92,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
Understand and respond to a changing coastline with this comprehensive reference
Coastal engineering concerns society’s interactions with coastlines and relates, for example, to coastal flooding, beach erosion, seawalls and breakwaters, and the design of marinas. As climate change drives sea level rise, coastal engineering is critical in responding to increased coastal flooding and receding shorelines. The need to develop coastal infrastructure while minimizing environmental impacts makes this a vital field.
An Introduction to Coastal Engineering offers a comprehensive overview of this subject, designed to bridge existing gaps in the general civil engineering literature. Covering all major aspects of coastal engineering, including ocean wave behaviour, structures, sediments, mixing processes, and modelling, the book emphasizes practical solutions and applications for students and practicing engineers alike. Thorough and rigorous, yet highly readable, the book is a must-own tool for developing solutions towards a sustainable coastal future.
An Introduction to Coastal Engineering readers will also find:
- Pertinent descriptions of wave theories, wave transformations, and random waves
- Detailed discussion of practical solutions, recent advancements in the field, and up-to-date data sources
- Worked-through examples and end-of-chapter problems with some written assignments
- A spreadsheet appendix containing a set of reference solutions
An Introduction to Coastal Engineering is ideal for students in upper-level undergraduate and graduate courses in coastal engineering, practicing coastal engineers, and other engineers engaged in coastal flood protection, waterfront development projects, and the minimization of environmental impacts along shorelines.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 575
Veröffentlichungsjahr: 2024
Ähnliche
Table of Contents
Cover
Table of Contents
Title Page
Copyright
Dedication
About the Author
Preface
About the Companion Website
1 Introduction
1.1 Scope of Coastal Engineering
1.2 Outline of Book
1.3 Example Projects
1.4 Evolution of Coastal Engineering and Future Trends
2 Regular Waves
2.1 Introduction
2.2 Boundary Value Problem
2.3 Linear Wave Theory
2.4 Wave Energy and Momentum
2.5 Waves with a Current
2.6 Extensions to Linear Wave Theory
2.7 Nonlinear Wave Theories
Problems
3 Wave Transformations
3.1 Wave Shoaling
3.2 Wave Refraction
3.3 Wave Diffraction
3.4 Standing Waves
3.5 Wave Reflection
3.6 Wave Transmission
3.7 Wave Attenuation
3.8 Waves of Maximum Height
3.9 Breaking Waves
3.10 Wave Runup
3.11 Numerical Models
Problems
4 Random Waves
4.1 Introduction
4.2 Probability Distribution of Wave Heights
4.3 Wave Spectra
4.4 Long-Term Variability of Storms
4.5 Extreme Value Analysis
4.6 EVA Alternatives and Extensions
4.7 Annual Wave Conditions
Problems
5 Winds
5.1 Introduction
5.2 Wind Data
5.3 Annual Wind Conditions
5.4 Design Wind Speeds
5.5 Wind Speed Correction Factors
5.6 Hurricanes
Problems
6 Wave Predictions
6.1 Introduction
6.2 Wave Hindcasting – Simplified Approach
6.3 Wave Hindcasting and Forecasting – Numerical Models
6.4 Ship Waves
6.5 Laboratory-Generated Waves
Problems
7 Long Waves, Water Levels, and Currents
7.1 Long Wave Theories
7.2 Tides
7.3 Tsunamis
7.4 Long Wave Oscillations
7.5 Storm Surge
7.6 Wave Setup
7.7 Sea Level Rise
7.8 Climate Change Impacts
7.9 Coastal Flood Levels
7.10 Coastal Currents
Problems
Notes
8 Coastal Structures
8.1 Introduction
8.2 Seawalls
8.3 Rubble-Mound Structures
8.4 Slender Structures
8.5 Large Structures
8.6 Floating Structures
8.7 Wave Impact Forces
8.8 Floating Breakwaters and Bridges
8.9 Other Loads
8.10 Renewable Energy Infrastructure
Problems
9 Coastal Processes
9.1 Introduction
9.2 Coastal Forms
9.3 Sediment Properties
9.4 Threshold of Sediment Motion
9.5 Beach Characteristics
9.6 Sediment Transport Processes
9.7 Bluff Erosion
9.8 Scour
9.9 Mitigation of Erosion and Accretion
9.10 Approaches to Shoreline Protection
9.11 Coastal Restoration
9.12 Coastal Management
Problems
10 Mixing Processes
10.1 Introduction
10.2 Advection–Diffusion Equation
10.3 Solutions to the Advection–Diffusion Equation
10.4 Diffusion and Dispersion Coefficients
10.5 Stratified Flows
10.6 Mixing in Estuaries
10.7 Estuarine Flushing
10.8 Salinity Intrusion in Estuaries
10.9 Turbulent Jets and Plumes
Problems
11 Design of Coastal Infrastructure
11.1 The Design Process
11.2 Accounting for Uncertainty
11.3 Selected Design Tools
11.4 Aspects of the Design of Coastal Structures
11.5 Design of Harbors and Marinas
Problems
12 Coastal Modeling
12.1 Overview
12.2 Numerical Models
12.3 Model Laws
12.4 Laboratory Models in Coastal Engineering
12.5 Laboratory Facilities
12.6 Wave Generation and Measurement
12.7 Field Measurements
Problems
A Reference Solutions
B List of Symbols
C Physical Constants
References
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1 Results of linear wave theory.
Table 2.2 Results of linear wave theory – deep water.
Table 2.3 Corresponding values of
kd
and
d
/
gT
2
for
kd
estimation.
Chapter 3
Table 3.1 Results for linear standing waves.
Table 3.2 Classification of breaking waves.
Chapter 4
Table 4.1 Selected extreme value probability distributions.
Chapter 5
Table 5.1 Classification of hurricanes by the Saffir–Simpson scale.
Chapter 6
Table 6.1 Threshold duration for various fetches and wind speeds.
Chapter 7
Table 7.1 Principal constituents of tidal records.
Chapter 8
Table 8.1 Formulae for Morison equation components.
Table 8.2 Closed-form solution for a large vertical circular cylinder.
Chapter 9
Table 9.1 Sediment size classification.
Chapter 11
Table 11.1 Five-point scales used for likelihood and consequence levels.
Table 11.2 ACSE wave agitation criteria.
Table 11.3 SCH wave agitation criteria.
List of Illustrations
Chapter 1
Figure 1.1 Coastal dike example project.
Figure 1.2 Types of coastal structure projects. (a) Seawall, (b) rubble-moun...
Figure 1.3 Coastal sediment transport example project.
Figure 1.4 Marina design example project.
Chapter 2
Figure 2.1 Definition sketch of a regular wave train.
Figure 2.2 Hyperbolic functions.
Figure 2.3 Water particle orbits and velocity amplitude profiles for various...
Figure 2.4 Spatial and time variations of a wave group.
Figure 2.5 Coordinate systems for waves with a current.
Figure 2.6 Coordinate system for oblique wave propagation.
Figure 2.7 Comparison of reference frames.
Figure 2.8 Profile of steep waves in shallow water.
Figure 2.9 Comparison of shallow wave profiles.
Chapter 3
Figure 3.1 Definition sketch of wave shoaling.
Figure 3.2 Wave shoaling relationships.
Figure 3.3 Illustration of wave refraction.
Figure 3.4 Definition sketch of wave refraction.
Figure 3.5 Wave direction changes due to wave refraction.
Figure 3.6 Examples of wave refraction. (a) Headland, (b) bay.
Figure 3.7 Definition sketch of wave refraction for a straight shoreline.
Figure 3.8 Refraction coefficient for a straight shoreline as a function of ...
Figure 3.9 Example application of a wave shoaling/refraction numerical model...
Figure 3.10 Illustration of wave diffraction around offshore breakwaters....
Figure 3.11 Example breakwaters and structures for which wave diffraction so...
Figure 3.12 Definition sketch of wave diffraction around a straight semi-inf...
Figure 3.13 Diffraction diagram for a straight semi-infinite breakwater for ...
Figure 3.14 Diffraction around a pair of overlapping breakwaters.
Figure 3.15 Definition sketch of standing waves.
Figure 3.16 Standing wave systems in closed and open-ended basins. (a) Close...
Figure 3.17 Sketch of partial wave reflection.
Figure 3.18 Sketch of oblique wave reflection.
Figure 3.19 Depiction of wave transmission past a barrier.
Figure 3.20 Friction factor as a function of wave Reynolds number Re
w
and re...
Figure 3.21 View of the surf zone, Honolulu, HI.
Figure 3.22 Illustration of types of breaking waves. (a) Spilling breaker, (...
Figure 3.23 Photographs of types of breaking waves. (a) Spilling breaker, (b...
Figure 3.24 Definition sketch of wave runup and wave setup.
Figure 3.25 Example application of a wave hindcast/transformation numerical ...
Chapter 4
Figure 4.1 Comparison of regular and random waves. (a) Regular, long-crested...
Figure 4.2 Sample of a random wave record.
Figure 4.3 The Rayleigh probability distribution. (a) Cumulative probability...
Figure 4.4 Plot of the Pierson–Moskowitz spectrum in dimensionless form.
Figure 4.5 Transfer function
T
F
(
f
) relating an output spectrum
S
x
(
f
) to an i...
Figure 4.6 Depiction of a directional wave spectrum.
Figure 4.7 Directional spreading function for various values of the spreadin...
Figure 4.8 Illustration of an EVA plot.
Figure 4.9 Illustration of a wave scatter diagram.
Chapter 5
Figure 5.1 Illustration of wind direction variation near Vancouver, BC.
Figure 5.2 Illustration of a wind rose.
Figure 5.3 Illustration of U
2m
occurrences in a wind record.
Figure 5.4 Satellite view of Hurricane Katrina, 2005.
Figure 5.5 Sketch of wind speed and pressure variations across a hurricane....
Figure 5.6 Graphic showing hurricane tracks meeting prescribed criteria.
Chapter 6
Figure 6.1 Distribution of the 99th percentile annual significant wave heigh...
Figure 6.2 Distribution of significant wave height (in ft) and wave directio...
Figure 6.3 Views of a ship-generated wave system. (a) Photograph of ship-gen...
Figure 6.4 Definition sketch of a ship-generated wave system.
Chapter 7
Figure 7.1 Illustration of diurnal, semidiurnal, and mixed tide level variat...
Figure 7.2 Views of tidal bores. (a) Tidal bore on the Qingtang River, China...
Figure 7.3 Tide record at Port Alberni, BC, during the arrival of the 1964 A...
Figure 7.4 Aspects of the 2004 Indian Ocean tsunami. (a) Initial drawdown an...
Figure 7.5 Example application of a tsunami model. (a) Vertical ground displ...
Figure 7.6 Illustration of aspects of a tsunami warning system. (a) Sensors ...
Figure 7.7 Amplification factor as a function of relative frequency for a ha...
Figure 7.8 Elevated waterfront houses at Outer Banks, NC.
Figure 7.9 Definition sketch of wind setup.
Figure 7.10 Depiction of wind setup and pressure setup components of hurrica...
Figure 7.11 Example application of a numerical model of storm surge inundati...
Figure 7.12 Definition sketch of wave setup.
Figure 7.13 Sea level rise measured by satellite altimetry.
Figure 7.14 Ground uplift rate for Canada.
Figure 7.15 Depiction of flood construction level components.
Figure 7.16 Gyres in ocean basins worldwide.
Chapter 8
Figure 8.1 Examples of categories of coastal structures. (a) Seawall – Vanco...
Figure 8.2 Definition sketch of normally incident linear waves at a vertical...
Figure 8.3 Definition sketch for the Miche-Rundgren/Sainflou methods.
Figure 8.4 Definition sketch for loads due to plunging breakers.
Figure 8.5 Definition sketch for the Goda formulation. (a) No overtopping (
h
Figure 8.6 Examples of rubble-mound structure sections. (a) Shoreline protec...
Figure 8.7 Examples of a rock breakwater and free-standing armor units. (a) ...
Figure 8.8 Reference flows used as a basis for development of the Morison eq...
Figure 8.9 Time variations of velocity and of Morison equation force and for...
Figure 8.10 Definition sketches for a pipe section and a vertical pile. (a) ...
Figure 8.11 Variation of force coefficients with
K
for various values of Re....
Figure 8.12 Variation of lift coefficient
C
L
with
K
for various values of Re...
Figure 8.13 Definition sketch for a vertical circular cylinder.
Figure 8.14 Variation of effective inertia coefficient
C
m
and phase angle
δ
...
Figure 8.15 Illustration of structure configurations for which diffraction s...
Figure 8.16 Definition sketch of a spring-mass-dashpot system.
Figure 8.17 Response curves for a single-degree-of-freedom system.
Figure 8.18 Definition sketch of structure motions.
Figure 8.19 Examples of wave impact loads on structures. (a) Horizontal cyli...
Figure 8.20 Examples of floating breakwater configurations.
Figure 8.21 Transmission coefficient
K
t
as a function of
B
/
L
.
Figure 8.22 Definition sketch for loads on fully and partially suspended cab...
Chapter 9
Figure 9.1 Photographs of various coastal forms. (a) Sand beach – Kailua, HI...
Figure 9.2 Shields parameter as a function of Reynolds number Re
*
.
Figure 9.3 Beach categories with respect to materials, steepness, width, and...
Figure 9.4 Illustration of summer and winter beach profiles.
Figure 9.5 Illustration of longshore current due to oblique waves approachin...
Figure 9.6 Solution of accretion profiles adjacent to a groin at various tim...
Figure 9.7 Types of erosion mitigation schemes. (a) Groin field – Rostock, G...
Figure 9.8 Selected shoreline protection schemes. (a) Revetment – Mundesley ...
Figure 9.9 Selected nature-based and hybrid shoreline protection schemes. (a...
Chapter 10
Figure 10.1 Solution of the one-dimensional diffusion equation for an instan...
Figure 10.2 Solution of the one-dimensional advection–diffusion equation for...
Figure 10.3 Representative vertical salinity distributions. (a) Two-layer fl...
Figure 10.4 Definition sketch of a simple jet or plume.
Figure 10.5 Schematic of an effluent plume associated with a multiport diffu...
Chapter 11
Figure 11.1 Illustration of a risk matrix.
Chapter 12
Figure 12.1 Photograph of a model study to assess downtime for a ship loadin...
Figure 12.2 Photographs of different kinds of laboratory facilities. (a) Wav...
Figure 12.3 Sketch of a wave flume.
Figure 12.4 Sketch of a wave basin.
Guide
Cover
Table of Contents
Title Page
Copyright
Dedication
About the Author
Preface
About the Companion Website
Begin Reading
A Reference Solutions
B List of Symbols
C Physical Constants
References
Index
WILEY END USER LICENSE AGREEMENT
Pages
iii
iv
v
xvii
xix
xx
xxi
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
283
284
285
286
287
288
289
291
293
294
295
297
298
299
300
301
302
303
304
305
306
307
An Introduction to Coastal Engineering
Michael Isaacson, P. Eng.
Professor Emeritus of Civil Engineering
The University of British Columbia, Canada
Copyright © 2025 by John Wiley & Sons, Inc. All rights reserved, including rights for text and data mining and training of artificial intelligence technologies or similar technologies.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission.
Trademarks: Wiley and the Wiley logo are trademarks or registered trademarks of John Wiley & Sons, Inc. and/or its affiliates in the United States and other countries and may not be used without written permission. All other trademarks are the property of their respective owners. John Wiley & Sons, Inc. is not associated with any product or vendor mentioned in this book.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.
Library of Congress Cataloging-in-Publication Data Applied for:
Hardback ISBN: 9781394257140
Cover Design: WileyCover Image: © Raul/Adobe Stock
To Sharon, Ben, Abby, Ian, and Jordanna.
About the Author
Dr. Michael Isaacson is Professor Emeritus of Civil Engineering at the University of British Columbia (UBC). He received his degrees from the University of Cambridge and has been active throughout his career in teaching, research, university service, professional service, and engineering practice.
Dr. Isaacson’s teaching contributions have included the delivery of a course in coastal engineering over several decades. His research interests relate to coasal engineering and he is the author/co-author of over 200 technical papers in his field. His service contributions have included roles as Head of Civil Engineering and Dean of Applied Science at UBC, journal editorships, and leadership roles on national and international professional committees and professional associations. Dr. Isaacson is the recipient of many national and international career achievement awards, best paper awards, and professional service awards.
Dr. Isaacson is a professional engineer and throughout his career has contributed to a wide range of local, national, and international engineering projects. These have included projects relating to design wave and wave load predictions, the motions of floating structures, assessments of tsunami and hurricane impacts, sedimentation studies, sea level rise and coastal flooding assessments, laboratory model testing, and the design of marinas and coastal infrastructure. Dr. Isaacson remains active in coastal engineering practice.
Preface
Coastal engineering relates to the solution of engineering problems in the coastal environment. It concerns, for example, the design of coastal structures such as breakwaters and seawalls, the design of harbors and marinas, understanding and addressing the consequences of sea level rise and coastal flooding, the assessment and control of sediment erosion and accretion, the design of shoreline protection schemes, and the assessment and controlled discharge of pollutants into the ocean environment.
This text stems from the author’s delivery of a combined undergraduate/graduate course in coastal engineering at the University of British Columbia over many years and from his engagement in coastal engineering practice over many years. It is intended to serve simultaneously as an introductory text directed to final-year undergraduate students, an advanced level text directed to graduate students, and a more general guide for practicing engineers engaged in coastal engineering projects. Some prior knowledge and understanding of applied mathematics and fluid mechanics is assumed.
The text includes the fundamental development of relevant concepts, a compendium of formulations, a series of illustrations and photographs (a number of the latter are drawn from British Columbia, reflecting the author’s experiences), a set of worked examples that illustrate basic calculations or the application of available spreadsheet solutions, and a set of problems and/or written assignments at the end of each chapter. The latter are written in generic form, and may be adapted by an instructor with respect to particular locations or circumstances.
The text contains three appendices. Appendix A, which may be downloaded as a spreadsheet from the text’s companion site, provides a set of reference solutions relevant to various formulations that are outlined in some chapters. Appendix B provides a list of the notation that is used and Appendix C provides values of physical constants that are used most frequently in coastal engineering. The text is based on the SI system of units.
In order that this text may serve simultaneously as an undergraduate and a graduate text, portions of the text considered to be at a more advanced level are contained within the symbols and , while problems or parts of problems considered to be at a more advanced level are preceded by the symbol ■.
This text is intended to serve as an introductory text and accordingly it does not encompass more advanced aspects of coastal engineering coverage. Thus, it does not provide detailed derivations of various formulae, but rather focuses on more general descriptions of how these are developed. While it makes extensive reference to numerical models, it does not provide sufficient information for the reader to be able to develop such models, which would rely on a thorough understanding of the underlying phenomena and of computational methods, nor to be able to utilize such models, which typically depends on users guides and specific training. Instead, the focus is on the development and use of spreadsheets to provide more fundamental solutions to coastal engineering problems. Finally, the text does not include a comprehensive list of references, but rather relies largely on references to other texts and manuals, sometimes implied, while more specific studies are only referenced as may be necessary.
Coastal engineering practice relies on access to and a reliance on various data sources, including wind records, tide records, hydrographic charts, bathymetry, and sea level rise projections. The format and availability of such sources vary worldwide. Where appropriate, commentary on access to such sources are provided, with a focus on data sources that are relevant to the United States and to Canada. If required, it is hoped that the reader may be able to find and utilize equivalent data sources in other countries as may be relevant.
The printed version of this text is in black-and-white, while the online version is in color. The reader is referred to the online version in order to view in color many of the figures including all those comprising photographs and graphics.
This work is an outcome of the author’s engagement over many years in coastal engineering research, teaching, and practice. Recognizing this, the author wishes to gratefully acknowledge the related collaborations, discussions, and contributions provided by his colleagues in the engineering profession and by faculty members and former students at the University of British Columbia and elsewhere.
The author hopes that the text contributes to the reader’s understanding of coastal engineering and ability to develop solutions to coastal engineering problems.
September 2024
Michael Isaacson
Vancouver, BC, Canada
About the Companion Website
This book is accompanied by a companion website:
www.wiley.com/go/coastalengineering
The website includes links that enable readers to download a spreadsheet Problem Data that contains data referred to in selected problems and a second spreadsheet Appendix A that contains a set of reference solutions. An additional link enables instructors to download relevant instructor resource materials.
1Introduction
1.1 Scope of Coastal Engineering
Coastal engineering relates to the solution of engineering problems in the coastal environment. It concerns, for example: the design of coastal structures such as breakwaters and seawalls, the design of harbors and marinas, understanding and addressing the consequences of sea level rise and coastal flooding, the assessment and control of sediment erosion and accretion, the design of shoreline protection schemes, and the assessment and controlled discharge of pollutants in the ocean environment.
Coastal engineering is a branch of civil engineering. Neighboring civil engineering subdisciplines that may be relevant to a coastal engineering project include hydraulic engineering, geotechnical engineering, structural engineering, and earthquake engineering. Neighboring disciplines include ocean engineering that relates to engineering projects in the ocean, uninfluenced by proximity to a coastline and often associated with offshore oil and gas recovery; naval architecture that relates to the design and operation of ships and marine vessels; and oceanography that relates to the scientific study of the oceans. Some subdisciplines of the sciences, such as nearshore oceanography and coastal geology, overlap with aspects of coastal engineering.
Several decades ago, Weigel (1964) and Ippen (1966) developed what may be regarded as foundational texts in coastal engineering. Subsequently, several texts on coastal engineering that have been available and widely used include Sorensen (2006), Sawaragi (2011), Reeve et al. (2018), and Kamphuis (2020), with each one providing different areas of emphasis. Many other texts focus on particular aspects of coastal engineering, such as marina design, port engineering, and coastal processes, or on topics within related disciplines such as ocean engineering and hydraulic engineering. In addition, various design manuals and guides that are relied upon in coastal engineering practice include the Coastal Engineering Manual (2002) and its predecessor the Shore Protection Manual (1984), the Rock Manual (2007), the Federal Emergency Management Agency’s (FEMA) Coastal Construction Manual (2011), and the EurOtop Manual (2018).
1.2 Outline of Book
Since ocean waves are usually the primary environmental consideration in coastal engineering, it is customary that a major part of a coastal engineering text relates to a description of waves. In this context, the chapter titles of the book are as follows:
Introduction
Regular Waves
Wave Transformations
Random Waves
Winds
Wave Predictions
Long Waves, Water Levels, and Currents
Coastal Structures
Coastal Processes
Mixing Processes
Design of Coastal Infrastructure
Coastal Modeling
More specifically, summaries of these chapters are given below.
Chapter 2, Regular Waves, describes the treatment of regular waves – that is, periodic waves that propagate over a horizontal seabed without a change in form. This chapter focuses on the development and application of linear wave theory.
Chapter 3, Wave Transformations, considers the transformation of waves associated with changes in water depth and with obstacles in the flow. Thus, the chapter treats wave shoaling, wave refraction, wave diffraction, wave reflection including standing waves, wave transmission, wave attenuation, wave breaking, and wave runup at a shoreline.
Chapter 4, Random Waves, recognizes the random nature of waves and distinguishes between the short-term variability of individual waves over a few hours and the long-term variability of different storms over several years, typically leading to design wave conditions with a specified return period.
Chapter 5, Winds, which serves as a prelude to Chapter 6, gives attention to descriptions of a wind climate, approaches to accessing and analyzing wind data needed for wave hindcasting, and a description of the wind field in a hurricane.
Chapter 6, Wave Predictions, describes the prediction of waves, with a focus on a simplified approach to wave hindcasting that provides estimates of wave conditions on the basis of available wind data. Other forms of wave prediction that are mentioned include ship waves and laboratory-generated waves.
Chapter 7, Long Waves, Water Levels, and Currents, outlines long waves, which include tides and tsunamis, and coastal flooding water levels and their components, which include storm surge and sea level rise. This chapter also indicates the impacts of climate change on coastal engineering practice and concludes with a summary of the kinds of currents that may be encountered.
Chapter 8, Coastal Structures, summarizes various categories of coastal structure and approaches to estimating wave loads on structures and wave interactions with structures. The kinds of structures considered include seawalls, rubble-mound structures, slender-member structures, large structures, and floating structures. This chapter also provides descriptions of the analysis of floating breakwaters and floating bridges and an outline of loads other than wave loads that may act on a coastal structure. The chapter concludes with a description of ocean-related renewable energy infrastructure.
Chapter 9, Coastal Processes, begins with an outline of the variety of coastal forms that may be encountered. It then describes in turn coastal sediments, the conditions for the onset of sediment movement under currents and waves, the movement of sediments near shorelines including sediment transport along beaches, bluff erosion, and scour, mitigation measures for addressing unwanted sediment erosion or accretion, approaches to shoreline protection, including reliance on nature-based methods, coastal restoration, and, finally, an introduction to coastal management.
Chapter 10, Mixing Processes, which is a primary topic of environmental fluid mechanics, describes fundamental solutions to the advection–diffusion equation and summarizes related topics that include stratified flows, mixing in estuaries, and jets and plumes.
Chapter 11, Design of Coastal Infrastructure, provides an introduction to the design process and approaches to accounting for uncertainty and outlines selected design tools including probability of failure analyses, risk assessment and management, permitting and approval requirements, and decision-making in design. It then summarizes selected design considerations with respect to coastal structures, including their modes of failure, design criteria, and aspects of detailed design. Finally, the chapter discusses the design of harbors and marinas, with attention given to available criteria for acceptable wave climate in marinas and to navigational considerations.
Chapter 12, Coastal Modeling, summarizes in a general way numerical modeling associated with various coastal engineering phenomena. It then outlines the underlying principles of model laws, describes different kinds of laboratory models used in coastal engineering, and provides descriptions of laboratory facilities, instrumentation, and measurements. The chapter concludes with a summary of field measurements relevant to coastal engineering.
1.3 Example Projects
In order to provide an appreciation of the breadth of coastal engineering practice, four generic example projects are now presented, along with an indication of the range of issues that may require consideration within each of these. (Any specific terminology that is used here is defined subsequently within the text.)
1.3.1 Coastal Flooding
The first generic project relates to an assessment of coastal flooding along a shoreline and the design of flood protection infrastructure. Figure 1.1 shows a coastal dike in Richmond, BC, used here as a basis for identifying key design parameters and associated issues.
The dike design entails the selection of the dike’s crest elevation and its sectional properties (e.g. crest width, seaward and landward slopes, rock size, vegetation, …) and a consideration of drainage and pump systems. The design is influenced by maximum water levels due to a combination of tides, sea level rise, storm surge and waves, and/or, if relevant, river flooding or tsunami flooding; a consideration of the probability and extent of any flooding in the context of uncertainty, risk, return period, and design life; wave runup and overtopping; and, finally, potential habitat enhancements, climate change impacts, permitting and approval requirements, and land use requirements. While Figure 1.1 indicates the case of a coastal dike, related coastal flooding projects may involve a seawall or other coastal defense along a shoreline, and a consideration of hurricane-prone areas for which storm surge is a major issue.
Figure 1.1 Coastal dike example project.
1.3.2 Coastal Structure Design
A second generic example relates to the design of different types of coastal structure. Figure 1.2 illustrates four types of structure: a seawall, rubble-mound shoreline protection, a piled pier, and a floating structure, with the figure used as a basis for identifying key design parameters and associated issues.
Common to all these are assessments of tides, water levels and wave climate, and an identification of return periods and suitable levels of uncertainty and risk. In addition, the design of seawalls entails a determination of wave loads, wave runup, and wave overtopping. Rubble-mound shoreline protection entails a determination of wave overtopping and a consideration of rubble-mound stability, which depends in part on the rubble-mound slope, rock type, and rock size. A piled structure entails a determination of wave loads on piles and the deck elevation. Finally, a floating structure usually entails an assessment of structure motions and mooring system and anchor design.
1.3.3 Sediment Transport
A third generic example project relates to an assessment of sediment transport along a beach or shoreline, along with any mitigation measures that may be undertaken. Figure 1.3 shows the shoreline at Spanish Banks in Vancouver, BC, used here as a basis for identifying key design parameters and associated issues.
Key considerations include sediment properties, the onshore–offshore transport of sediment during and between storms, the longshore transport of sediment (parallel to the shore), an assessment of beach slope and profile, and beach protection measures that may be introduced, such as groins, offshore breakwaters, and rock mounds. The design may be influenced by water levels, a consideration of uncertainty, risk, return period, and design life, and, finally, potential habitat enhancement, climate change impacts, and permitting and approval requirements.
Figure 1.2 Types of coastal structure projects. (a) Seawall, (b) rubble-mound shoreline protection, (c) piled pier, (d) floating structure.
Source: ShoreZone/CC BY 3.0.
Figure 1.3 Coastal sediment transport example project.
Source: ShoreZone/CC BY 3.0.
Figure 1.4 Marina design example project.
Source: Google Earth.
1.3.4 Marina Design
A fourth generic example project relates to the design of a marina. Figure 1.4 shows an example marina at Gibsons, BC, used here as a basis for identifying key design parameters and associated issues.
Key aspects of the design include the breakwater layout intended to achieve acceptable wave conditions within the marina, and the breakwater’s sectional design (i.e. crest elevation, crest width, slope, and rock type, size, and placement), so as to assure the effectiveness and stability of the breakwater. The design is influenced by design wave conditions approaching the marina; water levels (due to a combination of tides, sea level rise, storm surge, and waves); a consideration of uncertainty, risk, return period, and design life; and the infrastructure within the marina, including slips, moorage, and facilities, which depend in turn on vessel types, sizes, and numbers. A series of other issues that may require consideration include climate change impacts, permitting and approval requirements, sediment movement, currents, water quality and flushing, and navigability in the entrance channel. Economics, land transportation, and land-based issues are not normally considered within coastal engineering. In the case of ports or harbors that accommodate ships and larger vessels, many of the same considerations apply, but with an increased emphasis on a consideration of ship berths and moorings.
1.4 Evolution of Coastal Engineering and Future Trends
While coastal infrastructure had been built over many centuries, initially for coastal defense and harbor protection, it was not until the late 1800s that coastal infrastructure design was increasingly based on engineering principles, coinciding with the development of the engineering profession at that time. From the early 1900s onwards and certainly by the 1930s, key aspects of coastal engineering had become increasingly evident. These included an increased reliance on linkages to applied mathematics, fluid mechanics, oceanography, geology, and other disciplines; the application of scientific principles and scientific studies to areas such as coastal infrastructure design and beach protection; the introduction of laboratory modeling; an increasing number of papers published on coastal engineering topics; the formulation of early guidelines relating to coastal infrastructure and beach preservation; and the establishment of related organizations and research programs. For detailed information on the history and evolution of coastal engineering in various countries, the reader is referred to Kraus (1996).
Coastal engineering may be regarded as having evolved fully into a distinct discipline by the early 1950s. In 1950, the First Conference on Coastal Engineering was held in Long Beach, California with 35 invited papers. This conference series, which became known as the International Conferences on Coastal Engineering, is now held biennially in locations worldwide, with several hundred papers published in each set of proceedings.
Since the 1950s, there has been a continual broadening of the scope of coastal engineering from an initial focus on civil works projects so as to now include the development and application of advanced technologies over a range of areas such as dredging, beach nourishment, structural design, and port infrastructure and planning; the restoration, protection, and enhancement of coastal wetlands and other habitats, in collaboration with environmental engineers, environmental scientists, and biologists; and coastal management, with links to planning, geography, law, and other disciplines.
Associated with this broadening, a number of trends have emerged over the last few decades and are expected to evolve further into the future. These trends have arisen largely in response to emerging challenges, technological advances, and shifting societal values. A number of these are indicated below.
Computer modeling. Since the 1960s, the advent of computers and computer modeling has continually transformed coastal engineering practice. Today, computer modeling encompasses the use of spreadsheets to perform coastal engineering calculations, and the use of sophisticated computer models that can now describe highly complex and wide-ranging aspects of coastal engineering. This trend is expected to evolve further in the future. For example, AI (artificial intelligence) is already being relied upon in modeling coastal engineering phenomena.
Response to hazards and failures. Responses to natural disasters and failures, including devastation and damage arising from tsunamis, hurricanes, extreme storm surges, and extreme storms, along with lessons learned, have been the impetus to the establishment of research programs and to many research advances and design improvements, including the development of comprehensive design manuals that support engineering practice. As one of many examples, while breakwaters were being built in deeper water with larger artificial units in the 1960s and 1970s, extensive damage to a major breakwater at Port Sines, Portugal in 1978, which was associated with the strength, material properties, and shape of individual armor units, led to a major reconsideration of the design and deployment of very large armor units.
Probabilistic design. Another shift has placed a greater emphasis on approaches to accommodating uncertainties in the natural environment and in the design process. This has led to a more consistent approach to accounting for uncertainties in a project, an increased emphasis on probabilistic design, and the incorporation of risk assessment and risk management in projects.
Climate change. Of course, climate change and its impacts are now universally recognized, and approaches to accommodating the impacts of climate change have become an integral part of coastal engineering design. This is especially true with respect to the impacts of future sea level rise. Knowledge of climate change impacts is continually improving with respect to sea level rise, the intensity and behavior of hurricanes, extended ice-free seasons in Arctic regions, and wind and wave climates in different regions around the globe. Likewise, approaches to addressing these impacts, including adaptation measures along shorelines and probabilistic approaches to incorporating uncertainties, continue to evolve.
Range of applications. The range of applications of coastal engineering has been continually broadening, so that coastal engineers are increasingly engaged in areas such as the development of offshore renewable energy projects (involving also ocean engineering and other engineering disciplines), the restoration, protection, and enhancement of coastal wetlands and other habitats, and coastal management. Associated with this broadening are increased linkages with other disciplines such as geography, law, biology, business, and management, and increased interactions with planners, landscape architects, lawyers, biologists, developers, and government professionals. Three areas where such a broadening is occurring are highlighted below.
Nature-based shoreline protection. Associated with a greater recognition of the need to minimize environmental impacts, preserve habitats, and practice environmental stewardship, there has been an increased recognition of the need to incorporate coastal resilience into coastal engineering projects and to rely increasingly on “nature-based methods” of shoreline protection. Regulatory and permitting requirements increasingly require such considerations. While nature-based methods alone may be insufficient to provide adequate protection from shoreline erosion and flooding, the development of hybrid protection schemes that incorporate elements of traditional methods is becoming more common.
Coastal restoration. In a similar vein, there has been a major effort to support the recovery of degraded ecosystems and to support habitat protection and enhancement. This has led to a greater focus on the restoration of wetlands, salt marshes, and other coastal areas that support natural habitats. Engineers engaged in coastal restoration projects often collaborate with environmental biologists, landscape architects, and other professionals.
Coastal management and decision-making. Finally, coastal management, which refers to management activities relating to the coastal zone, is increasingly relied upon to assure an integrated and managed approach to potential interventions over broader areas of the coastal zone, with an emphasis on seeking sustainable solutions that are socially and environmentally responsible. Related to this, decision-making with respect to coastal development has become more inclusive and more reliant upon the engagement of user communities and all stakeholders. Accordingly, coastal engineering is becoming increasingly dependent on management, decision-making processes, and community and stakeholder interactions.
2Regular Waves
2.1 Introduction
This chapter provides a description of regular ocean waves, which refer to periodic water waves that propagate over a horizontal seabed without a change in form. The focus is on linear wave theory, which forms a foundation for much of coastal engineering practice. The governing equations of wave theory more generally and of linear wave theory in particular are described in turn, followed by the results of linear wave theory, the application of linear wave theory to determine the details of the associated flow, a consideration of energy and momentum aspects of waves, various extensions to linear wave theory, and, finally, an introduction to nonlinear wave theories.
In general terms, a wave theory provides a description of the fluid flow and related parameters associated with a periodic wave train propagating over a horizontal seabed without a change in form. Figure 2.1 provides a definition sketch of the wave form, identifying the coordinate systems (x, z) and (x, s) and various characteristic lengths associated with the wave train.
Fundamental parameters used to describe the wave train are as follows:
still water depth,
d
:
water depth if there is no wave activity
wave height,
H
:
vertical distance from wave trough to wave crest
wave period,
T
:
duration between successive wave crests crossing a fixed point
wave length,
L
:
horizontal distance between adjacent wave crests
The following parameters are derived from the above:
wave angular frequency,
ω
:
ω
= 2π/
T
wave number,
k
:
k
= 2π/
L
wave speed,
c
:
c
=
L
/
T
=
ω
/
k
The wave speed is also referred to as the wave celerity. A coordinate system (O, x, y, z) is defined with the origin O located at the still water level (denoted SWL) below a wave crest at time t = 0, x in the direction of wave propagation, y parallel to the wave crests, and z measured upwards from the SWL. That is, time t is defined such that t = 0 occurs when a wave crest crosses the z axis. A subsidiary vertical coordinate measured upwards from seabed is also used and is defined as s = z + d.
Figure 2.1 Definition sketch of a regular wave train.
A regular, periodic wave train is generally specified in terms of H, L, and d, or H, T, and d. Wave theory relates L and T and describes the fluid flow details, which refer to the free surface elevation that varies with x and t, the water particle displacement, velocity, and acceleration, and the pressure within the fluid, each of which varies with x, z, and t. These variables are as follows:
η
:
free surface elevation above the SWL
ξ
,
ζ
:
horizontal and vertical components of fluid displacements, respectively
u
,
w
:
horizontal and vertical components of fluid velocity, respectively
:
horizontal and vertical components of fluid acceleration, respectively
p
:
pressure (relative to atmospheric pressure)
It will be seen that, while the wave crests move along the x axis with speed c, the water particles themselves largely move in elliptic orbits.
2.2 Boundary Value Problem
A wave theory describes the details of a wave flow by developing and solving a boundary value problem for the unknown variables describing the flow. In fact, a solution to three of the flow variables, taken to be η(x, t), u(x, z, t), and w(x, z, t), is needed, since the remaining variables, including displacements, accelerations, and pressure, may be obtained from these.
The boundary value problem includes two equations of motion for the two variables u(x, z, t) and w(x, z, t) throughout the fluid region. However, as will be demonstrated below, it is possible and much more convenient to combine these into one variable, the velocity potential φ(x, z, t), that needs to satisfy a single equation of motion. The velocity potential is defined by:
That is, once φ(x, z, t) is known, u(x, z, t) and w(x, z, t) may be retrieved.
2.2.1 Assumptions
The boundary value problem may be developed on the basis of the following assumptions:
The fluid is incompressible and homogenous.
The fluid is inviscid and the flow is irrotational.
The seabed is horizontal and impermeable.
The flow is two-dimensional in the (
x
,
z
) plane, with waves that are periodic and of permanent form.
The water surface is uncontaminated and at constant pressure.
These assumptions may be used to develop the governing equations that are comprised of equations of motion and associated boundary conditions. In the following, the equations of motion and the boundary conditions are described in turn. These are then assembled so as to provide the complete set of governing equations for the wave flow.
2.2.2 Equations of Motion
The assumption that the fluid is homogenous (fluid density is invariable with location) and incompressible (fluid density is invariable with time) implies that the fluid mass within an elemental volume does not change. This leads to a continuity equation relating the velocity components u and w at any point in the fluid. The continuity condition for an incompressible fluid requires that:
The assumption that the fluid is inviscid implies that a fluid particle is not subjected to a shear stress. In this context, an irrotational flow refers to one in which fluid particles do not rotate around their own axes. This corresponds to the flow of an inviscid fluid (presuming that the particles are not initially rotating). Incidentally, a particle may well move in a circular orbit while not rotating about its own axis. The irrotationality condition requires that:
Rather than solve for the two unknowns u and w that need to satisfy the above two equations, it is possible to introduce a single variable, the velocity potential φ, as defined above. φ defined in this way satisfies the irrotationality condition, while the continuity equation leads to φ satisfying the Laplace equation throughout the fluid region:
2.2.3 Boundary Conditions
The various boundary conditions listed earlier are now developed. First, the condition that the seabed is horizontal and impermeable gives rise to a boundary condition at the seabed, which states that the vertical component of fluid velocity is zero at the seabed:
There are two boundary conditions along the free surface. The kinematic condition states that the fluid velocity at the free surface in a direction normal to the free surface is equal to the free surface velocity itself in that direction. This can be expressed in the alternative forms:
where nz is the z component of the unit vector normal to the free surface.
The condition that the free surface is uncontaminated and at constant pressure implies that surface tension effects are negligible (which is invariably the case, except for very short period waves under laboratory conditions), so that the pressure within the fluid along the free surface is equal to atmospheric pressure and so is constant. This leads to the dynamic boundary condition, which states that the pressure along the free surface, expressed in terms of the Bernoulli equation, is a constant. Thus:
where g is the gravitational constant and R is an arbitrary constant. Since φ may be redefined to incorporate any constant value without affecting the flow variables, one may take R = 0.
The requirement that the flow is two-dimensional in the (x, z) plane implies that the various flow variables do not change with the horizontal coordinate y that is parallel to the wave crests.
Finally, the assumption that the waves are periodic and of permanent form implies that flow variables such as the free surface elevation η and the velocity components u and w that vary with x and t may be considered to vary with a single compound variable x′ = x − ct, where c is the wave speed and x′ is distance measured relative to an origin that moves with a wave crest. Thus, when viewed from a wave crest, the free surface elevation does not vary with time so that it can be considered to be a function of x′ only: i.e. η(x, t) = η(x − ct). Likewise, the velocity potential φ can be written as φ(x, z, t) = φ(x – ct, z).
2.2.4 Governing Equations
Assembling the equations of motion and boundary conditions developed above, the governing equations become:
Laplace equation:
within the fluid region
Seabed condition:
at
z
= −
d
Kinematic FSBC:
at
z
=
η
Dynamic FSBC:
at
z
=
η
Permanent form:
φ
(
x
,
z
,
t
) =
φ
(
x
−
ct
,
z
)
where FSBC denotes a free surface boundary condition. In the above, permanent form refers to the requirement that the wave train moves steadily without its shape changing.
The boundary value problem defined by the above set of governing equations is nonlinear and its solution is rather complicated. This is because, first, the two FSBC’s apply on the initially unknown free surface location z = η, and, second, because the FSBC’s contain other nonlinearities corresponding to products of variables or their derivatives.
2.3 Linear Wave Theory
2.3.1 Governing Equations
An additional assumption is made in order to simplify the complete boundary value problem so as to enable a relatively straightforward and robust solution to be developed. This assumption is that the wave height H is small in relation to the other length scales of the wave flow, L and d. That is, H ≪ L, d. This simplification of the complete wave theory is referred to as linear wave theory, small amplitude wave theory, sinusoidal wave theory, or Airy wave theory.
There are two consequences of this additional assumption. First, the two free surface boundary conditions that apply on the initially unknown free surface location z = η may instead be applied directly at the known SWL, z = 0. Second, the nonlinear terms in the two free surface boundary conditions, corresponding to products of variables or their derivatives, are an order of magnitude smaller than the remaining terms and are therefore neglected.
Applying these to the complete boundary value problem, the governing equations for linear wave theory may be developed as follows:
Laplace equation:
within fluid region
Seabed condition:
at
z
= −
d
Kinematic FSBC:
at
z
= 0
Dynamic FSBC:
at
z
= 0
Permanent form:
φ
(
x
,
z
,
t
) =
φ
(
x
−
ct
,
z
)
where again FSBC denotes a free surface boundary condition. The two free surface boundary conditions may be rewritten such that one equation excludes η so that this equation along with the remaining governing equations may be solved directly for φ, while a second equation provides an explicit expression for η in terms of φ. These are, respectively:
2.3.2 Solution for Flow Field
The solution to the boundary value problem may readily be developed in order to obtain expressions for η and φ as follows:
Corresponding expressions for the various flow parameters (u, w, …) may then be developed from the above expression for φ.
However, for a given depth d, the wave period T and wave length L are related, so that either one, but not both, needs to be specified in defining a wave train. Thus, the solution provides also the linear dispersion relation that relates L and T on the basis of linear wave theory, so that one can be obtained if the other is known. Equivalently, the dispersion relation relates ω and k, or c and k. The various results of linear wave theory are given in Table 2.1.
Table 2.1 Results of linear wave theory.
Variable
Equation
Free surface elevation
Velocity potential
Dispersion relation
ω
2
=
gk
tanh(
kd
) or
Horizontal displacement
Vertical displacement
Horizontal velocity
Vertical velocity
Horizontal acceleration
Vertical acceleration
Pressure
Figure 2.2 Hyperbolic functions.
Background on hyperbolic functions. The hyperbolic functions sinh, cosh, and tanh appear extensively in Table 2.1. As background, summary information on these functions is now provided. The relevant hyperbolic functions are defined in terms of an argument z as follows:
Hyperbolic sine:
sinh(
z
) = [exp(
z
) − exp(−
z
)]
Hyperbolic cosine:
cosh(z) = [exp(
z
) + exp(−
z
)]
Hyperbolic tangent:
tanh(
z
) =
Each varies with the argument z as shown in Figure 2.2. Approximations for small and large values of the argument z are as follows:
For small
z
:
sinh(
z
) ≃ tanh(
z
) ≃
z
;
cosh(
z
) ≃ 1
For large
z
:
sinh(
z
) ≃ cosh(
z
) exp(
z
);
tanh(
z
) ≃ 1
2.3.3 Depth Parameter
The wave flow depends notably on the relative depth of the water, that is, the depth relative to the wave length, ranging from shallow water to deep water. Thus, a depth parameter kd (= 2πd/L), sometimes referred to as a relative depth parameter, appears in many of the formulae in Table 2.1. It is of interest to consider cases where the depth parameter is large, corresponding to deep-water waves, and is small, corresponding to shallow-water waves. Suitable approximations may be made for each of these ranges, leading to simplifications to the expressions given in Table 2.1. Between these two limiting cases, the waves are termed intermediate-depth waves, with the complete expressions in Table 2.1 then used. The deep-water and shallow-water approximations are now considered.
Deep-water waves. Deep-water waves correspond to conditions described by kd > π (i.e. d/L > 1/2). Large argument approximations to the hyperbolic functions then lead to simplified expressions in which the depth d is absent as expected. The subscript “o” is used to denote deep-water wave conditions. The dispersion relation may be simplified to the following alternatives:
Table 2.2 Results of linear wave theory – deep water.
Variable
Equation
Free surface elevation
Velocity potential
Dispersion relation
Horizontal displacement
Vertical displacement
Horizontal velocity
Vertical velocity
Horizontal acceleration
Vertical acceleration
Pressure
Thus, useful conversion formulae for Lo in terms of T include Lo (m) = 1.56 T2 and Lo (ft) = 5.12T2. Based on the above approximations, the various results of linear wave theory for deep-water waves are given in Table 2.2.
Shallow-water waves. Shallow-water waves correspond to conditions described by kd < π/10 (i.e. d/L < 1/20). Small argument approximations to the hyperbolic functions can be introduced and the dispersion relation may be simplified to:
However, beyond this simplification for c, it is convenient to continue to rely on the complete expressions for the various parameters given in Table 2.1.
2.3.4 Description of Results
The flow corresponding to Table 2.1 is now described in terms of the water particle orbits and the variations of the amplitudes of u and w with elevation. Thus, Figure 2.3 provides sketches of these for shallow-water waves, intermediate-depth waves, and deep-water waves.
Figure 2.3 Water particle orbits and velocity amplitude profiles for various relative depths.
The figure indicates how the water particles have elliptic orbits, with diameters that decay with reducing z or s. These range from flatter elliptic orbits for shallow-water waves, which are influenced by the seabed and which decay only gradually with depth, to circular orbits for deep-water waves, which are not influenced by the seabed and which decay with depth so as to vanish at close to half a wave length below the water surface.
2.3.5 Linear Dispersion Relation
The linear dispersion relation relates L and T (or ω and k) as indicated in Table 2.1. If L (or k) is known, then T (or ω) may be obtained directly from:
However, if T (or ω) is known, which is more often the case, then L (or k) needs to be obtained by an iterative or other approximate method. This is a fundamental requirement in most analyses of wave conditions. Three methods that may be adopted are outlined below.
Iterative method. One approach is to recast the dispersion relation into the form:
Table 2.3 Corresponding values of kd and d/gT2 for kd estimation.
d
/
gT
2
kd
d
/
gT
2
kd
d
/
gT
2
kd
Shallow-water limit:
0.018
0.957
0.060
2.407
0.020
1.024
0.065
2.595
0.001
0.200
0.022
1.090
0.070
2.785
0.002
0.285
0.024
1.156
0.075
2.976
0.003
0.351
0.026
1.222
0.080
3.169
0.004
0.408
0.028
1.288
0.085
3.364
0.006
0.507
0.030
1.354
0.090
3.559
0.008
0.593
0.035
1.520
0.095
3.755
0.010
0.673
0.040
1.690
0.100
3.951
0.012
0.748
0.045
1.864
0.105
4.147
0.014
0.819
0.050
2.042
Deep-water limit:
0.016
0.889
0.055
2.223
kd
= 4π
2
(
d
/
gT
2
)
This can readily be solved iteratively for kd using:
with X = kd, A = 4π2(d/gT2), and Xo taken as A or .
Look-up table. A simple approach is to rely on interpolation using Table 2.3, which shows corresponding kd and d/gT2 values. Exact formulae for kd for the shallow-water and deep-water cases are available and are included in the table.
Regression fit. Thirdly, a regression fit to various ranges of the exact equation has provided the following equation for kd in terms of d/gT2:
where .
Reference Solution A1 in Appendix A provides a spreadsheet solution to the linear dispersion relationship based on the iterative and regression fit methods. (The look-up table method is the simplest one to use without a spreadsheet.) This spreadsheet solution also includes the case of a coexisting current, to be considered in Section 2.5.
Example 2.1 Application of Linear Wave Theory
A wave train has a wave height H = 1.6 m and wave period T = 3.7 s at a location where the still water depth d = 7 m. On the basis of linear wave theory, calculate the wave length, the wave speed, the maximum horizontal velocity at mid-depth, and the orbital diameter of water particle motions at the seabed.
Solution
Specified parameters
g
= 9.80665 m/s
2
H
= 1.6 m
T
= 3.7 s
d
= 7.0 m
Solve for kd (three methods shown)
d
/
gT
2
= 0.0521
Method 1 – iteration method:
kd
= 2.1188
Method 2 – look-up table:
kd
= 2.1190
Method 3 – regression fit:
kd
= 2.1193
Wave length and speed
k
=
kd
/
d
= 0.3027/m
L
= 2π/
k
= 20.8 m
c
=
L
/
T
= 5.6 m/s
Maximum horizontal velocity at mid-depth
A formula for the required maximum velocity um may be developed from the formula for the horizontal velocity u given in Table 2.1 by taking cos(kx − ωt) = 1 for the amplitude or maximum value and s = d/2 for the mid-depth value. Thus:
um = = 0.5 m/s
Orbital diameter of water particle motions at the seabed
A formula for the required diameter do may be developed from the formula for the horizontal displacement ξ given in Table 2.1 by taking cos(kx − ωt) = 1 for the amplitude or maximum value and s = 0 for the value at the seabed, and then doubling this to convert from amplitude to diameter. Thus:
2.4 Wave Energy and Momentum
A description of the energy
