waves: Delft University of Technology, Department of Civil Engineering (in Dutch).

Battjes, J.A. (1986): Short Period Waves: Delft University of Technology, Department of Civil Engineering (in Duteh).

Berkhoff, J.C.W. (1981): Wave Penetration into Harbours, Comparison of Computation and Model Measurement: Delft Hydraulics, M1482.

Boussinesq, J. (1872): Theorie des Ondes et des Remous qui se Propagant le long d'un Canal Rectangulaire Horizontal, en communiquant au Liquide contenu dans ce canal des vitesses Sensiblement Paralleles de la

Surface au Fond: Journal de Math. Pures. et Appliques, Vol.17 Ser.2, pp 55-108.

Cokelet, E.D. (1977): Steep Gravity Waves in Water of Arbitrary Uniform Depth: Philos. Trans. Roy. Soc. Ser., A286, pp 183-230.

Dean, R.G. (1965): Stream Function Representation of Non-Linear Ocean Waves: J.Geophys.Res.70(18), pp 4561-4572.

Dean, R.G. (1970): Relative Validities of Water Wave Theories: Journal of the Waterways, Harbors and Coastal Engineering Division , American Society of Civil Engineers, Volume 96 (WW1) , pp 105-119.

Dingemans, M.W., M.J.F. Stive, A.J. Kuik, A.C. Radder and N. Booij (1984): Field and Laboratory Verification of the Wave Propagation Model CREDIZ: Proceedings 19th International Conference on Coastal

Engineering, Houston, Volume I, Chapter 80, pp 1178-1191.

Gerstner, F. (1802): Theorie der Wellen: Abhandlungen der Königlichen Bömishen Gese11schaft der Wissenschaften, Prague.

Irribarren, C.R. and C. Nogales (1949): Protection des Ports: XVII th Int. Nav. Congress, Section 11, Comm.4, Lisbon, pp 31-80.

Le Mehaute, B. (1976): An Introduction to Hydrodynamics and Water Waves: Springer Verlag, Dusse1dorf.

Pierson, W.J. (1950): The interpretation of Crossed Orthogona1s in Wave Refraction Phenomena: US Army Erosion Board, Tech. Memo. 21,

Washington, D.C.

Shore Protection Manual (1984): Volume I, Chapter 1/2/3, Coastal Engineering Research Center.

Swart, D.H. and C.C. Loubser (1978): Vocoidal Theory for all Non Breaking Waves: Proceedings 16th International Conference on Coastal Engineering, Hamburg, Volume I, Chapter 26, pp 467-486.

Wiegel, R.L. (1964): Oceanographical Engineering: Fluid Mechanics Series, Prentice-Hall, Englewood Cliffs, New Jersey.

3 SEDIMENT TRANSPORT

3.1 INTROpyCTION

Sediment transport p1ays an important ro1e in many co~sta1 engineering prob1ems. Frequent1y there is a shortage of material at some location

(undesired erosion); at other p1aces an overabundance of material can be equa11y troublesome (si1tation of a navigation channe1, for

examp1e). An important goal of coasta1 engineering research is to predict the sediment transport rates in the coasta1 region. Knowing

this transport, it is possible to predict both natura1 coast1ine changes and the inf1uence of man-made structures on the coasta1 zone. Compared to simi1ar predictions for rivers, coasta1 engineering

ca1cu1ations tend to be much more difficult; osci11ating water

movements under waves and the mu1titude of forces which cause currents increase the number of variables invo1ved considerab1y.

Transport can be defined most generally as a quantity of sediment which is moving (with a velocity). More specifica11y sediment transport in water (S) is defined as the product of a sediment concentration (c) and a velocity (V).

S - c V 3.1.1

If we are interested in the transport at one point Eq.3.1.1 shou1d be used. Usua11y however we are interested in the effect of sediment

transport on some given bottom area. It shou1d be obvious that a continuity principle can be applied to the volume w~~ch extends from the given bottom area to tpe water surface. Knowing the resu1ting sediment transport through the vertica1 boundaries, the bottom changes can be determi~ed. If the volume of sand which enters the area is bigger than the volume which 1eaves the area, accretion wi11 take p1ace. Vice versa, erosion wi11 occur. Our main interest is therefore the horizontal displacement of material through a given section in a given time.

In direct ana10gy to the situation in a river, the sediment transport rate can be described as the product of velocity, V, and sediment concentration, c, integrated over the water depth. In rivers, both V and c vary on1y very slow1y as functions of time, t, or horizontal position, x. They do vary as a function of e1evation, of course, and

this variation makes the use of the integra1 necessary. The equation for sediment transport in rivers is:

h

S -

I

c(z) V(z) dz

o