Investigation of the Performance

of Submerged Rubblemound

Breakwaters

by

Stuart R Seabrook

A thesis submitted to the Department of Civil Engineering

in confomity with the requirements for

a degree of Master of Science (Engineering)

Queen's University

Kingston, Ontario, Canada

August, 1997

Copyright O Stuart R. Seabrook, 1997

National Library 1+1 .,,da Bibliothèque nationale du Canada

Acquisitions and Acquisitions et Bibliographic Services services bibliographiques

395 Wellington Street 395. rue Wellington Ottawa ON K I A ON4 Ottawa ON K I A ON4 Canada Canada

rour Ne Voue dlérenw

Our üie None rdldrtMCB

The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, disûibute or sell reproduire, prêter' distribuer ou copies of this thesis in microfom, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/nlm, de

reproduction sur papier ou sur format électronique.

The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thése ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation.

Abstract

Submerged rubblemound breakwaters are being considered more fiequently in coastal

engineering design applications where naturaI and enWonrnentally sensitive solutions to

shoreline protection problems are required. Although a number of numerical and physical

studies on the performance of submerged breakwaters have been completed, there are still

relatively few practical tools for the design engineer. Those available are based on a limited

range of input design variables and therefore are insufficient in some cases.

This thesis presents the results of a number of 2-D and 3-D tests dehing the effect of depth of

submergence, crest width, breakwater slope and incident wave characteristics on the wave

transmission coefficient. A rubblemound breakwater section consisting of a core and 2 primary

amour layers was used in al1 tests, with a wide range of irregular and regular wave conditions.

The test resulrs indicate that the relative submergence and crest width are the most important

factors influencing the transmission process.

Alternative mathematical rnodels for the transmission coefficient were developed on the basis of

dimensional analysis considerations and graphical inspection of the 2-D test results. The

alternative models were subsequently evaluated on the basis of statistical measures and practical

design considerations. The proposed model predicts the 2-D test data well with an Et2 of 0.9 14

and removes the majority of the residual trends. Although prediction of the 3-D test data is not

as good, the model is expected to provide sufficient accuracy for preliminary design

requirements. Given the complex nature of the processes at submerged breakwaters, physical

model tests are recomrnended for confirmation of final design parameters.

The 3-D test results provided additional transmission data for consideration with respect to the

evaluation of the alternative mathematical models, as well as qualitative data reIating to

secondary hydrodynarnic processes (such as difiction, velocity patterns and beach

development) at submerged rubblemound breakwaters. Observations were made with respect to

velocity pattems and beach development behind the breakwater.

Acknowledgements

The preparation of this thesis and the associated experimentation have required the input of a

nurnber of individuals along the way. The author would like to take the opportunity to express

his gratitude for their assistance.

Dr. KeWi Hall has provided both fmancial and professional support throughout ths study. He

has served as a source of motivation and inspiration towards the topic and the profession as a

whole. His time and his efforts are greatly appreciated.

A note of appreciation is also due to Dave Tryon and Neil Porter of the Department of Civil

Engineering for their techical assistance and input at the coastal {ab, where the time required to

rnaster the apparatus can dampen the spirit of the keenest researcher.

As a fellow student and aspiring engineer, Jason Crowder paid his dues through back breaking

labour and monotonous data sampling in order that the 3-D testing could be completed on

schedule. His diligence is greatly appreciated.

Finally, Nadia Zurba, who provided the financial support such that the author could afford this

time away f?om the workplace. must be aclaiowledged. Her patience and persona1 support will

not be forgotten.

Table of Contents

Abstract

Acknowledgements

Table of Contents

List of Tables

List of Figures

Notation

Introduction

Wave Phenomena at Snbmerged Breakwaters

2.1 Physical Processes

2.2 Previous Studies 2.2.1 Physical Process Modelling 2.2.2 Numerical ModeIling 2.2.3 Physical Design Modelling

Research Objectives

Experimental Setup

4.1 Flurne (243) Testing 4.1.1 Physical Conditions 4.1.2 Model Scahg 4.1 -3 Breakwater Consûution 4.1.4 Wave Conditions 4.1.5 Instrumentation 4.1.6 Data Sampling and Analysis

4.2 Basin (343) Testing 4.2.1 Physical Conditions 4.2.2 Model Scaling 4.2.3 Breakwater Construction 4.2.4 Wave Conditions 4.2.5 Instrumentation 4.2.6 Data Sampling and Analysis

vii

5.0 Analysis of Wave Data

5.1 Overview

5.2 Zero Crosskg Analysis

5.3 Variance Spectral Density Anaiysis

5.4 Reflection Analysis

6.0 Discussion of Resuits

6.1 Flume Testing: Observations and Trmds 6.1.1 General 6.1-2 Tirne Series Transformations 6.1 -3 Spectral Transformations 6.1-4 Transmission 6.1 -5 Reflection 6.1.6 Cornparison with Existing Design Equations

6.2 Basin Testing: Observations and Trends 6.2.1 General 6.2.2 Time Series Transformations 6.2.3 Spectral Transformations 6.2.4 Transmission 6.2.5 Diffiction 6.2.6 Beach Development 6.2.7 Velocity Patterns

7.0 Analysis of Resolts

7.1 Transmission as a Function of Dimensionless Variables 7.1.1 Basic Dimensionless Variables 7.1.2 Effect of Perturbation due to Breakwater Obstruction 7.1.3 Overtopping Effect 7.1.4 Harmonic Generation 7.1.5 Frictional Effects 7.1.6 interna1 Flow m g ) Effects

7.2 Statistical Development of Design Equations 7.2.1 Analysis Techniques 7.2.2 Development of Alternative Design Equations 7.2.3 Evaluation of Alternative Design Equations

8.0 Conclusions and Recommendations

8.1 Conclusions

8.2 Recommendations

List of References

~ppendix A: Hypothetical Input Design Conditions for Reliminary Evafuation of Existing Models

~ppendix B: Test Breakwater Configurations and Stability Calculations

Appendix C: Wave Generation Routines and Characteristics

~ppendix D: Velocity Robe Calibration

~ppendix E: Selected 2-D Transmission Results

Appendix F: Selected 2-D Reflection Results

~ppendix G: 3-D Testing Transmission Results

~ppendix H: Scatterplot Comelation of Independent Variables

Appendix 1: Alternative Mathematical Models

Vita

List of Tables

Table Titie

4.1 2-D Testing Structure Configurations

4.2 2-D Testing - lrregular Wave Characteristics

4.3 3-D Testing Structure Configurations

4.4 3-D Testing - hegular Wave Characteristics

Page

45

46

56

57

List of Figures

Figure Title

Wave-Structure Interactions at Submerged Breakwaters Transmission at Permeable Submerged Breakwaters Harmonic Generation at Subrnerged Breakwaters Spectral Transformations at a Subrnerged Breakwater Distribution of Bound Hamonic Amplitude Stability Curves for Submerged Rubblemound Breakwaters Typical Design Curves - Seelig's Equation Transmission Coefficients at Submerged Breakwaters Ahrens' Reef Breakwater Transmission Curves Ahrens' Reef Reflection Coefficient van der Meer's Transmission Coefficient

2-0 Testing Configuration Water Level (Wave) Probe 3-D Testing at 90° 3-D Testing at 60" Typical BreWater Section Velocity Probe Effect of One Second Filter

Initial Conditions Testing Setup Reflection Analysis - Hume Testing Configuration : No Breakwater Time Series Record: &=û. 10 m; Tp=2.0 s; W1.5 rn; &=O. 10 m Regular Wave Senes: Hi=O. 10 m; T=1.5 s: B4.5 rn; ds=û.10 rn Effect of Penod (Tp) and Breaking Influence (djHJ on Spectral Transformations Effect of Crest Width (B) and Breaking Influence ( d P 3 on Spectral Transformations Effect of d, and H, on K, Effect of d, and Tp on K, Effect of B and Tp on Effectof OandTponK, Effect of D, and T, on Y Effect of d, and H, on K, Effect of d, and Tp on Effect of B and Tp on Effect of 0 and Tp on Y Effect of D, and Tp on Y Ahrens' Equation with Present Test Data van der Meer's Equation with Present Test Data Time Series - Basin T e ~ t H ~ 4 . 0 6 3 m; Tp=1.6 s; h4.28 m; P=90° 3-D Testing - Spectral Plots for 90" Incidence K, for 3-D Tests at 90" Incidence K., for 3-D Tests at 60' Incidence Final Beach Formation - 3-D Tests at 90" Incidence

Page

viii

Figure Title

Final Beach Formation - 3-D Tests at 60" Incidence Mean Velocity Vectors at O.2h:All Wave Conditions at P=90 O; h 4 . 19 m Mean Velocity Vectors at 0.2h:All Wave Conditions at P=90°; h4 .28 rn Mean Velocity Vectors at O.2h:M Wave Conditions at H O O; h a . 19 m Mean Velocity Vectors at 0.2h:AU Wave Conditions at 8 4 0 " ; h a 2 8 m

Effect of Relative Submergence Effect of Wave Reynolds No. Effect of Dimensionless Amour Size Effect of Dimensionless Wavelength Effect of Breakwater Slope Effect of Dimensionless Crest Width Effect of Breakwater Height Effect of Water Depth Perturbation Effect on K, Effect of Breaker Location Effect of Dimensionless Beat Length Effect of Friction Tenn Effect of Interna1 Flow K, vs. Estimate - Eqn. 7.12 Residuals vs. K, - Eqn. 7.12 Residuals vs. djHi - Eqn. 7.12 K, vs. Estimate - Eqn. 7.13 Residuals vs. d / Y - Eqn. 7.13 Normal Probability of Residuals - Eqn. 7.13

vs. Estimate - Eqn. 7.14 Residuals vs. d/Hi - Eqn. 7.14 Normal Probability of Residuals - Eqn. 7.14 Eqn. [7.13] Sensitivity to Parameter Estimates Eqn. 17.141 Sensitivity to Parameter Estimates Sensitiviiy to Input Variables (B = 0.3 m) Sensitivity to Input Variables (B = 3.5 m) Mode1 Cornparison with 3-D Test Results

Page

89 91 91 92 92

97 100 101 102 103 1 O3 104 105 107 1 O9 111 112 114 120 121 121 122 122 122 123 123 123 125 125 127 127 129

Notation

Variable Defmition Units

equivalent cross sectional area of breakwater

mest width of breakwater

dep th of submergence

nominal amour rnaterial diameter

nominal core matenal diameter

incident wave energy

transmitted wave energy

fiction factor

peak kequency of wave spechurn

dirnensionless fieeboard

Froude number

gravitational acceIeration

local water depth at breakwater

height of structure fkorn seabed

significant wave height fiom zero crossing analysis

incident wave height (assumed same as and bi)

fictional head Ioss

zero moment wave height fiom spectral analysis

wave number

dissipation coeficient

reflection coefficient

Variable Definition

transmission coefficient

deepwater wavelength

wavelength in local depth

wavelength computed using lenear wave theory

porosity of breakwater matmal

stability number for rubblemound breakwater

spectral stability number for rubblemound breakwater

Reynold's number

rubblemound breakwater damage level indicator

peak period of wave specm

velocity

Weber number

breaker distance along breakwater crest

n' dimensionkss variable of a given phenomenon

spectral wave parameter (Phillip's constant)

angle of wave incidence

speciral wave parameter (peak enhancement factor)

surface rougnhness size

watw surface displacernent

angle of breakwater face

scale of modelled phenomenon

beat length of specific wave fiequency

dynamic viscosity

(ml

( P a 4

Variable Defrnition

characteristic breaker location parameter

kinematic viscosity

surf similariîy parameter

modified surf similarity parameter

pi constant

density of air

density of Stone

density of water

suface tension

dimensionless function of modelled phenornenon

Units

xii

1.0 Introduction

The coastal zone is a delicate yet volatile and dynamic area. It is the area in which the majority

of a water body's kinetic energy is dissipated through wave breakuig, runup and bed friction.

The most significant result of these processes is the erosion and subsequent transport of the shore

and beach matenals. This littoral zone is very important to the public for economic and social

reasons and to wildlife for habitat and food supply purposes. It is therefore very important that

the coastal zone be protected and maintained so that these considerations are addressed in a

compatible and effective fashion.

Engineered shoreline protection measures have taken many forms in the past; some have met

with certain degrees of success while others have been obvious failures. Although the most

evident failures are those related to stnictural integrïty, there are many more subtle failures. the

effects of which are often not discovered for a number of years. These subtle failures are ofien

directly reIated to the disniption or destruction of natural habitats and processes. Therefore. the

focus of shoreline protection design has been expanding to include features which protect against

and compensate for any loss of natural habitat areas.

Conventional sub-aerial (surface piercing) breakwaters are typically designed to totally eliminate

wave energy in their lee for a particular design wave condition. Often, this design wave

condition is relativeiy severe and as a result, the structure must be very large. Where a minor

degree of overtopping is permissible, the physical requirements of the breakwater are only

slightly reduced. As such, these structures must be massive to withstand the impact forces of

breaking waves, and often they al1 but eliminate water circulation in the protected area.

Although this calm condition developed in the lee of the breakwater protects the shoreline and

can provide for good mooring conditions, under certain circurnstances, the hydrodynamics

developed at sub-aerial breakwaters and the associated shoreline response can degrade water

quality and natural habitat W.S. Amy Corps, 1984).

in recent years, increased artention has been given to a class of breakwaters collectively termed

submerged breakwaters. As indicated by the name, these structures are constructed below a

specified design water level. In cornparison to sub-aerial breakwaters, submerged structures

permit the passage of some wave energy and in turn allow for circuIation along the shoreline

zone at the cost of a reduced level of protection. In addition to providing environmental benefits.

these stmctures have also found applications as preliminary defense rneasures in extreme wave

climates where they reduce the wave forces on the primary defense structures (Cornett et al.,

1994). This approach can be particularly beneficial in areas where a primary defense structure

has sustained some damage or in areas where there is a projected long-term relative increase in

sea Ievels and wave heights.

The class of submerged breakwaters includes nurnerous specific types including active

submerged, flexible membrane, fixed wave baniers, reef breakwaters or conventional

rubblemound submerged breakwaters. Although there has been a considerable amount of

research performed on these various structures, the physical processes are very complex and

there are still several outstanding questions, especially with respect to practical design guidelines

and procedures.

The focus of this thesis is on the practical application of conventionai rubblemound submerged

breakwaters (hereafter referred to simply as submerged breakwaters). More specifically, design

guidelines for determinhg breakwater geometry will be suggested such that a desired level of

wave transmission can be achieved.

The successful application of any design guidelines requires a basic understanding of the theory

and assurnptions employed in their developrnent. To this end, the following research objectives

have been defined:

1. To review the physical processes takmg place at submerged breakwaters as defined by

previous investigations and by physical observations of tests carried out in this study.

2. To perform 2-dimensional (2-D) hydraulic model tests for a wide range of breakwater

geometries and compare the results with existing design equations to assess their effectiveness in

estimating wave transmission.

3. To modie or redefine the existing design equations as necessary to provide a simple design

tool which is based on the physical processes occurring at submerged breakwaters and therefore

applicable over a wider range of design conditions.

4. To perform a limited number of 3-dimensionaI (343) hydraulic model tests to assess the

effectiveness of the proposed design equations in a 3-D setting.

5. To make qualitative observations of beach development, difiction and velocity patterns in

the lee of submerged breakwaters.

These research objectives were firlfilled through a thorough review of literature reIating to

submerged breakwater testing and related physical processes, followed by a comprehensive 2-D

and 3-D physical mode1 testing program at the Queen's University Coastal Engineering Research

Laboratory (QUCERL).

A variety of structure geometries with varying depths of submergence, crest widths, dopes and

amour Stone diameters were tested under a range of regular and irregular wave conditions in a

I .O m wide flume at QUCERL. Transmission coefficients were measured for ail tests and

subjected to graphical and statistical anaiysis to develop a design equation (mathematical model)

for wave transmission, based on parameters representing the physical processes occuring at

submerged breakwaters.

A set of 3-D tests was also camied out in the three-dimensional wave basin at QUCERL.

Transmission coefficients were measured for al1 tests. Qualitative observations were made

regarding other hydrodynamic processes behind submerged breakwaters on the basis of velocity

measurements and beach developments. Measured transmission coefficients were used to assess

the effectiveness of the equations developed fkom the 2-D test results. Modifications to the

equations were considered in Iight of this comparison.

This thesis is generally presented in the order of experimentation. A summary of the literature

review is presented, followed by a discussion of the experimental setup and data sampling and

analysis techniques. Results are presented graphically and statistically, leading to the

development of an improved design equation. Cornparison of the design equation predictions

with 3-D test results provides the basis for modifications, if necessary, ultimately leading to

concIusions and recornmendations of this thesis.

2.0 Wave Phenomena at Submerged Breakwaters

2.1 Physical Processes

The physical processes involved in the wave-structure interaction at submerged breakwaters are

very complex. The complexities arise fiom two distinct sources: the extreme variability of the

local wave climate and the numerous physical processes associated with the interaction of any

specific wave climate with the unique characteristics of a given submerged breakwater

configuration. It is this interaction which dictates the degree and rate of wave transformation at

a submerged structure and thereby defmes the resulting wave transmission, reflection and energy

dissipation. The energy dissipation at the structure, in tum. defines the forces which will

influence the stability of the breakwater.

Basic physical processes at a submerged breakwater can generally be separated into three

regions. These regions are shown in Figure 2.1 for a typical submerged breakwater

configuration with the notation for physical variables used throughout this report. The three

regions are described briefly in the following paragraphs.

Region 1 is located over the seaward slope of the structure. In this region the incident wave field

is complex and irregular in nature, varying spatially and temporally at any gwen location.

incident waves may exhibit varying degrees of non Iinearities due to the complex forces which

have generated ttiem and they may be transmitted on a water surface which fluctuates with a

much longer period due to tidal or surge effects. Shoaling and rehction by the IocaI

bathymetric conditions offshore of the breakwater increase the non-linearity of the wave field.

Rtgion 1 Rtgion 2 .- Region 3 %

Figure 2.1 Wave-Structure Interactions at Submerged Breakwaters

in Region 1, the incident wave encounters the rising face of the submerged breakwater and

begins to shoal. As a result of this shoaling, bound harmonic waves are generated (Beji and

Batqes, 1993 j. Some of the incident wave energy is reflected and this reflection has been found

to be strongly dependent on the depth of submergence (Ahrens, 1987; van der Meer, 1991) and

somewhat dcpendent on the slope of the structure mattatri et al., 1979).

Wave breakmg characteristics at bars and artificial reefs are largely influenced by the slope of

the forward face of the structure and the depth of submergence as well as the offshore velocity

across the breakwater crest (Smith and Kraus, 1990). Other phenornena found to occur in this

region in laboratory tests include the build-up of low-fiequency wave heights, thought to be the

result of interaction between incident bond long waves, and an offshore curent developed by a

set-up of water leveI behind the breakwater (Petti and Ruol, 1991 and 1993; Liberatore and Petti,

1993).

Region 2 is located on the crest of the breakwater. In this region, energy fiom the fundamental

wave frequency is transferred to higher harmonic fkequencies (Driscoll, Dairymple and Grilli,

1993). Beji and Baqes (1993) desmie this phenornenon (for long waves propagating over a

bar) as a rapid flow of energy fiom the primary wave to higher harmonics, generating

"dispersive wave tails" which appear to travel with nearly the sarne celerity as the prïmary

waves. For the case of submerged breakwatm, where the incident wave transformations will be

more abrupt, the process may be affected to some degree by the influence of reflected waves and

wave b reahg ont0 the crest of the structure.

As the waveform passes over the breakwater mes& energy is also dissipated through fictionai

resistance. Because the breakwater is penneable, larninar and turbulent flows within the

structure also result in some energy dissipation.

Region 3, located beyond the shoreward limits of the breakwater crest is a transition to deeper

water where the higher frequency wave components developed in Regions 1 and 2 separate fiom

the fwidamental components and mvel with their own celerity (Beji and Battjes, 1993). This

process generally results in a broad energy spectnun in the lee of the breakwater with a

decreased characteristic wave height (Hm,) and reduced peak wave period (Tp). Again, this

process has been observed in investigations involving spectral analysis of wave transformations

at submerged breakwaters (Petti and Ruol, 1993; Cornett, Mansard and Funke, 1994).

Studies of velocity fields at submerged breakwaters show the development of eddies, indicating

flow separation and subsequent energy losses immediately shoreward of the breakwater crest

(Ting and k m , 1994). Other general phenornena that may occur in the vicinity of submerged

breakwaters include a local setup of water levels in the lee of the breakwater due to a net mass

flux of water over the breakwater crest, and wave current interactions as an offshore curent

(return flow) is developed in the surface region over the breakwater (Fetti and Ruol, 199 1 ) or in

gaps between segrnented breakwater structures (Fulford 1985).

It is therefore evident that there are nurnerous factors associated with the incident wave field. the

breakwater geometry and the local bathymetry which c m effect the wave transformation and the

resulting energy transmission, reflection and dissipation at a submerged breakwater. Frevious

physical and numerical investigations have been perfonned in an effort to concisely defme ihe

effects of these various factors. A general review of these investigations and their findings is

provided in the following sections.

2.2 Previous Studies

Due to the complexity of the wave-structure interaction at submerged breakwaters and the

number of variables which affect the physical processes involved, it is not yet possible to fully

describe these processes with currently available theory. Physical modelling is therefore an

important tool in studying submerged breakwaters and the associated wave nansformations.

Numerous physical modelling studies of submerged breakwaters (or obstacles) have been

performed in recent years. Durùig this period, there have been a number of technological

advancements in the field of physical modelling and data sarnpling and analysis which have

improved the quality of the results and the development of the associated theories. The most

significant developments which have served to improve the state of the art include advances in

wave generation techniques to include irregular and 3-D wave generation potential, adsorption of

reflected waves by wave generators and the use of numerical processing capabilities to utilize

more complex numerical methods, such as Fourier analysis techniques.

Early physical models (Dick and Brebner, 1969; Dattatri et al.. 1979) focused on the observation

of general transformations of regular waves at submerged structures. More recent investigations

have considered spectral transformations of irregular waves and have focused more closely on

the specific processes occuming at the submerged structure (Petti and RuoI, 199 1 and 1993 :

Lîberatore and Petti, 1993; Driscoll, Dalryrnple and Grilli, 1993). Numerical models have also

been developed in an effort to mode1 the transformation of an incident wave train over a

submerged breakwater or bar (Yasuda and Kara, 1991 ; Ohyama and Nadoaka, 1993; Cruz. Isobe

and Watanabe. 1993).

Although these studies have irnproved the understanding of the specific interactions between an

incident wave train and a submerged breakwater, they still rely on a number of assumptions and

simplifications and they do not provide a simple approach for design applications. Given the

uncertainty in the underlying assumptions and input data required to accurately predict the wave

transformations and physical interactions using a relatively complex numerical mode], it is often

more desirable to use a less complex empirically developed technique to design a submerged

breakwater.

Therefore, there has been some development in this area regarding the prediction of breakwater

stability and îransmission and reflection characteristics as a function of breakwater design and

incident wave characteristics (Goda et al., 1967; Goda, 1969; Seelig, 1980; van der Meer, 199 1;

Comett, Mansard and Funke, 1994). These equations provide a relatively simple rnethod of

determinhg design parameters from avaiIable input parametem. However, some of the

equations are based on tests involving a limited range of input variables and do not give due

consideration to the processes which have been identified by other investigations.

A brief summay of previous studies and their significant findirigs is provided in the following

sections. The review is organized according to the focus of the investigations.

2.2.1 Physical Process Modeilhg

The scope of physical process modelling is more specific in nature than physical design

modelling in that the specific principles underlying wave transmission are sought rather than

general numerical relationships. Incident, reflected and transmitted wave trains, as well as wave

breaking characteristics, have been observed and docwnented by a number of authors as

described below. Recendy, much attention has been given to the cornparison of incident wave

spectra, refiected wave spectra and transrnitted wave spectra and the variations in these spectral

diagrarns with changes in breakwater configurations.

A s previously noted, the fust studies of subrnerged breakwaters considered, in a general manner,

the physical processes that occurred when a rnonoch.romatic incident wave train encountered a

submerged breakwater.

Dick and Brebner (1 969) tested submerged rectangular breakwaters of various crest widths. A

permeable breakwater was represented by a block of nested tubes. The authors observed energy

losses at the submerged breakwater due to wave breakmg and turbulence, as well as a transfer of

energy to fiequemies higher than that of the incident wave. Plots of the transmission and

reflection coefficients reveal an undulating pattern with a maximum reflection coefficient

occurring when the incident wave has the same period as a standing wave on the crest of the

breakwater. A definite minimum transmission coefficient was observed for the permeable

structure while the solid structure did not exhiiit such a characteristic.

Dattatri, Raman and Shankar (1979) perfomied a number of tests with various shapes of

submerged breakwaters under regdar wave attack. Solid and permeable nrbble breakwaters

were tested and the influence of a number of physical breakwater pararneters and incident wave

characteristics on the transmission coefficient were evaluated. The investigation revealed that

the most important parameters a f f e c ~ g wave transmission are the crest width and the depth of

submergence as shown in Figure 2.2.

1 Relative crest width, ( B I 1 1

Figure 2.2 Transmission at Permeable Submerged Breakwaters

Source (Damtri et al, 1978)

The shape of the breakwater was found to affect reflection considerably but had little effect on

transmission. Porosity of the breakwater was not f o n d to be an important factor for either

process.

More recent studies have taken advantage of the ability to genwate irregular waves and perfonn

spectral analysis on the incident and ûansmitted wave series. One of the most obvious and

significant observations of al1 physical process models is the transfer of energy fkom the

fundamental wave fi-equency to higher harmonic frequencies. This phmomenon has been

observed and anaIyzed from a spectral perspective by numerous authors (Petti and Ruol, 199 1 ;

Liberatore and Petti, 1993; Dr~scoll, Dalrymple and Grilli, 1993).

Peni and Ru01 (1991 and 1993) considered short (c 0.5 fp : where f, is the wave fiequency

associated with the peak energy of the specûum) and long (> 0.5 $) wave transformations at

submerged breakwaters. The short waves were not significantly affected by the breakwater and

the rneasured wave characteristics satisfied an assumed conhuity type relation:

This relation is based on the assumption that incident wave energy (Ei) is transmitted (E,),

reflected (EJor dissipated (E,) (ie. E, = Er + E, +E,). Since the wave energy is proportional to

the square of the wave height (E a Hz), the following generalization c m be made.

E , Er =, + + = 1 Et Et El

H: - 4 where K: = - - - (similady for Kr ; Kd) H: El

The Iong wave components did not satism this relationshp and it was found that there was a

significant buildup of low fiequency bound Iong waves on the offshore side of the structure.

Zero crossing analysis of the Iong wave record showed the significant wave height to increase to

approximately Nvice the incident wave height immediately in fiont of the breakwater, decreasing

sharply behind the structure. Analysis showed K, to be between 0.35 and 0.45 with Kr values of

about I .O. It was concluded that the bound long waves were interacting with an offshore velocity

which in effect was preventing them fiom passing. The offshore velocity was generated by a

water level setup behind the breakwater and the long waves found behind the structure were

presumed to be a result of regeneration by the short waves passing the breakwater or seiche

between the structure and the spending beach.

These results show h t it is important to consider the testing setup when interpreting test results

and conclusions. Although the spectral degradation at submerged breakwaters can be expected

in a prototype situation, the phenornena associated with the offshore current and seiche behind

the breakwater may not occur in a 3-D setting where energy may be msferred m the longshore

direction. These processes mut be considered when interpreting test results and should be

considered on a site specific basis for design applications.

Driscoll, Dalrymple and Grilli (1993) performed tests on submerged rectangular impermeable

breakwaters with regular monochrornatic incident waves and through extensive instrumentation

plotted the spatial amplitude variation of the fundamental fiequency as well as the 2nd, 3rd and

4th harmonies. The results indicate that, over the structure, there is a bnef increase in the

amplitude of the fundamental wave frequency and then a decrease. The hmonic amplitudes, on

the other hand, grow over the width of the breakwater crest and continue to modulate in the lee

of the breakwater. The results of these tests, as shown in Figure 2.3, depict the harmonic

generation process discussed below. The hdamental wave shoals on the fkont face of the

breakwater, increasing in amplitude, but when on the crest, its energy is transferred to hmonic

fiequency components.

Beji and BattJes (1993) conducted laboratory experirnents with breaking and non-breaking waves

over a submerged bar in an effort to determine the role of wave breaking in the process of high

frequency energy generation. hegular waves were generated such that a short-wave signal

($ = 1.0 Hz) and a long-wave signal ($ = 0.4 Hz) were tested. For each of the characteristic

frequencies, significant wave heights were generated such that non-breakmg waves, spilling

waves and plunging waves were produced. The results indicate that there is very little spectral

transformation in the case of the short waves. The long wave spectra, however, reveal

considerable transformation as energy is nansfemed to higher fiequencies. Results from these

tests are shown in Figure 2.4 below.

rn f r o m bock of obstocle

Spa~ ia l evolution of the first four liarinonics iu the exptrimentd data; ( * ) = l s ~ tiariiionic, ( )=2iid harmonic, ( a )=3rd harrnonic, ( 4 )=4tli harnionic. Fine daslied liaes are spiines fit to data points, long dashed lines show bou tidaries of obstacle.

Figure 2 3 Harmonic Generation at Submergeci Breakwaters

(Source: Driscoll et al, 1993)

Considering two conditions (non-breaking and plunging breakers) there seems to be little

difference in the shapes of the specü-a. The breaking process, instead, appears to simply reduce

the overall spectral intensiv. The authors consider these results to support the development of a

numencal model which combines a weakly non-linear, non-dissipative model, such as the

Boussinesq model, to account for the high frequency generation process, with a semi-empirical

dissipation relation to account for the breaking processes.

It is important to consider the processes discussed in the above paragraphs when developing a

design relationship for submerged breakwater transmission. Given the wide range of variability

in incident wave and breakwater conditions, it is almost impossible to test al1 possibIe

conditions. Therefore, it is not sufficient for a design relationship to simply satisQ the available

test data; it must also physically represent the processes which have been identified in other

Numerical models Vary widely in their approach (and inherent assumptions) to the problem of

predicting wave transmission at submerged breakwaters. Some models are limited to

transmission of regular waves or solitary waves over gently sloping bottom topographies while

others are developed to model the transformations of fui1 irregular wave trains over rapidly

varying p e ~ o u s structures. Perhaps the most common shortcoming of existing numerical

modelling techniques is the inability to combine the breakmg wave process with the non-linear

dispersive spectral transformations occurring in non-breaking conditions.

A discussion of numerical modelling of wave transformations at submerged breakwaters is best

sumrnarized according to the mathematical approximations utilized. A review of the literature

indicates that the majonty of the approaches fa11 within two main groups: those employing

Boussinesq or Korteweg de Vries (KdV) equations and those employing a fûlly non-linear

potential flow theory approach which is typically solved using the Boundary Element Method

(BEM). Although a few other approaches will be discussed here, these two groups encompass

the majority of numerical modelling works presently being pursued.

A numencal mode1 of reflection and transmission of monochromatic waves at a submerged

impermeable breakwater was developed by Kobayashi and Wurjanto (1989). The model

employed vertically integrated Iinear equations of continuity and momentum, combined with an

energy dissipation equation, to account for wave breakmg and fiction. This model provided

transmission and reflection coefficients consistent with limited physical testing data but was

limited to monochromatic incident waves and impermeable boundaries.

Rojanakamthom et al. (1991) proposed a model based on a mild slope equa!ion for flow over a

porous bed. An analytical solution was derived for monochromatic progressive waves and the

resulting equation was modified to account for wave breaking on a porous structure. The model

was extended to irregular waves through analysis of the transformation of each individual wave

component. The model predicted the root mean square water surface displacernent (q-)

satisfactorily for a set of cornparison physical test data. There is no information, however.

regarding the spatial distriiution of water surface in the area behind the breakwater.

Losada, Silva and Losada (1996) presented a model for directional irregular wave trains incident

to porous submerged breakwaters. This model is based on the extended rnild dope equation for

waves propagating on a porous iayer but no allowance is made to account for wave breaking.

The mode1 compares well with physical test data, except where wave breakmg occurs.

A number of authors have used the fully non-linear potential flow theory in an effort to

successfully model wave transformations at submerged breakwaters. The advantage of this

approach is that the dynamic fkee sinface boundary permit5 development of a fully non-linear

wave field. The model is, therefore, found to provide a reIatively good representation of the

dispersive waves in the lee of the breakwater. The most significant problem with this approach

is that is assumes potential flow and there fore c m o t accurately model those cases where large

breaking waves result in significant flow sepration.

The non-linear potential flow approach was used to model the transformation of a solitary wave

at a subrnerged breakwater by Cooker and Peregrine (1989), Yasuda and Hara (199 l), GnlIi and

Svendson (1991) and Lee, Chang and Zhuang (1993). Hara, Yasuda and Sakikabara (1 993) use

this solitary wave model to detemine breakhg characteristics for a number of solitary wave

forms and submerged breakwater geometries. The results were analyzed using regression

techniques to defme the incident critical wave height, the breakpoint and the breaker height.

Grilli, Losada and Martin (1 994) performed similar analyses and found reasonable results except

where there was considerable flow sepration due to breaking waves. As previously noted, these

models appear to provide relatively good representation of wave transfomuitions up to wave

breaking. Beyond this point, however, the results may not be reliable.

The fblly non-linear potential flow theory was extended to continuous wave trains by a number

of authors to observe the generation and dispersion of harmonic waves at submerged

breakwaters. Ohyama and Nadaoka (1 993) developed a time dependent BEM for potential flow

with non-reflective open boundaq conditions and showed that there is a signifiant transfer of

energy fkom the fundamental wave to harmonic fiequencies. Comparing modelling results for a

subrnerged step and a submerged dyke, the authors predicted that the amount of energy

transferred ùito free wave components will Vary appreciable with the breakwater crest width.

This phenornenon is defmed by the beat length ( A 9 as expressed in Massel, (1983):

where k, and ko are wave numbers for free waves in the 1" harmonic and fundamental fiequency

respectively. The effect of the beat length on the numerical results is shown in Figure 2.5 beIow :

Figure 2.5 (b) shows the case where B/A : = 1 .O. It can be seen that where B/A = (1.2, ...) the

amount of energy transferred to the 1" harmonic V-Y, = 2) is minimized. When B/A ', = ( i/2, 3/2,

...) the energy transferred is minimized. The crest width relative to the beat length (B/A ',) is

therefore suggested to be a defmitive parameter (Mei and Ünlûata, 1972).

(a) Stepped Bottom

(b) Rectangular Dike

Figure 2.5 Distribution of Boxtnd Earmonic Ampiitude

(Source: Ohyarna and Nadoaka, 1993)

Driscoll, Dalrymple and Gnlli (1993) compared experïmental results to numerical results

generated using a linear scattering model and the extended BEM model developed by Gnlli and

Svendsen ( 2 99 1 ). The resuits indicated that the Iinear scattering model predicted the reflection

coefficient relatively well but tended to over predict the transmission coefficient due to the

omission of energy transfer to higher harmonies. The BEM model simulated the spatial

amplitude modulations in the breakwater shadow relatively well except where there was

significant flow separation in the physical tests.

C m , Isobe and Watanabe (1993) derived non-linear second order Boussinesq-type equations

which accounted for wave breaking through the use of a dissipation factor. The numerical

results were f o n d to be comparable to physical test results in the area approaching the

breakwater and on the breakwater crest. The dispersion of the transmitted wave in deeper water

in the shadow of the breakwater is not well represented. The breakrng wave profiles were

predicted well prior to the area of wave disassociation.

Boussinesq equations with improved dispersion char;tcteristics were developed by Beji and

Battjes (1 994) and compared to physical test data for non-breaking waves. The results compared

favorably with the physical test data, even in the deeper water in the lee of the breakwater where

harmonic decomposition occurs. Consideration of physical test data for breaking and non-

breaking waves indicated that the breaking process simply scales the transmitted energy spectra

and has Iittle effect on the dispersion process. Therefore, the authors predict that the model

noted here couid be fitted with a semi-empirical model to simulate the energy dissipation

associated with wave breaking.

As indicated in the preceding discussion, there are numerous numerical methods available to the

engineer but there are limitations associated with each one. It is also important to consider that

limited physical test data has been used to veriQ the various numerical models and it is therefore

very difficult to define the true capabilities of a particular numerical method. Although full

numerical modelling of wave transformation at submerged breakwaters has not been possible

until recently due to computational requirements, recent improvements in computational speed

and memory availability with a reduction in costs have made numerical modelling much more

feasible for the design engineer. Complex computationaI techniques and detailed graphical

presentation of wave transformations are possible with relatively inexpensive computing

facilities. It is important, however, that the results of the mode1 are not perceived to be more

accurate and dependable than the input data and assumptions.

Given the uncertainties associated with the present state-of-the-art in numerical modelling

techniques, it may be most appropriate for the design engineer to consider more general design

equations developed fiom physical test data. A number of design equations, based on the

physical modelling work of a number of authors, are avaitable. These equations are discussed

briefly in the following section.

2.23 Physical Design Modeiiîng

Physical design modelling provides a relatively simple approach to defining performance

characteristics of a subrnerged breakwater. Typically, performance characteristics of interest

include the transmission coefficient (K,), the reflection coefficient (K,) and a stability critenon

such as the stability number (N,). These characteristics are generally described as a h c t i o n of

readily available incident wave and structure characteristics.

Although the focus of this thesis is related to the transmission and reflection charactenstics of

submerged breakwaters, the mode1 breakwaters were designed to remaùi statically stable during

testing. Therefore, it is important to provide a brief discussion of stability equations available to

the design engmeer.

(a) Breakwater Stability

The most comprehensive relationships developed specifically for stability of submerged

rubbIemound breakwaters have been developed by van der Meer (199 1) and by Vidal et al.

(1993). As previously indicated, the customary approach to defining breakwater stability is

through the use of a stability number which relates the ability of the structure to withstand a

certain amount of damage to the incident wave conditions and armour charactenstics.

Van der Meer's (1 99 1) analysis of the stability of statically stable (conventional nibblemound)

submerged breakwaters defined the armour stability in terms of a spectral stability number (N,)

on the basis of Ahrens' work (1 987). The spectral stability number d i f f a fiom the conventional

sub-aerial breakwater stability number (N,) in that it includes the effect of incident wave period.

Using limited test data from van der Meer (1988) and Gilver and Smensen (1986), a design

equation for stability of submerged rubblemound breakwatm was derived as follows:

hl - = (2.1 + 0.19 cxp (-0.14~:) h

where h, and h are as defined in Figure 2.1, S represents a predetemiined allowable damage

levei and N, * is the spectral stability number.

Where: Hs = significant wave height; L~ = mial1 amplitude wave length based on T, and depth at toe of structure; D,, = nominal diameter of stone (D, = (MdpJ'"); A = relative density (A = p h , -1); Mm = average Stone mass (50% on mass distribution curve); Ps = mass density of amour stone; and P, = mass density of water.

Given the limited data used to develop this equation, it should be considered as a preliminary

approximation.

Stability analysis of submerged and low-crested rubblemound breakwaters was also undertaken

by Vidal et ai. (1993). Three-dimensional tests were performed and the stability of stones was

evaluated for four areas: the crest, the fiont slope, the back slope and a total section (including

crest, front slope and back slope). The results were presented for each breakwater section using

the conventional stability number (N,) and the dimensionless relative freeboard (F,) where:

An example of the stability design c w e s for each section, assuming a given damage level is

shown in Figure 2.6. The total dope curve reflects damage contributions fiom al1 other sections

and is therefore the most appropriate conservative estimate of stone size. It m y , however, give

considerable over estimations of amour requirements for specific areas of the breakwater under

certain fieeboard conditions.

Although other estimators of m o u r stone stability are available, the methods discussed above

are the simplest and most general methods available. Both approaches were used to estimate

armour requirements for the laboratory testing undertaken for this thesis. A bnef discussion of

the results is included in Chapter 4.

- - - - -

Figure 2.6 Stabiiity Cumes for Submergeci Rubblemound Breakwaters

(Source: Vidal et al. (1994))

(b) Wave Transmission and Reflection

PhysicaI design rnodelling of wave transmission and reflection at submerged breakwaters has

been undertaken by relatively few authors, Therefore, although the volume of test data generated

to date is substantial, some test variables have not yet been fully explored. Previous testing

results and analyses have predicted that the most important variables affecting wave transmission

at submerged breakwaters are relative submergeme and =est width (Goda, 1969; Tanaka, 1976;

Seelig, 1980 and van der Meer, 199 1).

One of the most comprehensive physical rnodelling exercises performed for analyzing the wave

transmission at breakwaters was perfonned by Seelig (1980). In this set of laboratory tests, a

total of 19 breakwater sûuctures of various configurations and construction materials were tested

with regular monochromatic waves. Many of the structures were also tested under a number of

irregular wave conditions. The tests included both sub-aerial and submerged breakwaters and

transmission coefficients were defined for transmission by overtopping (Km) and for transmission

through the breakwater (K,). The the overall transmission coefficient is defmed as K, where:

The transmission coefficient for overtoppmg was fond to be convenimtly expressed as a

hct ion of the ratio of m u p to freeboard, where runup is predicted using a formula by Hudson

(with present notation):

where R is the runup, H is the mean incident wave height (ie. 63% of H,), L,, is the deep water

24

wavelength and F is the freeboard (F = h,-h). Using the results of this calculation. the F/R ratio

cm be detemined and the overtopping transmission coefficient or submerged breakwaters is

subsequently dehed as:

F F B b

O-' ' ; 0.885 - 53.2 p91 K = C (1 - -) - (1 - 2C) where C = 0.51 - - R hl hl

The tests by Seelig (and tests by Saville (1963), in Seelig, 1980) indicate that the breakwater

freeboard and runup have a major influence on transmission by overtopping while crest width

has a very minor influence.

Transmission through a permeable sub-aerial structure is estimated on the basis of the head

across a representative rectangular breakwater cross section. This is an iterative procedure,

where the initial head is assumed to be equal to the wave nmup, and using nomographs based on

experimental &ta by Madsen and White (1976), the transmission coefficient is estimated and the

head loss is revised iteratively until the calcuIations converge. This method is found to be

acceptable for various amour and wave conditions. However, no mode1 is proposed for

estimating the transmission tl~ough submerged permeable breakwaters. Data cornparhg similar

permeable and impermeable structures shows that the total transmission coefficient approaches

the overtopping transmission coefficient as the structure becomes more submerged. Seetig

postdates that for Mz, r 1.2, the transmission coefficient through the structure is negligible.

Given a hypothetical incident wave height, period and sîructure geometry (summarized in

Appendix A) , K, has been plotted for a range of crest widths and sumergence depths using

Seelig's equation. These values are shown in Figure 2.7. Inspection of the plot shows that the

relationship is not maintained within the appropriate physical limits of K, (0.0 to 1 .O). The

hypothetical input parameters are outside the range indicated for BA, but are not physically

unreasonable. The relationship therefore is shown to be inadequate for large crest widths and

large relative depths of subrnergence.

Submerged Breakwater Transmission 1 Seelig, j980

- - - - -

Figure 2.7 Typical Design Curves - Seelig's Equation

Seelig ( 1 979) also presents a design equation for K, as defined by Goda et al. (1967) and Goda

(1969).

x h s - h il, -h K, = 0.5 ( 1 - Sin [-( - + p ) ] ) ; a - P < - % p - a [2.1 O ] 2a H, 4

The a and P terms in this equation are empincal coefficients, and therefore, relative

subrnergence ((h,-h)/& ) is suggested to be the most important variable affecting ûansmission.

A physical mode1 test of an existing submerged breakwater at Santa Monica, California

26

(damaged by the storm El Nino in 1983) was undertaken and is documented in Adams and Sonu

(1 986). The test detmined the m i s s i o n coefficient for the damaged breakwater and the

results were compared to previous studies perfonned b y Tanaka ( 1 976). Although Tanaka's

work could not be obtained for this study, some of his results relating wave transmission

coefficients (H, /Mo ') at submerged and low-crested breakwaters to dimensionless neeboard

(R/H, ') and dimensionless crest width (Bao) were presented in Adams and Sonu.

Design curves based on monochromatic wave tests, for four dimensionless crest widths rangmg

From 0.025 to 0.10, are reproduced in Figure 2.8 below. Li this figure, H, 'is the transmitted

wave height, Ho 'is the deep water unreflected wave height, B is the crest width, L, 'is the deep

water wave length and R is the fieeboard (-d,).

H,'/H,'

-2.5 -2.0 -1 5 -1 .O 9 .5 O 0.5 1 -0 1.5 2.0 2.5

R/H,' 7

Figure 2.8 Transmission Coefficients at Submerged Breakwaters

(Source: Tanaka (1976))

Although the submerged breakwater modelled by Adams and Sonu had an irregular cross section

due to the storm damage to the prototype, the physical test results generally supported Tanaka's

design curves for the conditions tested. The small discrepancies obsewed by Adams and Sonu

were attributed to the fact that they used significant wave height to define K, while Tanaka used

monochromatic wave height.

A specific class of submerged b r e h a t e r known as a reef breakwater was tested by J.P. Ahrens

(1988). The reef breakwater is a homogeneous nrbblemound structure designed to reshape to a

stable profile afier initial construction. The main advantages of the structure are that it can be

designed using a Stone gradation which is readily available and it can be constructed by simply

dumping matenal from a barge to a specific elevation and top width.

Although the nature of a reef breakwater is somewhat different than a subrnerged rubblernound

structure, the transmission and reflection equations developed by k e n s were assurned to be

valid for rubblemound structures. Consequently, it is assumed that the homogeneous nature of

the reef breakwater does not significantly change the transmission and reflection process, and

that the final stable shape of a reef breakwater is approximately trapezoidal. These assumptions

yill be made in this thesis, only for the purpose of comparisons.

A number of physical modelling tests were undertaken by Ahrens using irregular waves. The

reshaping characteristics and stability of the structure were assessed first, and subseguently, the

transmission coefficient of the stable structure was evaluated. A mathematical relationship was

developed to predict the equilibrium profile of the breakwater and the equilibrium crest height

was then used to detemine the hydraulic performance characteristics such as transmission,

reflection and energy dissipation.

The design equation denved by Ahrens (1987) can be stated:

where : D,, = nominal diameter of breakwater material (m) Ar = average cross xctional area of breakwater (m2) 41 = local wavelength based on linear-wave theory (m) F = fieeboard (ie. -4) (m) and al1 other variables are as defined in Figure 2.1.

The coefficients for this equation were empirically detemined such that C, = 1.188, = 0.26 1.

C, = 0.529 and C, = 0.0055 1. In order to use this equation with the presmt test data, the nominal

diameter of the core matenal (4) was substituted for D,,. This substitution requires that the

effects of the amour material of the rubblemound structure are ignored. Therefore, the equation

is expected to predict conservatively if the primary effect of D,, is nlated to transmission

through the breakwater , or liberally if related to the fiction effect of the surface layers. It

should be noted that Kr is defuied by Ahrens to be Hr /H, where H, is the zero moment wave

height in the same location as the probe measuring transrnitted wave height, but without the

breakwater in place.

Assuming that a reef breakwater would reshape to a trapezoidal section, the equation of Ahrens

may be extended approximately to submerged rubblemound breakwaters As previously noted,

this extension would require that there is little difference in performance between a

homogeneous and non-homogeneous structure. Nevertheless, extending Ahrens' equation to the

hypothetical variables used to denve Figure 2.7, the predicted transmission coefficient cm be

detemined as a function of crest width and depth of submergence as shown in Figure 2.9.

Without the benefit of test data for cornparison, only general comments can be made regarding

the relationship. The values appear to be generally consistent with previous findings, showing

that Kr increases wîth increasing depth of submergence and decreases with increased crest width.

However, some transmission coefficients for the larger crest widths and Iarger relative

submergence may be too small. This mode1 will be discussed fhrther in Chapter 6.

Subrnerged Breakwater Transmission Ahrens, 1987

Figure 2.9 Ahrens' Reef Breakwater Transmission Curves

Reflection fiom submerged breakwaters is also an important factor as ultimately. the wave

energy that is not transmitted or dissipated on the structure is reflected. Reflected wave energy

can be a problem at breakwaters as it may cause navigational problems or increase erosion in the

vicinity of the breakwater. Ahrens used the method of Goda and Suzuki (1976) to resolve

incident and reflected spectra and define a reflection coefficient Kr such that:

where Er and E, are reflected and incident wave spectra energy respectively.

Based on regression analysis of test results, Ahrens (1987) defmes an equation for the refiection

coefficient as:

h c2 4 F Kr = exp [C, (-) + - + c, (7) + c, (-)l =P hl (-1 8

4 0

where C, = -6.774, = -0.293, C, = -0.086 and C, = 0.0833. Assuming the same hypothetical

testing condition as that used to derive Figure 2.9 above, a plot of Kr as a fimction of crest width

and depth of submergence is shown in Figure 2-10.

Su bmerged Breakwater Reflection Ahrens, 1987

1

Figure 2.10 Ahrens' Reef Reflection Coefficient

Inspection of this plot again shows the significance of crest width and submerged depth in the

relationships developed by Ahrens. However, it is contrary to intuition that K, should decrease

with increased crest width. This is most Iikely due to the assumption relating crest width to the

cross sectional area t em (A,) used in Ahrens' equation. Ahrens defines A, based on the "as built"

section but defînes the related dimensionless ratio (Am in tenns of the reshaped crest height

(h,). In Ahrens' tests, increasing A, would in t u . increase the stability of his structure and the

resulting reshaped h, would be higher. Assuming that an increased B can be represented by an

increase in A, while maintainhg h, constant, as assumed here, may introduce some error.

hother detailed set of mode1 tests was perfonned over a number of years at Delft Hydraulics

(van der Meer and Pilarc yk, 1990; van der Meer, 199 1; van d a Meer and Angremond, 19%:

Van der Meer and Daemen, 1994) for development of design equations for wave transmission at

a variety of low crested and submerged breakt-vaters.

Van der Meer plotted the transmission coefficient (Kt = H, / H i ) as a function of several

dimensionless parameten representing mcident wave and breakwater characteristics. Linear

regression was performed for most relationshsps and ultimately combined to express K, in terms

of a number of parameters. The wave transmission coefficient (Kt) at the statically stable

submerged breakwater was found to be cornplex, being a fùnction of relative submergence

(d, / D,), relative wave height (H,/D,&), deep water wave steepness (S, = 2rHs /gT;) and

relative crest width (B /D,). nie thal relationship proposed (expressed in ternis of parameten

defrned in this report) is:

+ 0 K , = a - 4 o a

w here: H, a = 0.031 - - 0.24 D5oo

It is interesting to note that Kr is a Iinear fûnction of al1 dirnensionless variables except the

relative crest width (B/D5J.

Using the same hypothetical incident wave conditions and structure characteristics as those used

for Figure 2.10 above, van der Meer's relahonship is plotted as a fûnction of crest width and

depth of submergence in Figure 2.11. Van der Meer's equation is reported to be vaiid for :

I < H , / D , < 6 and 0.01<s,<0.05

Although the incident conditions are rnaintained withm these limits fol the rnost part, the

predicted K, does nct remain within acceptable physical limits (ie. O < K, c 1.0). It is suspected

that the broad range of crest widths assurned for the hypothetical input conditions cannot be

accounted for by this design equation. Therefore, it is evident that this design equation is not

appropriate for a broad range of conditions.

Submerged Breakwater Transmission van der Meer, 1991

Figure 2.11 Van der Meer's Transmission Coefficient

Considering the evaluation of the existing design equations presented above, there is an obvious

need to develop a more robust design equation for predicting the transmission coefficient at

submerged rubblemound breakwatm. The equation should be valid for a larger range of design

conditions and the form of the equation should reflect the physical processes which are

occ&g. Although the equations discussed above do not hlly satisfy these criteria, they have

been proven to be effective over certain ranges of data and do provide an indication of the

important parameters. The ability of the existing design equations to predict the test results

presented in this report will be discussed in Chapter 6.

It is important to note from the discussion in this section that the transmission coefficient at

submerged breakwaters has been found, by most authors, to be strongly influenced by the

relative depth of submergence. The second most important variable would appear to be a

dirnensionless form of the crest width. Other dimensionless variable combinations are used in

the various equations to improve the agreement with the test data. In general, it has been shown

that:

where n represents the breakwater porosity, L represents a characteristic wavelength and the

remaining variables are as shown in Figure 2.1. There has been Iittle effort to relate these

proportional relationships to the physical processes taking place at submerged breakwaters. As a

result, the equations fit the test data for which they were developed, but are not applicable over a

broad range of input conditions and some are not bounded at the physical limits of Kr.

In order to develop a design relationship for K, in terms of the physical processes occuning at

submerged breakwaters, it is necessary to consider al1 of the variables affecting the transmission

process and determine physically relevant f o m of these variables. In general, the transrnitted

wave height H, can be expressed as a function of a number of independent variables as follows:

where p, g and p are density, gravitational acceleration and fluid viscosity respectively and the

other variables are as previously defined.

It is generally accepted that the transmission coefficient. Kr , is d e h e d as the ratio of tranmitted

wave height (Hl) to the incident wave height (4.). Therefore, using customary dimmsional

analysis techmques, we can choose p, g and Hi as basic or repeating variables and defme K, as a

dimensionless variable on the basis of the previous statement. Therefore,

where X , = P

In [2.17] and [2.18], q5 represents a dimensioniess function, and the dimensionless variables

within the parenthesis may or may not have any significant physical meaning. It is possible,

however, to take any combination of these dimensionless variables to f o m other physically

relevant dimensionless variables if there is some prior understanding of the factors affecting Kr.

This prior understanding is provided, to some degree, by the extension of Iinear wave theories to

the physîcal processes at submerged breakwaters and by the results of previous investigations.

However, these investigations do not consider a suffkient range of the independent variables

influencing K, as noted in [2.17]. This is shown by the inability of existing models to predict Kt

over a wide range of hypothetical design conditions. Therefore, based on the need for a simple

design equation expressing Kr as a fiinction of relevant dimensionless variables and the

limitations associated with previous investigations, a number of research objectives have been

established as described in the following section.

3.0 Research Objectives

The review of existing literatwe on submerged breakwaters has s h o w the physical processes

which take place as incident waves interact with the breakwater to be very complex. In general,

there is çorne degree of confidence that the processes cm be described qualitatively on the basis

of physical modelling results and observations. However, there is currently no theory that can be

applied numerically to model these processes completely and accurately under al1 conditions.

Therefore, it is necessary from an engineering design perspective, to rely on physical modelling

data to predict breakwater performance characteristics. Although there are inconsistencies and

enors introduced by the conditions imposed on the physical models, the rnodels are developed

such that these mors are minimized for the processes being studied.

Design equations provide relatively simple relationships based on data derived from the actual

physical processes in question and are, therefore, an invaluable tool. In order for the design

equations to be useful and robust, they must be based on a sufficiently large range of testing data.

It is also important that their fonn and the variables that they include are physically relevant to

the physical processes occurring.

There are a few problems associated with the state-of-the art design equations presently available

to the design engmeer. It has been noted, in the previous section, that physical model tests to date

have not provided sufficient variation in crest width to adequately assess its effect on wave

transformations at submerged breakwaters. It is also questionable as to whether the

dirnensionless variables representing the effect of crest width are physically based given the

limited test data. Finally, al1 of the physical rnodelling of wave transmission discussed above

has been undertaken in a two-dimensional environment. Under these conditions, phenornena

unique to individuaI f l u e testing apparatus may influence the physical observations and test

results to some degree.

The objectives of this thesis are therefore:

To review and document the physical processes taking place at submerged breakwaters

as defmed by previous investigations and by the physical observations made in this

study .

To test a sufficiently wide range of submerged breakwater crest widths under a relatively

Iarge range of incident wave conditions in a 2-D setting in order to assess the validity of

the exisbig design equations.

To extend and modi@ the existing design equations as necessary to provide some

physical basis for the dimensionless parameters utilized. The assessment of reasonable

dimensionless variables will consider the results of previous physical process modelling

and the most promising numerical rnodelling.

To pcrfom 3-D testing for a number of conditions similar to those tested in the 2-D

apparatus to assess the validity of the proposed equations for application in more

realistic 3-D environments.

To make qualitative observations of other hydrodynamic processes, such as beach

development, diffraction effects and velocity distributions at submerged breakwaters.

Experimental Setup

4.1 Fiume (2-D) Testing

The 2-D t e s ~ g was undertaken in a wave flume at QUCERL. The flume is 47.0 m long, 1 .O m

wide and 1.2 m deep and is equipped with a flapper-type wave paddle, fixed at the floor of the

flume. The paddle is siniated in a small sump at the end of the flume and is capable of

generating regular and irregular waves .

4.1.1 Physical Conditions

The submerged breakwater testing in the flurne took place on a flat platform located at the top of

a 1 : 10 beach slope. The beach and platform were supported by a1uminum channel supports along

the sides of the flume and by cross supports. Most of the beach and platform was constructed

from 12 mm plywood, painted with pool paint to reduce fiction effects and to prolong the life of

the structure. The platform directly under the breakwater was constructed with 19 mm plywood

to minimize movement of the breakwater and platform under the dynamic pressure conditions

generated by the incident and hansmitted waves.

The beach and p l a h allowed the structures to be tested in relatively shallow water with a

large range of incident wave heights. The breakwater was consûucted with a core of angular

Stone with a relatively coarse gravelly core material (D, = 0.0 1 7 m) and two layers of primary

armour Stones (Dm = 0.059 m). Typical breakwater material distribution curves are included in

Appendix B. The front toe of the breakwater was located 0.10 m fiom the top of the plywood

beach slope for al1 test cases.

Reflection from the back wall of the flurne was minirnized by using a 1: 10 permeable beach

constnicted of layered, coarse filter fabric in fiont of a porous concrete block matruc to dissipate

energy through turbulent friction. The reflection characteristics of the test setup were determined

by analysis of the wave characteristics in the fiume without any breakwater in place. The

reflection characteristics of the test setup are documented and discussed in more detail in

Chapter 6.

Water surface time series were measured immediately in fiont of the breakwater and 2.0 m

behind the breakwater crest using five-probe arrays in =ch location. The probe array pemits

separation of incident and reflected wave trains on the basis of linear wave theory. This is

discussed in more detail in Chapter 6.

The testing routine was designed to cover a wide range of physical variables relating tu the

breakwater and incident wave characteristics. Each submerged breakwater was tested at five

different water depths to assess the effect of depth of submergence on the transmission

characteristics. The vanous breakwater configurations tested are discussed in the following

section. A typical test setup for the 2-D tests is shown in Figure 4.1.

c 47.Um

Wavt Absorbing Beach

Figure 4.1 2-D Testing Configuration

4.1.2 Mode1 Scaling

In order ta test the effect of a large range of physical parameters, it is necessary to mode1 a

variety of wave heights and periods. Therefore, the physical modelling parameters were selected

to permit as large an incident wave as possible given the mechanical limitations of the wave

generator. It was also desirable to keep the test structure relatively small in order to rninimize

matenal requirements and construction tirne.

Since the model represents no specific prototype çtnicture, the mode1 scale (defined below) can

be considered to be I: 1. Modelling variables are selected such that the effect of scale is

minimized for the physical processes modelled. As a result, the model could in fact represent

larger protome conditions. Assuming typical hydraulic rnodelling scales in the order of 1 :20 to

1 :30, the range of conditions tested yield prototype conditions typical of those encountered on

the Great Lakes. Therefore, the proposed testing program provides a comprehensive assessment

of the performance of submerged breakwaters.

in reality, there are physical limitations which determine the effectiveness of a model test in

representing prototype conditions. These limitations are the result of complex interactions

between the fluid and various structural materials and the resulting forces generated. In order for

a model to accurately represent the prototype situation, the relative effect of these various forces

and interactions m u t be the same in the model as in the prototype. This condition is achieved by

ensuring dynamic similarity between the model and the prototype and is best descnbed through

dimensional analysis. Although the theory of dimensional analysis will not be discussed in detail

here, the dimensional similarity of the model to an assumed prototype situation is presented.

Defining the scale of the model as the ratio of the physical length of an object in the prototype to

the physical length of the same object in the model, the scale of any parameter or phenornenon in

the mode1 can be simply defined as:

where /1, represents the scale of a paranieter a. The subscripts p and rn represent prototype and

model respectively. For example, the length scale (L) in a model, which is in effect the

defmition of the model scale, can be defined as AL = LJL,. SUriilarly, the flow scale (AQ) of a

model could be defined as AQ = Q&

In order for a model and a protoiype to be dynamically similar, the scale of al1 dimensionless

parameters must be unity. That is:

where Y, is the dimensionless form of a particular quantity a. Dimensionless parameters are

important as they descnbe the relative forces which dnve the processes being modelled.

Typicall y, there are three dimensionless pararneters which desmie the dynamics of a hydraulic

model. These pararneters are:

i ) Froude Number (FJ:

ii ) Reynolds Number (RJ:

iii) Weber Number (W,):

The Froude , Reynolds and Weber Nurnbers represent the ratios of inertial forces to

gravitational forces, inertial forces to viscous forces and inertial forces to surface tension forces

respectively.

Inspection of the equations will show that if water is used in the model test (which is usually the

case), then gravity, fluid density and fluid viscosity will be the same in the model and prototype.

Therefore, it would be impossible to ensure that:

Wave transformations due to bathymetric changes and the inertial stability of armour units are

d e h e d by the Froude criterion and are often the most significant processes modelled.

Therefore, it is imperative that this criterion is met in any model studying these processes.

The Reynolds nurnber criterion governs viscous effects such as wave transformations due to bed

fnction and stability against flow induced forces such as drag and shear. Saîis-g this critenon

is very important when modelling interna1 flow within a breakwater or transmission through a

structure. However, where the effects of fnction are small reIative to gravitational forces and the

Reynolds nurnber for the model can be maintained within a similar range as that in the prototype,

thus maintaining similar drag coefficients, Reynolds number similarity is not as important

(Dalryrnple, 1985). In accordance with this consideration, it is generally recommended that the

Reynolds nurnber in physical hydraulic models be maintained such that:

R e r 3 . 0 X I O J

where the characteristic length used is that of the armour uni& and the velocity is based on the

significant wave height ((g*H~*?@ai and b e l , 1969). The Reynolds number was computed

for the various conditions imposed in the 2-D tests with the following results.

4.13XI04 s R e s8.26X104

Therefore, the model is expected to provide acceptable results with respect to viscous effects.

Modelling effects due to incorrect Weber number shilarity are typically insignificant when the

wavelengths of concem are much greater than 2 cm (Dalrymple. 1985). Given the relatively

large wavelengths considered in this study, the effects of surface tension (O) are assumed to be

uisignificant relative to the inertial fluid forces.

It is therefore expected that the mode1 results can adequately represent the transmission

characteristics resulting from Froude scale phenornena (ie. transmission due to wave breaking

and wave transformation due to depth limited effects). Transmission characteristics dependent

on Reynolds scaIe phenornena are not represented as accurately but the rnodelling results are

considered to be acceptable given that the Reynolds number is maintained above recommended

limits. It is also assumed that the processes govemed by the Reynolds number are less

important than those governed by the Froude number with respect to wave transformations and

transmission characteristics.

4.1.3 Breakwater Construction

The breakwater construction materials were selected on the bais of the incident wave

characteristics. It is necessary to minimize (or eliminate if possible) damage to the breakwater

during testing. Therefore, it was necessary to choose an mour size which wouId be stable

under the most severe wave conditions tested. The stability equations of van der Meer (1 99 1)

and Vidal et a1.(1993), as discussed in Section 2.2.3 above, were used to estimate the necessary

amour size. These equations incorporate a factor representing the acceptable damage level.

The factor representing the minimum acceptable damage level was selected for these

calculations which are provided in Appendix B of this report.

The amour size required to prevent damage to the breakwater under the most severe testing

conditions (ie. largest waves with minimum breakwater submergence) is as follows:

van der Meer: D , = 0.051 m (WJh = 360 g ) Vidal et al. : = 0.057 m (WJk = 500 g)

A gradation about this Dm was selected in accordance with the Shore Protection Manual (SPM)

(U.S. Army Corps of Engineers, 1984) recommendation for cover layer design such that the

armour Stones selected satisfi:

0.75D5& s D, s 1.25 D,,

Core material was then selected such that:

0.01 W5& s WC s o.1Ows,

where the subscripts a and c represent armour and core matenal respectively. A test stmcture

was first constructed using armour material based on the van der Meer equation. The structure

was not stable enough for testing. The amour was then replaced with Stone satisfjnng the

equation of Vidal et al. (1 993). This structure was stable for al1 but the most severe cases (W 10

and W 1 1 as defined in Section 4.1.4) at submergence depths of 0.0 rn and 0.05 m. For these

cases, tests were not completed.

A nurnber of breakwater geometries (see Figure 2.1 for definition of terms) were modelled

under various depths and incident wave conditions to determine the effect of the breakwater crest

width and front face slope on the transmission and reflection characteristics of the structure. Two

additional structures were tested with srnaller armour to assess the effect of v q m g surface

roughness coefficients. A summaxy of the structure geometries tested is provided in TabIe 4.1.

Table 4.1 2-D Testing Structure Configurations

The breakwaters were d l constructed in "the Qy". An outline of the desired structure was drawn

on the fluxne wall, the core material was dumped and shaped to give the desired slopes and the

armour stones were individually hand placed. Placement of armour stones was deliberate but

stones were not "fi" into the armour layer. This method generally provides a realistic

representation of the effort made to place armour stones in prototype conditions.

Structure*

B (m)

Cbt 8

D,, (m)

4.1.4 Wave Conditions

AI1 breakwater structures were tested under regular and irregular incident wave conditions. The

regular wave tests serve to confurn some of the wave transformation processes documented in

previous studies. These tests were not used to determine transmission coefficients. however, due

to theû unrealistic nature with respect to design applications.

Al1 wave signais were generated using the GEDAP Real l ime Control Wave Generation

Package (Miles, 1990). The package consists of k e e modules. The PARSPEC module

generates a wave spectmm based on defining charactenstics such as significant wave height,

peak period, peakedness factor and shape factors. RWSYN generates a wave train fiom this

spectrurn through inverse Fourier tmnsforms. RWREP2 then grnerates a voltage signal for a

specific wave generator based on the input wave miin and characteristics of the specific wave

generator as indicated in the wave machine calibration file.

45

Note: All stnicturcs w m wtd ar ci, = 0.0 m, 0.05 m 0.10 m. 0.15 rn and 0.20 m

GI

0.3

1.5

0.059

G2

0.6

i5

0.059

G3

1.5

1.5

0.059

G4

2.5

1.5

0.059

GZ

3.5

1.5

0.059

G6

2.5

5

0.059

G7

1.5

5

0.059

G8

0.6

5

0.059

G9

0.3

5

0.059

Ci10

0.6

3

0.059

G12

2.5

1.5

0.037

1

G11

2.5

3

0.059

Ci13

0.6

1.5

0.037

Regular wave signals are generated directly by the R W P 2 muhne given the wave height and

period. The signal generation routine again references the wave machine calibration tile to

detennine the required voltage signal.

The calibration of the wave paddle sirnply involves measuring the displacement (stroke) of the

paddle when subjected to a standard calibration signal. The calibration signal cycles h m the

minimum voltage applied to the servo mechanics (-5 volts) to the maximum voltage (+5 volts).

Based on the corresponding minimum and maximum displacements, the depth of water at the

paddle and the gap between the bottom of the paddle and the floor, the GEDAP package

detemines the stroke and stroke rate (ie. voltage and slew rate) required to gmerate the desired

wave train. The wave paddle was calibrated before testing began. A full description of the

GEDAP system and calibration procedures is documented by Pelletier (1990).

Al1 imgular wave spectra were Jonswap spectra with or = 0.0081 and y=3.3. These are typical

charactenstic parameters representing the area under the energy density spectrum and the

peakedness of the spechum respectively, and are the default values of the GEDAP PARSPEC

routine. Although diffaent spectral generating parameters may have some effect on the test

results, the sensitivity of these parameters is not addressed in this report. The spectral

characteristics of the incident target spectra are presented in Table 4.2.

Table 4.2. 2-D Testing - Irregular Wave Characteristics

WO8

0.15

WI 1

0.20

W09

0.15

Wave Set 1

Hmo(m)*

W10

0.20

; T P ~

W03

0.05

i .2

WO1

0.05

NOR: Wavc characttristics are targct values - those measured in flume may Vary to sorne d e p .

1.5 1.5

WOS

0.10

W04

0.10

W02

0.05

2.0 L

2.0 1.5

W06

0.10

W07

0.15

2.0 1.5 1.2 2.0 1.2

ReguIar wave signals were generated for 10 cm waves onIy, but for al1 three periods noted in

Table 4.2 above. Wave generation parameters and incident wave characteristics are included in

Appendix C.

Due to the dynamics of the wave generator and the physical characeristics of the wave flume,

the generated spectra (and characteristic parameters) Vary to some degree fiom the target spectra.

The variation is due to a number of influences including:

(i) elecironic conversion of the digital wave generator input signal to an analog voltage

signal whch is subsequently electronically filtered to a finite nurnber of voltages,

(ii) losses associated with the hydraulic and mechanical response of the wave paddle,

(iii) physical characteristics of the wave flume.

Each signal was modified slightly by applying an amplification factor in an effort to achieve the

target specmim. It is, however, reasonable to assume that the target spectrum is not satisfied

exactly . Typical mors in characteristic parameters may be in the order of 5 to 10 %. Signa1

generation routines and input parameters are included in Appendix C of this report.

For the sake of convenience, some results may be expressed in tem of the target spectrum

characteristics rather than the measured spectral characteristics where the differences do not

significantly influence interpretation of the results.

4.1.5 Instrumentation

The wave flume was outfitted with 10 wave probes to measure the water surface time series. The

wave probes are capacitive type water level gauges sampling water slirface elevations at 20 Hz.

Technical documentation for the wave probes is provided in Wiegert and Edwards (198 1). The

wave probes consist of the bow whch provides an open circuit between the b m s reference post

and the insulated probe ''wire-wrap". The water provides the medium to close the circuit and the

changing height of the water in the bow provides a variation in resistance, thereby varying the

circuit's ability to charge the reference capacitor of the probe's circuitry. A general schematic of

the probe and bow is shown in Figure 4.2.

Figure 4.2 Water Level (Wave) Probe

The probes were initially calibrated at three different water levels on the bow to provide a

conversion from voltage to water surface elevation. The probes are reported to be linear in their

response with linearity errors generally less than 1.5% (Wiegert and Edwards, 198 I j. The

linearity of the bows was confinned during calibration of the probes. Linearity can be

dependent on the wire used to make the probes and can be affected to some degree by buildup of

deposits on the probe wires. Care was taken to keep the probe wires as dean as possible during

testing.

The resolution of the probes is reported to be better than 1 mm (Wiegert and Edwards. 198 1 )

over the 15 cm (6") range. The calibration parameters for the probes were found to be relatively

sensitive to the water temperature as evidenced by the variation in calibration parameters

determined fiom day to day. Therefore, during periods immediately following a change of water

in the flume, full calibration of the probes was camied out more fiequently.

Ten wave probes were used in total, arranged in arrays of five with the first anay located

immediately in ffont of the test breakwater and the second array located 2 meters behind the rear

toe of the test structure. The array configuration allows sepration of the incident and reflected

wave trains through the GEDAP refiection analysis as discussed in Section 4.1.6.

Al1 probes were set to the 60 cm (24") measuring range given the potential size of the incident

waves. Large bows (about 60 cm) were used on the fun array while smaller bows (about 30

cm) were used on the second array. This inconsistency was simply due to a shortage of the

larger bows. Although using the probes at the largest range setting may result in some loss of

resolution, the large range was necessary to sample the desired range of incident wave

parameters and the loss in resolution is not expected to affect the results considerably.

4.1.6 Data Sampling and Andysis

As previously noted, data were sampled fiorn the wave probes at a rate of 20 Hz. Sampling was

not initiated until2 minutes afier the wave paddle was started in order to allow some

stabilization of the physical effects of the flume on the wave characteristics. The duration of

sampling for each test was set equal to the duration of the wave s ipa l to msure that the desired

target spectral characteristics were attained. Following each test, the flme was allowed to settle

until no residual agitation was evident.

Collection and analysis of the testing results were performed using the National Research

Council of Canada's GEDAP software package. The basic analysis consisted of a spectral and

zero crossing analysis of each of the individual wave probe records to define the dynamic

characteristics of the water surface behmd and in fiont of the submerged breakwater. In

addition, a reflection anaiysis was performed at each of the two probe arrays to separate the

incident and reflected wave bains. Therefore, the incident wave train fiom the k t array would

define H, and the incident wave train fiom the second array would defme H,. The reflechon

analysis is based on a least squares approach to defining the incident and reflected specûa using

3 of the 5 probes in each array (Mansard and Funke, 1987).

The transmission coefficient for a given breakwater is defmed as:

%O,

K, = - where : Hm,, = H m o ,

Transrnitted zero -m ornent wave heighr l4-51

Incident zero -mornenr wave height

The significant wave height (H') was used instead of the zero-moment wave height in some cases

when was not considered to be representative of the wave ctiaracteristics; this situation arose

in the 3-D testing as discussed in Section 4.2.6. Otha parameters used in the characterization of

the wave climate, but not directly derived fiom the spectral or zero-crossing analyses, (such as

local wave length associated with peak penod; LJ are based on linear mal1 amplitude wave

4.2 Basin ( 3-D ) Testing

4.2.1 Physical Conditions

The 3-D testing was undertaken in the three-dimensional wave basin at QUCERL. The basin is

approximately 30 m X 25 m X 1.2 m and is equipped with a piston action wave paddle capable

of generating regular and irreguiar waves. The paddle consists of three separate 3.5 m sections.

producing a wave board 10.5 m long. The paddle can be rnoved throughout the basin to permit a

variety of testing configurations and incident wave angles.

The present testing was undertaken using two of the paddle segments providing a 7.0 rn wide

testing area. A concrete beach was constructed at a dope of 1 : 10 to provide energy dissipation

and to muiimize reflection in the testing area. The submerged breakwater structures were

constructed on the flat floor of the basin approximaatefy 2.0 rn offshore of the concrete beach toe.

This configuration permitted the measurement of ûansmitted wave characteristics and velocities

in a relatively large area behind the structure. No measurements were taken on the 1 : 10 beach

dope.

The breakwaters were tested with incident waves at 90' and 60" to the shoreline. Wave guides

constnicted at 90" to the paddle, fkom the paddie to the beach. maintain the wave energy within a

constant width section and minirnize the effect of disturbances outside the testing area on the

incident wave conditions.

The 3-D tests were instnimented with 8 wave probes and 2 velocity probes. Due to the complex

multi-directional nature of reflection processes in a 3-D setting, no attempt was made to separate

incident and reflected waves. Therefore, only one wave probe (Probe 5) was iocated offshore of

the breakwater to define the incident wave conditions. The remaining probes were arranged

behind the breakwater (as shown in Figures 4.3 and 4.4) in order to define the effects of

diffraction on the transmitted wave field.

The velocity probes were relocated a number of times during certain wave sets to provide a

better indication of the velocity patterns in the sheltered zone. Multiple velocity measurements

were made during testing with signals W02, WOS, W06, W07 and WlO. Each of the probes was

moved to three different locations during the test and at each location velocity rneasurements

were made a three depths (2 cm, 0.2 h and 0.7 h) measured fiom the basin floor.

The typical basin setup for incident angles of 90" and 60" is defined in Figures 4.3 and 4.4

respectively. Velocity and wave probe locations are noted accordingly. The frst velocity

measurements were consistently made directly behind the breakwater (locations 10 1 & 20 1)

2 cm fiom the basin floor. Subsequent velociv rneasurements were taken in the order defined by

the location codes (ie. 10 1, 102, 103 etc.). The subsequent velocity measurements were bken

without re-initialking the wave signal : that is to say that after the first samples were taken from

velocity probes 1 and 2 at locations 10 1 and 201 respectively, the probes were moved to

locations 102 and 202 while wave generation continued and sampling was then re-initiated. This

approach results in sorne disturbance to the testing since it is physically necessary to enter the

t e s ~ g basin to relocate the probes. This disturbance dissipates quickly, however, and is not

expected to influence the resdts.

- -2om 7.m-

Figure 4 3 3-D Testing at 90"

Figure 4.4 3-D Testing at 60'

Sand was placed on the beach to visually assess the effect of the submerged breakwater on beach

53

development and transport processes. The sand @,=0.50 mm) was distriîuted prior to each test

condition (ie. new water level) such that sand extended fiom slightly above the static water level

to the toe of the concrete beach. Therefore, the thiclmess of the beach was slightiy greater for

lower water Ievels. The sand was not redistrhuted until the water level was changed again and

therefore, beach development takes place over a range of incident wave conditions and durations.

These factors are expected to influence the £bal distriiution of the sand to some degree and

therefore, the results are only qualitative in nature.

4.2.2 Mode1 Scaling

Physical limitations associated with the basin mode1 did not permit testing at the same scale as

that used in the flume tests. The basin's wave paddle is not capable of producing waves as large

as those tested in the flume and there was insufficient armour available (Dm = 0.059 m) to

construct the test breakwater section- Therefore, the basin tests were scaled on the basis of a

smaller m o u - (D,, = 0.037 m) resulting in a geometric scale of 0.63. in other words, the ratio

of al1 characteristic lengths in the basin (3-D tests) to those in the flume (2-D tests) is 0.63. Al1

geometry, wave characteristics, amour Stone sizes etc. were scaled on this basis.

As previously noted, it is important to maintain a Reynolds number above 3.0 X 104 in physical

hydraulic models in order to minimize the viscous scale effects. Using the D, of the amiou.

used in the basin tests as a characteristic length, the Reynolds numbers for the basin tests lie

within the following range:

2.06X104 SR, ~ 4 . 1 1 XIO'

Therefore, Re for some tests may be slightly outside the recommended range and sorne viscous

effects may be improperly represented, although their influence is expected to be minimal. It is

therefore important to interpret the results with this fact in mind.

4.23 Breakwater Construction

Al1 breakwaters tested in the basin w m constnicted from a relatively fine core material (D, =

0.004 m) and two Iayers of armour (D, = 0.037 m). A section of plasticized fly screen was laid

between the amour and the core material to permit easier construction. The porosity of the fly

screen was approximately 0.75 which is expected to be much more porous than the core material.

Therefore, the fly screen is not expected to affect the results. Gradation curves and relevant

calculations are provided in Appendix B of this report. A photo of a breakwater used in the

basin tests is shown in Figure 4.5.

Figure 4 5 Typical Breakwater Section

Al1 breakwaters were constnicted to a length of 3.0 m. As previously indicated, other

dimensions were selected to coincide with structures tested in the flume, scaled according to the

basin testing scale. Crest widths of 0.19 m. 0.38 m and 0.95 m were rnodelled representhg 2-D

testing crest widths of 0.3 m, 0.6 m and 1.5 m respectively. Six different configurations were

tested and are summarized in Table 4.3. Configurations BO and B7, not s h o w in the table,

represent the case with no breakwater tested at incident angles( P ) of 90" and 60" respectively.

These tests were used to verify the incident wave conditions and to provide general reference

observations for the "no-structure" tests.

Table 4.3, 3-D Testing Structure Configurations

4.2.4 Wave Conditions

As previously noted, the basin tests were scaled relative to the flume tests and therefore the

incident wave conditions were predetermined. However, the breakwaters tested in the 2-D

setting were constructed on a platform and incident waves were generated in approximately 0.75

m to 0.95 m of water. In the basin, the breakwaters were constructed on the basin floor and the

waves were generated in approximately O. I9 rn to 0.28 m of water. Therefore, the largest waves

of the 2-D test series could not be generated due to depth limitations. This depth limitation is an

important consideration when comparing the 2-D and 3-D testing results because the shallow

depth in the basin will result in more wave breaking and non-linear wave properhes than

observed in the flume. The spectmI characteristics of the incident waves cm, therefore, be

expected to be different for the two testing conditions. This discrepancy is discussed in more

detail in subsequent sections.

Structure

B (ml

P

BI

0.3 8

90

B2

0.95

90

Cot 8

D,,(m) Note: Al1 smictures werc testcd at d, = 0.032 m. 0.063 m and 0.126 m.

I .5

0.037

1.5

0.037

B3

0.19

90

1.5

0.037

84

0.19

60

1.5

0.037

B5

0.3 8

60

B6

0.95

60

1.5

0.037

1.5

0.037

Al1 of the wave trains tested were irregular, generated fiom a Jonswap s p e c m with a = 0.008 1

and y = 3.3. The signais tested are sumarized in Table 4.4 below.

Table 4.4. 3-D Testing - Irregular Wave Characttistics

4.2.5 Instrumentation

The water Ievel probes used in the 2-D tests were also used in the 3-D basin tests. Al1 water

level probes used in the 3-D testing were equipped with a 30 cm *wave bow and the probe

circuit boards were al1 set to the 30 cm (12") operating range.

k Wave Set

Hmo (m)*

Tp(s)*

Velocity probes were supplied by National Research Council of Canada (NRC). The probes are

custom-made at NRC and they measure cment velocities as a function of electromagnetic

current which is generated as the water moves across the face of the sensor. A general schematic

of the probe is shown in Figure 4.6.

The power supply for each velocity probe is equipped with a one second filter. One of the

probes was tested (afier calibration as described below) with the filter in (on) and out (off) to

assess its effect on the sampled data. The result of the cornparison is shown in Figure 4.7 below.

The data indicates that the filter removes much of the larger velocity fluctuations, assumed to be

associated with the short wave energy. Another effect of the mter is to introduce a periodic drift

to the filtered data. This drift could be due to the electronic effect of the filter itself, external

noise or low fiequency energy generated in the test.

Noie: Wave characteristics arc appximatc - those mcasurcd in basin may vary to ~ m w degm.

WO1

0.032

0.95

W07

0.063

1.59

W06

0.063

1-19

W08

0.063

1.98

W02

0.032

1.19

W09

0.095

0.95

W03

0.032

1.59

W04

0.032

1.98

W10

0.095

1.19

W05

0.063

0.95

_I

W I 1

0.095

1.59

Given the magnitude of some of the higher fiequency energy, it was considered desirable to keep

this energy in the sampled data and filter it later, if necessary, using the GEDAP anafysis

package. Ttierefore, the probe power supplies were used with the filter tumed off (out).

+-- Robe Cicuitry

Figure 4.6 Velocity Probe

The probes were calïbrated by towing at a constant velocity in a 0.6 m wave flume at QUCERL

and measuring the voltage developed for a given tow velocity. The probes were towed at various

orientations until a maximum mean voltage was estabiished. This orientation was noted and

assigned an X or Y axis. Subsequently, the probes were towed at three different speeds along

each of the four defined axes to provide calibration parameters. A zero velocity measurement

was also taken for each axis. 'Ihis calibration data was input to the GEDAP RTC-CMIB

routine for the appropnate channels and was not adjusted for the remainder of the tests. The

calibration data for the velocity probes is included m Appendix D of ths thesis. As indicated by

the data, the probes are Iinear in nature and were calibrated as such.

( b ) Fiter Out

Figure 4.7 Effect of One Second Filter

4.2.6 Data Sampling and Analysis

All data sampling was undertaken through the GEDAP (RTC-DAS) data acquisition system at

20 Hz. The eight wave probes were sampled on channels 1 through 8 and the velocity probes

were sampled on channels 9 through 11 . Because the data output jack format fiom the velocity

probe power supplies is not compatible with the amplifier used for the wave probes, an adaptor

was constructed by QUCERL technical staff.

As in the flurne tests, the duration of each sarnpling period was set equal to the duration of the

wave min tested and wave generation was initiated 2 minutes before sampling to allow some

stabilization of the wave characteristics. AIthough velocity sampling was performed at various

locations for each wave signal without allowing the basin to seîîie between samples, a settling

period was provided between each wave signal.

Due to the complex multi-directional nature of reflection in a 3-D testing situation, no reflection

analysis was performed for the basin tests. Individual probes were used to represent the

dynarnics of the water surface at a given point and comparisons were made on this basis. The

transmission coefficient is, therefore, defined at each probe location based on the average

transmitted significant wave height (H,). Probe 5, located offshore, was used to defme the

incident wave height-

The significant wave height was used to d e h e Kt in the 3-D tests due to the complex reflection

within the test area which could not be separated from incident energy by the GEDAP analysis,

and due to the seepage of energy into the testing area fiom the surroundhg basin. As previously

indicated, Hm, provides a characteristic wave height based on the spectral energy. In the 3-D

tests, the total energy behind the breakwater is not entirely the result of transmission fiom the

seaward side of the breakwater and as a result, H,, may not provide as representative a

characteristic wave height as does H,.

5.0 Analysis of Wave Data

5.1 Overview

Al1 data from 2-D and 3 - 0 tests were analyzed using the GEDAP data analysis software package

(Miles, I W O ) . Analysis of general statistics, zero nossing analysis (ZCA) and variance spectral

density analysis (VSD) were completed for both 2-D and 3-D tests. Reflection analysis was

performed for 2-D tests only. A bnef discussion of the significant aspects of the analysis

techniques, as applied in this study, is provided in the following sections.

5.2 Zero Crossing Analysis

The zero crossing analysis was performed using default parameters as specified by the GEDAP

software package. Because ZCA requires at least 50 points per average zero crossing period and

the sampling fiequency used in the data sampling was 20 Hz, the analysis of rnost signals

required interpolation for additional points. This interpolation is performed by GEDAP usïng

the cubic spline technique and a local parabolic fitting technique to define wave crests and

troughs respectively.

The most important parameters derived fiom ZCA are output with each time series plot in the

test data voIumes (separate to this thesis) . Typically, where parameters are computed on the

basis of zero up-crossing and zero dom-crossing techniques, the average of these results has

been presented. Although the zero-moment wave height was used to define K, for the 2-D tests,

due to reflection analysis requirements, the significant wave height (H') derived from ZCA was

used to define Kt for the 3-D tests. The reason for this approach is to minimize the impact of the

long wave and short wave energy arising from the general testing configuration and seepage into

and out of the testing area. This energy is not a function of the breakwater transmission and

reflection characteristics, but tends to affect the rneasured spectra1 energy, thereby increasing the

computed Hm,.

53 Variance Spectral Density Andysis

The variance spectral density analysis provides numaous parameters defining the wave field on

the basis of its wave energy spectm. It is very important to remember that some of the spectral

based characteristics assume a narrow spectrum (Rayleigh Distribution) and as a result, there

may be some degree of error associated with the parameter estimate. However, given the power

of spectral analysis techniques, analysis of wave data ofkm assumes a Rayleigh distribution of

wave heights and as a result, zero-moment wave height (X,) and peak penod (5) are commonly

used parameters. The VSD analysis used in this study assumed default settings as provided by

the GEDAP software, with the exception of the upper cutoff fiequency, which was defined as 4.0

Hz in order to maintain consistency in plotting.

Important spectral based parameters associated with the tests are included with the ZCA

parameters and time series plots in the test data volumes previously noted. Spectral plots derived

fiom the VSD analysis are provided for selected 2-D tests and al1 3-D tests in the test data

volumes. When intqreting this data, it is important to consider that the zero-moment wave

height (Hm,) is based on the total energy of the rneasured spectnun, regardess of the spectral

spreading. Therefore, H,, is a good indicator of the energy in a given area but not a particulariy

good indicator of a characteristic wave height if the spectnim is very broad.

The effect of the spectral spreading on the estimate of I&,,, is overlooked in the case of the 2-D

testing since the energy behind the breakwater is assurned to be due to msmission across the

breakwater and it is necessary to use the spectral measure in order to separate the incident fiom

reflected wave components. The implications of these assumptions mus& however, be

considered care full y when interpreting the test results.

5.4 Refiection Analysis

Reflection analysis was perfomed for the 2-D tests only, given the limitations associated with

the analysis technique and the multi-directional characteristics of the wave field in the 3-D tests.

The analysis is based on a least squares analysis of data fiom 3 probes, selected fiom a possible 5

probes. The optimum 3 probes are selected by the GEDAP routine PRBSP on the b a i s of

cnteria relating probe spacing to wavelength. Cross spectral analysis is then perfomed for the 3

probes by the GEDAP routine XSPEC, and incident and reflected energy is computed for a

number of discrete fiequency bands.

Reflection coefficients are interpolated for a number of defined fiequencies and output with

incident and reflected spectra, the reflection coefficient spec tm, error threshold values and a

smoothed reflection coefficient spectrum. Typical output for this analysis consists of incident

and reflected spectra, error threshold values and the associated characteristic parameters. The

average reflection coefficient presented in this study is simply the ratio of the reflected H,, to the

incident Hm,. Therefore, the effects of higher reflection coefficients at high frequencies and the

effects on spectral distribution are not evident in the representation of K,

Discussion of Results

6.1 Flume Testing : Observations and Trends

6.1.1 General

The initial flume setup was tested without a breakwater structure in place in order to determine

the hydrodynamic characteristics of the fluxne and the plywood beach. This testing was atso

used to evaluate incident wave characteristics in front of the plywood beach and platform. The

testing setup for this initial condition is s h o w in Figure 6.1.

l

Wavc Absorbing k c h

Figure 6.1 Initial Conditions Testing Setup

The entire set of wave signals (See Table 4.2) was tested with this setup. During the larger wave

series. some wave breaking was observed on the beach slope. Although existing numerical

analysis techniques, including those used in the GEDAP analysis packages discussed herein,

cannot accurately account for breaking wave phenomena, it is important to test a wide range of

incident characteristics. Therefore, some breaking was considered to be acceptable.

Reflection analysis at the two probe amys shown in Figure 6.1 provided the incident and

reflected wave characteristics in these regions, thereby defming the reflection fkom the plywood

tesbig pfatform and the flurne wall. Reflection characteristics of the testing configuration are

shown in Figure 6.2 where K,. is the reflection coefficient ( H , J and h/H, is the relative depth.

2-DTest Configuration - No Breakwater Reflection Characteristics - Array 1

a - Probe Array 1

2-DTest Configuration - No Breakwater Reflection C haracteristics - Array 2

b - Probe Array 2

Figure 6.2 Reflection Analpis - Flume Testing Configuration : No Breakwater

The reflection coefficient for the entire test apparatus was computed at the l* probe array. The

reflection coefficient generally ranged from 5% to 15%' increasing as the relative submergence

of the platforni was reduced. This result was expected given the increased obstruction of the

beach to wave transmission as the depth on the beach was reduced.

A reflection coefficient was also computed at the 2"6 anay, representing only the reflection

charactenstics of the rear wall of the flume. This reflection coefficient was generally higher than

that measured at the 1 ' array, increasing considerably with decreasing relative submergence of

the platfonn. The incident wave in this case was the wave transmitted ont0 the platforrn area and

it was somewhat smaller than that measured at the l* array due to reflection fiom the plywood

beach and wave breaking on the platform. The energy spectra of the incident waves measured at

the 2" array contained more energy at higher frequencies than those of the 1' array. It is

therefore expected that the relatively high reflection from the rear wall of the flume is due to the

inability of the absorption beach and concrete block rnatrix to damp out the high frequency

energy. It is also important to consider that the computation performed by the GEDAP analysis

package will include some degree of error as it is not designed to accommodate the non-linear

wave characteristics associated with breaking waves.

Following the tests of the basic flume configuration, the first breakwater structure (G1) was

constnicted, using an amour size as determined using van der Meer's (1991) stability formula.

The suggested amour stone (D, = 0.05 1 m) was not stable with the larger waves. Considerable

damage was done to the fiont face and crest of the breakwater. Given the extensive damage,

Vidal et al. 's (1 994) stability formula was used to determine a new D, (0.057 m).

Some damage was also sustained using this larger stone but only with the largest waves at low

submergence depths. As a resulf the H' = 20cm waves were not tested at submergence depths

less than 10 cm. During the remauider of the testing, only rninor damage was sustained by the

breakwater on occasion. This damage generally consisted of rnovement of one or two Stones on

the breakwater which were replaced before the following test.

6.1.2 Time Series Transformations

Induced wave breaking and generation of higher fiequency wave components on the crest of the

breakwater were visually evident. Return flow velocities, generated by currents which were

developed by a srna11 water level setup behind the breakwater, were also evident under certain

incident wave conditions. Wave breaking occurred mainly on the fiont slope and crest of the

structure but occasionally, with a narrow crested breakwater, wave breaking would occur behind

the structure.

Inspection of the t h e serîes record and associated statistical and zero-crossing parameters, show

a nurnber of changes taking place across the breakwater. In general, there is a reduction in the

wave heights but an increase in wave frequency behind the structure. Thîs is evident in Figure

6.3 which shows time senes fiom Probes 5 (incident) and 10 (transmitted) as noted in Figure 4.1.

There is a mean water level setup behind the breakwater due to the mass transfer across the crest

whch is produced by the wave breaking. This mean setup is likely a function of the 2-D test

condition and will give rise to a prolonged offshore velocîty across the breakwater during

penods of wave down rush. This is expected to have some influence on the incident waves,

including possibly the buildup of low fiequency wave energy in &ont of the breakwater (Petti

and Ruol, 1991 and f 993; Lîberatore and Petti, 1993). Although this process will have some

effect on the transmission of the longer waves, it is a characteristic of 2-D testing and cannot be

avoided in this case.

Other transformations of the wave time series are not readily evident upon visual inspection of

the irregular wave series plot. Inspection of a typical regular wave senes, however, immediately

shows the harmonic decomposiiion phenommon as well as wave steepening and breakmg on the

breakwater. A typical regular wave series plot is shown in Figure 6.4.

Because regular waves do not provide realistic test resuits for practical engineering design

applications, they were not considered in great detail in this study, except to confinri the general

be haviour of submerged breakwatm stnxctures.

Typical Water Level Record 2-D Tesfing : lrregular Waves

0.06 1 I l l

O 10 20 30 40 50 l ime (s)

- - - --

Figure 63 Time Seria Record : &,, = 0.10 m; T, = 2.0 s; B = 1.5 m; d, = 0.10 m

Typical Water Level Record 2-D Testing : Regular Waves

0.1 -- 1

I

O 5 10 15 20 25 30 Tim e (s)

-- - - -

Figure 6.4 Reguiar Wave Series (H, = 0.10 m; T = 1.5 s; B = 15 m; d, = 0.10 rn )

6.13. Spectral Transformations

Spectral analysis of incident and transrnitîed wave series is quite effective in showing the

influence of submerged breakwaters on inegular waves. Generally, the results obtained during

these tests are in agreement with results presented by previous authors. There is a dehi te shift

in the energy s p e c m as the waves pass the breakwater, whether wave breaking is occurring or

not. This is best shown in the senes of plots of Figure 6.5. In this figure. the presence of wave

breaking is associated with the relative depth of submergence ( d a which has been shown to be

an important variable in previous investigations; large relative submergence will result in less

wave breaking than srnaIl relative submergence as observed during testing.

The plots support the results presented by Beji and Battjes (1993), showing the effect of period

on the transformation process and the effect of wave breaking on the spectral transformations.

ï h e transfer of energy to higher frequency wave components is seen to increase considerably

with increasing period as indicated by comparison of Figures 6.5b and 6.Sd. The effect of wave

breaking is not so pronounced however, as indicated by comparison of Figures 6.5a and 6 3 ,

where the amount of energy in the higher frequency components is similar. The effect of the

wave brealung is evident in the reduction of energy near the peak frequency. The increase in

higher frequency energy between Figures 6 . 5 ~ and 6.5d is expected to be due to the non-linear

wave transformations and harmonic generation rather than the wave breaking process.

The effect of crest width on spectral transformations is also evident in the case of both breaking

and non-breaking waves. It is evident that the increased crest width results in a reduction of the

transmission coefficient in both breakmg and non-breaking conditions. This is possibly due to a

number of processes:

(1) turbulent hction, especially in the case of breaking waves;

(ii) harmonic generation, in the case of breakmg and non-breaking waves; and

(iii) energy loss due to flow within the breakwater armour, especially in the case of srnall

subrnergence depths.

It is important to note that the reflection coefficient is relatively unaffected by the change in crest

width as will be noted in Section 6.1 .S. Figure 6.6 shows the effect of the crest width on the

spectral transformation for particular tests with breaking (dm = 1 .O) and non-breahng (djH, =

4.0) waves.

It is evident fiom these figures that d/H, and B are important parameters in defining the extent of

spectral transformations at subrnerged breakwaters and therefore will strongly influence the

transmission coefficient K, . The importance of these variables on the transmission process will

be discussed firrther in following sections.

(a) - Non-Breaking H, = 0.05 m;Tp = 1.2 s ; d/H, = 4.0

(b) - Breaking Hl = 0.05 m;TP = 1.2 s; d m , = 1.0

( c ) - Non-Breaking Hl = 0.05 m;T, = 2.0 s; dm, = 4.0

(d) - Breaking H, = 0.05 m;Tp = 2.0 q dm, = 1 .O

Figure 6.5 Effect of Period (T,) and Breaking Influence ( d m on Spectral Transformations

INCIDENT H,(m) : 0.058 SPEC T, (s) : 2.133

TRANSMïlTED H, (m) : 0.047 - - - - - - - - - - SPECT, (s) : 2.133

0.0 4.0 Frwency

(a) - Non-Breaking B = 0 3 m; H, = 0.05 rn; d m , = 4.0

tNCIDEM HM, (m) : 0.05 1 - SPEC T, (s) . 2.133

TFUNSMI77ZD H, (rn) : 0.034 - - - - - - - - -- SPEC T,, (SI : 2.133

MCIDENT H,(m) : 0.050 SPEC T, (s) : 2.133

TRANSMiTlED H,(m) :0.032 - - - - - - - - - - SPEC T, (s) : 2.133

(b) - Breaking B = 0 3 rn; H, = 0.05 m; d p , = 1.0

INCIDENT H,(m) : 0.048 SPEC T, (s) : 2.133

TRANSMITTED H, (m) : 0.0 10 - SPEC T, (s) : 2.133

( c ) - Non-Breaking B = 3.5 m; 18, = 0.05 rn; d/H, = 4.0

(d) - Breaking B=35 m; H,=O.OSm; d,/H,= 1.0

Figure 6.6 Effect of Crest Width (B) and Breaking Influence (dJ on Spectral Transformations

6.1.4. Transmission

Although previous investigations have defmed numerous physical processes important in

defming the transmission characteristics of submerged breakwatm, there is not a clear

consensus as to which variables are most important. Therefore, the effects of various incident

wave characteristics and structure geometries on transmission in the 2-D tests were investigated

graphically to identi@ the variables which have most influence on the transmission process.

This is a simple approach to defining the variables which should be considered for further

analysis. The effects of various dimensionless variables on K, are investigated in Chapter 7.

The transmission coefficient K, was ploned as a function of submergence depth, incident wave

height, incident wave period, crest width, breakwater dope and amour diameter. Selected plots

are s h o w in Figures 6.7 through 6.11 ; additional plots of these variables are provided in

Appendix E. Inspection of the figures generally reveals the trends suggested in previous studies,

with the exception of the effect of D,. The presmt test resuits indicate a slight increase in

transmission as the armour size is increased.

Su bmerged Breakwater Transmission

Figure 6.7 Effect of d, and H, on K,

-- - - -- -

Submerged Breakwater Transmission Kt vs ds (B=0.6 m, Hs - 0.10 rn )

Figure 6.8 Effect of d, and T, on Y

Su bmerged Breakwater Transmission Kt vs B (dsz0.05 nt, He - 0.10 m)

Figure 6.9 Effect of B and T, on K,

Submerged Breakwater Transmission Kt vs Slope (ds=0.05 m, Hs - OdO m)

Figure 6.10 Effect of 8 and T, on 6

Submerged Breakwater Transmission Kt vs US0 (ds=0.05 m, Hs - 0.10 m)

-

Figure 6.1 1 Effect of D, and T, on Y

Although some of the trends are very weak, the effect of submergence, significant wave height

and crest width are relatively strong. The effect of wave period (or wavelength) is evident but

does not appear to be large. The effect of breakwater slope and arrnour diameter are not as

evident and gwen the variabili~ in the plots, it is not possible to visually confirm the presence of

a trend. Statistical analysis of test variables is more appropriate for detennining the significance

of trends arnong these parameters. This is discussed M e r in Chapter 7.

6.1.5 Reflection

Reflection fiom submerged breakwaters is important to the transmission process because it

reduces the wave energy available for transmission. It is therefore important to consider the

effect of the various testing variables on the measured reflection coefficient if the individual

physical processes are to be understood. In effect, the definition of the transmission coefficient

K, is a representation of the Iosses due to energy dissipation and reflection at the breakwater.

Conservation of energy suggests that the incident wave energy is reflected, transmitted and

dissipated such that:

where E, E,. Er and E, are transmitted, incident, reflected and dissipated energy measures

respectively. Therefore, K, (equivalent to Et '/ E, ") can be expressed as a function of the

reflected and dissipated energy as follows.

Consideration of the reflection coefficient K, is therefore necessary to fully understand the

transmission process.

Selected plots of the reflection coefficient as a function of various test variables are presented in

Figures 6.12 through 6.17; additional plots are included in Appendk F. The plots indicate that

the reflection coefficient is most sensitive to the submergeme depth and the wave period, but in

general, does not v q as widely as the transrmssion coefficient. The reflection coefficient fiom

the test breakwater remained between approximately 5% to 30% for al1 2-D tests.

Su bmerged Breakwater Reflection Kr vs do (8.0.6 ml Tp - 2.0 s)

Figure 6.12 Effect of d, and E, on K,

1 Submerged Breakwater Reflection Kr vs B (ds=0.05 ml Hs - 010 m)

Figure 6.14 Effect of B and T, on K,

Submerged Breakwater Reflection

- ppppp

Figure 6.13 Effect of d, and T, on I&

Su brnerged Brea kwater Reflection Kr vs Siope (ds-0.05 m, Hs - 0.10 m)

.. . .

Figure 6-15 Effect of 0 and T, on y

Submerged Braakwater Reflection Kr vs D5O (ds=O.05 m, Hs - 0.10 m)

Figure 6.16 Effect of D , and T, on K,

It is important to note that the 1" probe array is situated immediately in fiont of the breakwater.

This location muiirnizes the effect of reflection from the plywood beach on the test results.

Therefore, the measured reflection coefficient can be assumed to be primarily due to the test

breakwater. Although a srnail amount of the reflection measured in front of the breakwater may

be originating fiom the end wall of the flume, (ie. reflection of the transmitted wave) passing

back over the breakwater, it is difficult to separate these effects. nimefore, for the purposes of

analysis in this study, the reflection measured in fiont of the breakwater will be assumed to be a

result of the breakwater obstruction only.

6.1.6 Cornparison with Existing Design Equations

In light of the above discussions, it is evident that the design of subrnerged breakwaters should

include consideration of the depth of submergence, incident wave height and the breakwater

crest width as prirnary parameters. The existing design equations developed by Seelig, Ahrens

and van der Meer do in fact incorporate al1 of these parameters, but their innuence is based on

testing for a limited range of breakwater configurations. In particular, crest width has not been

varied over a wide range in previous tests. The general expected pnformance of these equations

for a variety of design conditions was discussed in Section 2.2.3.

A cornparison of these equations with the present test data is given in this section. Since Seelig 's

equation was developed fiom such a mial1 amount of test data and does not perform well outside

a very small range of input conditions, only Ahrens' and van der Meer's equations are

considered further.

(a) Ahrens' Equation

The design equation derived by Ahrem (1987), as previously noted, is specifically developed for

predicting the transmission past reef-type breakwaters and, therefore, cannot be expected to

predict K, well. However, the equation was fond to be well botmded, and in general, the trends

predicted are consistent with existing theones. The predicted Kt is compared to obsmed Kr for

the present test data (with D, = 0.059 m and a front face dope of 1: 1.5 ) in Figure 6.17. The

data are plotted for the five crest widths tested.

The results indicate that the equation predicts K, well for narrow crest widths but in general,

underestimates the transmission for wider smctures. This is not surprising since the crest width

is not expticitly accounted for in the equation. Given the large degree of scatter in the plot for

the wide crest structures, it seems possible that the equation misrepresents the effects of a

number of the influencing variables. The underestimation of K, for wide structures would be

consistent with a misrepresentation of the transmission through the armour of a conventional

rubblemound structure. Although it is not possible to isolate the various sources of the

discrepancies between the estirnates and the observed data, it is suffïcient to show that Ahrens'

equation does not represent the present test data well.

Corn parison of Test Data and Ahrens' Equation

- - - -

Figure 6.17 Ahrens' Equation with Present Test Data

(b) van der Meer's Equation

As noted in Section 2.2.3, the design equation developed by van der Meer (1991) was based on

the results of tests by many authors, including Ahrens, and under various input design conditions

remains relatively welI bounded for narrow crest widths but for larger widths, it gives

unreasonable estimates of K,. Transmission coefficients fkom the present test data are compared

to those predicted by van der Meer's equation in Figure 6.18.

The cornparison is relatively good for the narrow structures, although under certain conditions,

van der Meer's equation predicts transmission coefficients greater than unity. This could be

because some of the present test conditions produce parameters outside the range used in the

development of the design equation. As expected, transmission over wider structures is not

predicted accurately because the equation produces negative values of K,. The strong linear

grouping of the coefficients does suggest that the equation may represent the physical processes

relatively well except for those processes related to crest width. It is also evident that the

equation tends to overestimate KI when Kt is large, indicated by the upward curvature of the

plots. This effect could be due to improper representation of transmission mechanisms which

become more important under conditions which result in higha transmission coefficients.

Corn parison of Test Data and van der Meer's Equation

I

B = 0.3 m O

B = 0.6 m A

B = 1.5 m x

B - 2.5 m w

6 = 3.5 rn - Meas. = Obs.

- - - - - - - - - - -

Figure 6.18 Van der Meer's Equation with Present Test Data

6.2 Basin Testing: Observations and Trends

6.2.1 General

As in the 2-D tests, the incident wave conditions were f is t tested without a breakwater structure.

The testing without structures provided some insight hto ihe effects of the test setup on the wave

generation and provided background data to assess the effects of the breakwater structures on

such processes as beach development, wave difkction and wave induced velocities.

Interpretation of the 3-D testmg results m u t take into cons iddon the complexity of the

wave/stmcnire interactions and the physical differaices betwem the basin and flume testing

conditions. The 3-D setting results in a rnultidirectional reflection phenornenon as well as

diffraction around the breakwater heads. The wave analysis package is not capabIe of

separating incident and refiected waves in such a testing condition and therefore. reflection

analysis was not performed in the basin. Visual observations are also more difficult to assess in

the basin tests due to the wave interactions.

It is also important to consider the effects of the water depth in the basin on the testirig results.

Incident waves in the 2-D tests were generated in about 1 .W m of water and were shoaled to the

breakwater over a 1 : 10 beach slope. Therefore, there was very little wave breakmg offshore of

the breakwater and the assumption of irrotational linear motion in this region may be considered

acceptable. In the basin tests, however, the waves were generated in about 0.3* m of water and

there was considerable wave breaking between the wave paddle and the breakwater structure.

The nature of the wave energy and velocities under these conditions cannot be assumed to be

irrotationa1 and linear. Furthemore, the physical interactions between a breakwater and incident

wave series under these conditions may differ fiom that in the 2-D setting due to the change in

vertical energy and velocity distributions within the wave set.

These factors may account for some of the differences between the 2-D and 3-D transmission

responses. Although the 3-D tests possibly represent a more realistic design condition, the

inability to separate the incident wave from reflected and refiacted components makes it dificult

to define the transmission accurately. The consequences of these dificulties will be discwed

further in Chapter 7.

6.2.2 T i e Series Transformations

The effects of the submerged breakwaters on wave breaking and harmonic decomposition of the

incident wave irain were visually evident and apparent in the t h e series data. These effects are

not as clear as in the 2-D tests, however, due to the contribution of wave energy fiom diffraction

and reflection in the 3-D setting. It is also important to note that the distance between probes

measuring incident and tranmitted waves is longer in the basin than in the flume, causing a

larger shift in the phase between records. A typical the series plot for a basin test is s h o w in

Figure 6.19. The decomposition effects of the breakwater are much less evident in this plot

when compared to Figure 6.3 but visual observations c o n h e d the harmonic generation

process.

Typical Wafer Level Record 3-0 Tesfing : Irregular Waves

o. 1 - E =O.OS

r Q

" 0 b Q) - rn s-0.05 i f 1 . l

l

1 I I I -0.1 -,

O 10 20 30 40 50 Tïm e (s)

Figure 6.19 Tirne Series - Basin Test

H,=O.O63 m ; Tp=l .6s ; h=O28m; P=90°

Overall, the time series data provided littie information as a result of the cornplexity of the wave

field.

6.23 Spectral Transformations

Inspection of the measured spectra at the various locations behind the submerged breakwater

shows the spatial effect of the structure on reducmg wave energy. in general, the spectral

broadening behind the structure is evident and is more pronounced immediately behind the

breakwater than near the head. When the incident waves are not 90" to the structure, the greatest

spectral decomposition shifts away fiom the breakwater head which is subjected to the most

direct incident wave attack Therefore, it is evident that the difficted waves are significantly

influencing the wave conditions in the protected zone. A typical set of spectra for a 3-D test is

shown in Figure 6.20.

6.2.4 Transmission

The ûansmitted wave height in the 3-D tests is a function of the location behind the breakwater.

As indicated in the previous section, the broadening of the spectra is generaIly most evident

where the contributions fkom diffiaction are rninimized. Therefore, the transmission coefficient

is expected to be minimized in these areas as well. As discussed in Section 6.2.5, the difhction

patterns were not readily evident during testing, and therefore, it has been assumed that the

difhction will behave in a sirnilar (but not identical) rnanner to that observed and documented

for surface piercing breakwaters (US. h y Corps, 1984).

Transmission coefficients were computed for al1 wave gauging locations behind the breakwater.

The effects of relative submergence, crest width and incident angle on K, at two probe locations,

are shown in Figures 6.21 and 6.22 below. Sunilar plots for al1 probe locations are provided in

Appendix G. As previously noted, K, is reduced with increasing crest width and with decreased

diffiaction effects, assurning dif ict ion patterns are sunilar in nature to those observed for

surface piercing breakwatm. No specific physical relationship relating Kt to spatial

parameters has been attempted at this tirne as it is beyond the scope of the present snidy. It is

sufficient to note that fiirther investigation regarding this aspect of the design is necessary and

that under specific design conditions, physical modelling will most likely be necessary to address

this issue.

Cornparison of K, as a fùnction of djH, for 2-D testing (see Figure 7.1) and 3-D testing (see

Appendix G) show similar trends with respect to d/H, aIthough transmission coefficients greater

than 1 .O were recorded in the 3-D tests. Th~s is possibly due to the combined effects of

transmission, diffraction and reflection in the Lee of the breakwater. The increase in K, with

decreasing B in the 3-D tests is also evident when Kt is relatively mal1 but is not always evident

when Kr is large.

3-D Testing : Submerged Breakwaters Effect of Crest Width (90 Deg ; P2)

f 4 -- ---------------------------

Figure 6.21 K, for 3-D Tests at 90" Incidence

3-D Testing : Submerged Breakwaters Effect of Crest Width (60 Deg ; Pa)

Figure 6.22 K, for 3-D Tests at 60" incidence

6.2.5 Diffraction

ûriguially, it was anticipated that difhction patterns might be obvious enough

assess qualitatively in an attempt to define the effect of spatial variation behind

to document and

the breakwater

on Kr. It quickly became evident during testing, however, that the complexity of the wave

conditions in the basin prohibited such an attempt. Some diffraction was observed during

testing, eçpecially with lower submergmce depths as would be expected.

It is expected that, in general, the difhction patterns b e h d the submerged breakwater will be

somewhat simiIar to those behind a conventional rubblemound structure. Although the general

pattern may be similar, the degree of wave height reduction and the magnitude of change in

direction of the propagation vector will be considerably smaller. The contribution of îhe

diffracted energy to the transmitted field is expected to increase the transmission coefficient and

therefore, areas expected to exhibit low levels of diffracted energy are expected to exhibit lower

Kr values. This statement is a generality and is intended only as an observation, put forih for

fùrther consideration.

6.2.6 Beach Development

Observations of beach development during testmg also serve to confÏrm the effects of difiction

in the shidy. in cases where offshore d a c e piercing breakwaters have been used to protect

beaches, the formation of tombolos often occurs. These are projecting h g m of the beach

which may ultirnately join the offshore breakwater, cutting off longshore transport. The

formation of these structures is the result of reduction of local longshore transport and the

transport effects of locally diffiacted waves. The onset of simdar foxmations were noted during

the 3-D testing. Rojecting beach f o m and adjacent ripple formations show the effect of the

diffracted wave field.

During the 90 O incidence tests, the beach development behind the breakwater was relatively

syrnrnetrical with respect to the structure. At high water levels, the final beach form was less

distinct than at low water levels and was assumed to be due to the reduced impacts associated

with larger submergence depths. As well, the beach forms initially developed as two separate

projecting formations, one at each end of the breakwater, ultimately developing one larger

projection at the rnid-point of the breakwater by the completion of the tesfing. A typical beach

development after testing at 90" incidence is shown in Figure 6.23.

Initially, the diffracted waves moved the sand into the protected areas fiom the ends of the

beach. As the tests progressed, this process continued until the formation was centred behind the

breakwater. Given the limited sand volume on the beach, the formation is limited in size. It is

very important to consider that this observation is qualitative in nature only. There was no effort

to minimize scale effects associated with the sand transport phenornenon and, as a result, only

the general nature of the beach development can be considered to be representative of prototype

conditions.

1 Figure 633

Final Beach Formation - 3-D testing at 90' Incidence

Testing at 60" incidence resulted in a considerably different beach development At large

submergence depths, there was vexy little deveIopment of beach formations; instead, the majority

of the sand was washed off the beach in the direction of incident wave propagation. When the

submergence was small, however, a projecting formation was developed. The location of the

formation moved downdrift as the depth of submergence was increased. This formation was

developed with sand eroded from the updrifi end of the beach and deposited behind the

breakwater. Under smaller submergence, it appeared that reduced wave energy behind the

structure prevented migration of the beach form downdrifi. Typical beach development after

testing at 60" incidence is shown in Figure 6.24.

Waves diffracted around the downdrift head of the breakwater developed bedform ripples on the

88

downdrïfi beach. indicating a tendency for the diffraction process to stabilize the beach

formation to some degree, further inhiiiting its movemt downdrift. Again, it must be

recognized that in prototype conditions, the process would behave much differently. One

significant influence in the prototype would be the effect of longshore transport which is not

fully developed under the present testing conditions.

It is interesting to note, however, that the location of the beach form changes with changes in

submergence depth, providing incident waves are not 90' to the breakwater. Therefore, it is

possible that dynarnic water level and wave atîack conditions would prevent the development of

a permanent tombolo-type formation.

Figure 6.24 Final Beach Formation - 3-D Testing at 60' Incidence

6.2.7 Velocity Patterns

A comprehmsive analysis of the velocity patterns behind submerged breakwaters would be a

sipificant task and is beyond the scope of the present objectives. A prelimuiary inspection of

the basic trends was undertaken, however, to complement the 3-D transmission data. The

velocify data presented in this section was sampled at a depth of 0.2h.

Assuming linear wave properties, the mean particle velocity of a wave train in deep water would

be zero. in reality, however, there is mal1 mean velocity in the direction of wave propagation

associated with a net mass transport (Kamphuis, 1995). In shalow water where the local

bathymetry affects the direction of wave propagation and specrtral characteristics, the velocities

will be influenced accordingly. Such transformations occur in the region of a subrnerged

breakwater and as a result, the velocity patterns will be affected. In general, the mean velocity

vectors associated with the 3-D tests were quite variable fiom test to test but t h e were some

consistencies worth noting.

For al1 tests, the direction of the mean velocity vectors was relatively consistent. The velocities

along the toe of the concrete beach were typically directed offshore and this effect was more

pronounced in areas which were less protected by the breakwater. This velocity is possibly the

result of reflection of long waves from the beach, an effect which is not strong in the lee of the

breakwater since these waves are dissipated or reflected by the submerged breakwater.

The velocity probes located approximately mid-way between the toe of the beach and the

submerged breakwater showed less consistent results, especially in the case of 90" incidence

(see Figures 6.25 and 6.26). The probe located cIosest to the head of the breakwater was the

most consistent, and typically showed a mean velocity into the protected area. This could be

associated with the effects of difhction on the direction of wave propagation in this area. The

probes located in the protected area were less consistent when more protection is provided (ie.

with lower submergence or a wider crest width). This is likely because under more protected

conditions, srnail fluctuations have a !arger effect on the mean velocity.

~ i ~ u r e - 6 2 5 Mean Velocity Vectors at 02h

AU Wave Conditions at Q = 90"; h = 0.19 rn

Figure 6.26 Mean Velocity Vectors at 0.2h

Ail Wave Conditions at P = 90"; h = 0.28 m

Figiire 6.27 Mean Velocity Vectors at O.2h

Al1 Wave Conditions at P = 60"; h = 0.19 m

Figure 6.28 Mean Velocity Vectors at 0.2h

Ali Wave Conditions at P = 60'; h = 0.28 m

The effect of submergence was evident in testing at 60' incidence as well, as shown in Figures

6.27 and 6.28. The tests at low water (h4.19 m) showed mean velocity directions which are

more consistent with the assumed diffiction effects than tests at high water (h=0.28 m). The

probes dong the updrift end of the beach contuiued to exhibit an offshore trend.

The relative submergence appears to be a significant factor with respect to the mean velocity

behmd the breakwater for testing at both 90" and 60"; the effect of cmt width is not as evident.

It is important to realize, however, that these observations are based on only the mean particle

velocity, and consider only a small portion of the test data. Consideration of a11 test data and its

spectral characteristics would be necessary to provide any conciusive comments.

7.0 Analysis of Results

7.1 Transmission as a Function of Dimensionless Variables

The development of a design equation for K, has been approached k o u g h graphica1 and

statistical analyses of the test results. AIthough a design equation can be constructed directly

fiom graphical analyses, it becomes cumbersome when numerous independent variables are

concerned. Therefore, this chapter presents graphical analyses of a number of dimensionIess

variables defined on the basis of the test results, as well as dirnensionless variables put forth in

previous investigations, in order to provide a qualitative assessment of the transmission process.

This assessment provides a preliminary understanding of the most important variables and

provides a base to build upon with statistical analysis techniques. The statistical analysis

techniques are discussed in more detail in Section 7.2.

7.1.1 Basic Dimensioniess Variables

Previous studies investigating the transmission process at submerged breakwaters have identified

some of the more important variables. As noted in t2.151, it is s h o w that:

This statement is generally supported by the results obtained fiom 2-D testing in this study, with

the possible exception of the D,, trend.. More importantly, however, it has been indicated that

the depth of subrnergence d, is possibly the most important factor. Inspection of the test results

in Appendix E supports this observation and also shows that H, and B are relatively important

parameters as well.

The expression of a design equation for Kt requires that the physical processes be expressed in

t m s of dimensionless variables. AIthough the most effective form of these dimensionless

variables is not irnmediately hown. the basic dimensionless variables derived from an

expression of al1 the independent variables influencing the transmission process (reiterated tiom

[2.17]) are:

As previously indicated, those variables noted in 17.21 may or may not be physically relevant.

Therefore. it is customary to combine a number of these variables together to form alternative

dimensionless variables which are physically relevant. The formation of these relevant

dimensionless variables requires some pnor knowledge of the transmission process and the

factors which are most influential. This pnor knowledge is provided in part by the results of

previous investigations and by the observations noted in Chapter 6.

Conventional dimensional analysis techniques could be used to develop a design equation on the

basis of linear analysis of Log-Log plots for various combinations of the dimensionless

variables. The analysis undertaken in this thesis, however, considered various dimensionIess

variables in graphical form to assess their ability to uniquely defme K,. Non-linear regression

analysis was then undertaken assuming a variety alternative design equations in various forms in

an effort to maximize the fit of the equation while controlling the K, estimate limits.

Considering expressions [7.1] and [7.2], it seems reasonable to assume that (dm) will have

some physical relevance to the transmission process. Assuming that wave msformations on

the face of a submerged breakwater are similar to those on a beach, the relative submergence is

analogous to the solitary wave breaking criterion where the ratio of the breaking wave height to

the depth of b reahg (Hfl,) is approximately 0.78. or to wave breaking criteria for irregular

waves brealung on a beach where the ratio of HS#, is 0.56e3'" (where H, is the significant

breaker height and rn is the beach slope)(Kamphuis, 2991). Aiternatively, a similar relationship

describing the stable wave height on a shelf (Horikawa and Kuo, 1966) suggests that the ratio of

the stable wave height to the shelf depth (Hjd,) is between 0.35 and 0.4. Although the exact

form of this breaking relationship is not h o w n for the case of a permeable. steeply sloping

submerged breakwater, the dependence of Kr on d/H, is easily shown.

The relative submergence was therefore used as a preliminary plotting variable to consider the

effects of the other independent variables. Plotting the 2-D testing results as a function of the

relative submergence (Figure 7.1) confirms its importance as a defining variable. There are two

very important observations that can be made fiom the &ta in Figure 7.1. First, although the

data groups very well using d/H, as a defining variable, the data is in a relatively wide band,

indicating that there are other independent variables which significantly affect the observed K,.

Second, there appears to be a considerable change in the plot characteristics around 1 .O 5 d/H,

2.0. This would indicate a possible change in the physical processes which are rnost important in

defrning Kr. Below this range of d f i , the relative submergence seems to have a considerable

influence on y, while beyond this range, the influence is reduced.

Assuming that wave breaking is the only process affecting wave transformations at a submerged

breakwater, and furthemore, assuming that the solitary wave depth limited breaker criterion is

appropnate for submerged breakwaters, then the maximum transmitted wave height would be

equal to breaker height defined by the depth of water on the breakwater crest. It is worth noting

that the irregular wave breaking criterion or the relationship for stable waves on a shelf could

also be used to show a relationship between K, and d f i . Although the numerical constants of

the relationship would be slightly d i f f m t , the general cornparison is just as valid. Therefore. in

terms of the solitary wave breaking criterion, it can be stated:

given Hb = 0.78 hb where Hg = breaker height; hb = depth of breaking

assuming (81 Hb = H, (ii) hb = d, for submerged breakwaters : H, = 0.78 d, providing d, forces breoking

and as a result:

2-D Submerged Breakwater Tests Kt vs. ds/Hi

Figure 7.1 Effect of Relative Submergence

For the case of irregular waves, ignoring for the present time the effects of groupiness, setup and

wave interaction with the r e m flow evident in 2-D tests. the presmce of a range of incident

wave heights complicates this expression. In this case. H, and H, represent a characteristic

(typically significant) wave height for the incident and transmitted spectra.

When d/H, is very srnall, the majority of the incident waves will break and the condition rnay

approach that of a series of solitary waves. If breaking were the onIy factor influencing K,, the

expression presented above may then be appropnate. As d/H, increases, however. fewer waves

break and the influence of d e , is reduced, and at some point where only a few waves break, Kr

would be virtually independent of d p , . An approximation of this point c m be made assurning a

Rayleigh type incident spectra and assuming that an incident wave will not break if:

Given that H, (or Hm, in the case of a Rayleigh distribution) provide an approximation of the

average of the highest 1/3 of the waves in a given wave series, it rnay be assumed that if the

significant wave height is not broken at a submerged breakwater, then the majority of the waves

in the train will pass unbroken. As a result, the transmission coefficient will be relatively high

and increasing the water depth at the breakwater will not significantly affect the number of

waves breakmg, and therefore, should not significantIy influence K,. If this were m e , the

influence of the relative subrnergence ( d f i ) on K, would be expected to be reduced beyond the

value of d/H, which causes breaking of the significant wave height.

Therefore, where H, is represented by Hi in [7.4], it can be stated that Kr should not be

significantly influenced by a change in relative submergence where:

This observation is somewhat consistent with the data presented in Figure 7.1. Realistically,

however, the breaking influence is not the only factor governing K, and therefore, at smalI d B ,

values, the dope of K, vs. dm, is not expected to be exactly 0.78 as predicted by the solitary

wave breaking cnterion or 0.4 as indicated by relationship for the stable wave height propagating

on a shelf. Furthermore, there is no single point where the influence of wave breakmg is lost.

The influence of the other dimensionless parameters of 17.21 are considered graphicaIIy in the

following figures. Because a number of the variables are not easily controlled during testing,

many of the dimensionless variables considered are defined over specific ranges. The size of the

range chosen for plotting will influence the grouping of the data and, therefore, range sizes were

selected to ensure sufficient detail.

Figure 7.2 shows the variation of Kr with (pH,(H&')/p, a form of Reynold's number (RJHJ

based on the incident wave height. This form of Reynold's number incorporates a velocity t e m

(g-HJ" proportional to the orbital velocities (based on linear wave theory), and therefore

represents the ratio of inertial forces associated with the waves to the viscous forces of the fluid.

This variable does not appear to have any discemable influence on K,. The grouping of the

various Reynolds number ranges along the d/H, a i s c m be amibuted to the common

occurrence of H, in both of the independent variable terms.

2-D Submerged Breakwater Tests Kt vs. dslHi and Re(Hi)

Figure 7.2 Ef fe t of Wave Reynold's No.

Figure 7.3 shows the effect of the dimensionless amour size ( D 5 H , ) on K, As in Figure 7.2,

rhere is no defînite trend associated with the variation in D-/H, and the grouping along the d/H,

axis again is due to the effect of H, in both independent variables. It is difficult to determine if

D, /H, has any physical relevance in the transmission process at submerged breakwaters. It is

antîcipated that the armour diameter may affect processes related to fictional and drag resistance

as water flows across and w i t h the amour layer. However, the innuence of H, in such

processes may be less important than other variables whch couid be used with D, to fom a

representative dimensionless variable. The plot indicates that any such relationships are not well

represented by this variable.

2-D Submerged Breakwater Tes1 Kt vs. ds/Hi and DSOalHi

Figure 7.3 Effect of Dimensionless Armour Sue

The effect of the breakwater porosity (n) was not investigated in this thesis. Although it may

have some influence on Kt when d, is small, Seelig (1 980) suggested that for submerged

breakwaters (especially for Wh, 2 1.2) the transmission through the structure is negligible.

Therefore. it is expected that this omission will not significantly affect the resulting equation.

The effect of the dimensionless local wavelength (WHi) (based on linear wave theory) is shown

in Figure 7.4. This variable is in fact the inverse of the local wave steepness. Although previous

authors have indicated that the transmission at submerged breakwaters increases with increased

period which is proportional to wavelength, it is not evident in this plot. The plots of

dimensional variables in Appendix E show a very weak trend with wave penod and this small

effect appears to be lost in the formulation of this dimensionless variable. Nevertheless, it is

apparent that the relative effect of wavelength is small:

2-D Submerged Breakwater Kt vs. ds/Hi and LIHi

Tests

Figure 7.4 Effect of Dimensioniess WaveIength

Figure 7.5 shows the influence of the breakwater slope on the îransmission coefficient. The plot

shows the data fiom the various breakwater structures to be well mixed and, there fore, there

appears to be no effect. The breakwater slope does have a mal1 effect on the reflection

coefficient as shown in Appendix F, but in cornparison to K,, Kr is relatively constant over the

range of variables tested. Thmfore, the negligible influence on the test data is not surprising.

The effect of the crest width on K, is shown in Figure 7.6. A definite trend is evident in this

plot, with increased transmission associated with the smaller relative crest widths. The

suitability of the dimensionless parameter B/H, is questionable. It is possible that this term could

be representative of an overtopping effect, with low B/Hi conditions resulting in waves breaking

into the lee of the breakwater and high B 4 conditions resulting in waves breaking onto the

nest. In non-breaking conditions, however, the significance of this term is not obvious.

Alternative dimensionless variables relating to crest width are discussed in following sections.

2-D Submerged Breakwater Tests Kt vs. dslHi and Cot Theta

Figure 7.5 Effect of Breakwater Slope

2-D Submerged Breakwater Tea Kt vs. dslHi and BIHi

Figure 7.6 Effect of Dimensionless Crest Width

The effect of relative breakwater height and relative water depth are indicated in Figures 7.7 and

103

7.8 respectively. Similar to Figures 7.2 through 7.5. there is no evidence of any significant

relation between these variables and K,. Although the structure height (h,) undoubtedly has an

effect on the transmission and perhaps more importandy on the reflection process, the shucture

height with respect to the water depth (hW would be a more dennitive variable than hjH,.

Therefore. the influence of the breakwater height has been lost to some degree in this

dimensionless variable.

It is important to note that the structure height was not varied diauig the tests. and furthmore,

the structure height was always significant with respect to the water depth. This condition is

expected to be the reason for the relatively constant reflection coefficients. The fact that K, for

the 2-0 tests reached a maximum of about 0.9& when relative submergence was large and

relative crest width was mal1 is consistent with the measured reflection coefficients of about

10% under these conditions.

2-D Submerged Breakwater Kt vs. dslHi and hslHi

Tests

- - - -

Figure 7.7 Effect of Breakwater Height

2-D Submerged Breakwater Tests Kt vs. dslHi and hIHi r

Figure 7.8 Effect of Water Depth

The relative water depth (h/X3 is to some degree an indicator of the non-linearity of the wave

field if the water depth is relatively shallow, although non-linear effects are better represented by

alternative variables as discussed in the following sections. When the water is deep with respect

to the wave height, however, this variable has no physical significance. As indicated in Figure

7.8, there is no apparent relation betwem Kt and h 4 .

Considenng the basic dimensionless variables plotted in the figures above, it c m be seen that

some form of the relative submergence and some form of the relative crest width are important

in defning a design equation for K,. It is also possible that those dimensionless variables which

were not seen to have any significant effects. could have some influence if expressed in an

altemate dimensionless forrn. Therefore, a nurnber of alternative dimensionless variables

associated with the relative submergence and crest width are discussed in the following sections.

It is assumed that transmission is due to the combined (additive or multiple) effect of a number

of rnechanisms which permit aiergy transmission over the breakwater crest or through the

material of the structure. The following variables are suggested, based on the assumption that

they are representative of individual components of the overall energy transmission mechanim.

Variables considered to be representative of the effects of perturbation due to obstruction (surf

similarity parameter), effects of overtopping (or wave breaker location), effects of hannonic

generation and the effects of niction due to flow across the breakwater and drag due to flow

within the permeable structure have been considered in the following discussion. Al1 variables

have been shown to be consistent with the basic dimensionless variables presented in [2.183.

Those variables considered to be most promising are considmd for staîistical anaiysis in Section

7.1.2 Effect of Perturbation due to Breakwater Obstruction

The effect of perturbation caused by the submerged breakwater on the incident wave breaker

characteristics was investigated by Ham, Yasuda and Sakakibara (1992). A modified surf

similarity parameter ( 5," ) was defined for the interaction of solitary waves with submerged

trapezoidal breakwaters. This parameter appears to incorporate the effects of the relative

submergence and the crest width in its formulation.

Since this parameter was developed on the basis of solitary waves, some discrepancy with the

present experimental results is expected givm the dynamic water level fluctuations and

velocities associated with irregular waves and the related effects on breaking characteristics.

Nevertheless, the parameter was plotted as a function of K, in order to assess any evident trends.

The results are presented in Figure 7.9.

The plot shows that there is a good visual correiation between the transmission coefficient and

the modified surf similarity parameter aiîhough the data does f o m a relatively broad band. This

parameter is included in the statistical analysis in Section 7.2 in order to quantitatively assess its

ability to predict Kr.

Submerged Breakwater Transmission Kt vs Surf Sim ilarity Parameter

O. 5 1 1.5 2 2.5 3 3.5 SSP (Hara,Yosada end Sekakibura, 1992)

Figure 7.9 Perturbation Effect on Y

7.13 Overtopping Effect

Transmission at conventional srnface piercing breakwatm is largely due to the effect of waves

overtopping the structure. Observations of wave breakmg at submerged breakwaters with maIl

submergence depths during the present testing program indicate that transmission due to wave

breaking across the structure is an important factor for these structures as well. in the case of

wider structures, where the wave breaks somewhere along the crest of the structure, it is still

possible that the location of the breaker can be an important variable since the hydrodynarnic

regime changes visibly at the breaking point.

Therefore, two dimensionless variables whch are assumed to be representative of this

overtopping effect have been considered. The fist variable simply relates the breakwater crest

width to the incident wave height (B&) and therefore, does not account for the variable effects

of local depth and wavelength on the wave breaking process. The considerable effect of the

local depth on the breaking process m u t be isolated in order to graphically assess the

relationship between K, and MI,. This variable is one of the basic dimensionIess variables

presented in [7.2] and K, was plotîed as a fimction of B/H, in Figure 7.6.

The figure shows that there is a strong relationship between Kt and B/H, when the depth of

submergence is small as would be expected since this is the condition under which the majority

of wave breaking would be taking place. Further analysis of this variable (B/HJ is undertaken in

Section 7.2.

A more complex dimensionless variable describing the breaker location for solitary waves

incident to submerged rectangular dikes was developed by Hara, Yasuda and Sakikabara (1993).

The dimensionless b r e h position was defined in terms of a characteristic variable ( p ') such

that:

Xb where - = 1 0 . 7 8 ~ ' - 4.14 0.6 5 r 1.03 h

- xb - - 1.94~' + 5.01 1.03 s P' i 2.2 h

Using these relationships, where X, is the distance from the top corner of the seaward edge of the

breakwater to the breaking point, another dirnensionless breaker location parameter can be

developed for X,/. sirnply by multiplying those above by WB. It was assurned that the range of

p 'over whch these relationships are valid could be extended to 0.32 s p ' s 1 .O3 to include the

entire present test data set. It was also assurned that the reiationship is equally valid for those

waves breaking on the breakwater as it is for those breaking behmd it. These assumptions

cannot justified but are made only in order to assess the potential of this existing variable to

define Kr.

Figure 7.10 shows K, as a function of X@, and it is evident that there is a relatively strong

correlation between the variables. Sepration of the data on the basis of d, was not undertaken

for this plot as it is included inherently in the h / h term in the development of the parameter p '.

This parameter is investigated statistically in Section 7.2, bearing in mind the assumptions made

here in order to extend its range of application.

2-D Submerged Breakwater Tests Kt vs. dsllii and XblB r

- - - - - - - -- - - - - - -

Figure 7.10 Effect of Breaker Location

7.1.4 Harmoaic Generation

A dimensiontess variable representing the effect of hamonic generation on transrnitted wave

height was presmted by Mei and Onliiata (1972) for regular waves. as discwsed previously in

Section 2.2.2. This parameter was defined as B/&' where A' (BEAT2) is a beat Imgth. as

previously defined. Assuming that this parameter can be extended to irregular waves using a

beat length based on T, of an irregular wave spec- the dimmsionless beat length of the

second harmonic was computed for the various test conditions. Because the relative amplitudes

of the harmonic fkequencies appear to be approximately periodic in space, (see Figure 2.5) the

tranmission coefficient was plotted against the absolute value of the sine of the beat length

(ABS[Sin BA,,. In effect, this cornparison would relate the transmission coefficiemt to the

relative energy in the second harmonic fkquency. The resulting plot is show in Figure 7.11.

The plot shows that there is generally no dependence of Kt on BI&'. This result is not surprising,

however, as this variable simply defines the distribution of energy between the fundamental and

harmonic frequencies. Since Kt is computed on the basis of the entire spectral energy, the

transfer of energy between frequencies will not affect the computed Kr unless there is an

accornpanying loss of energy. Therefore, this result suggests that there is insignificant energy

loss as energy is transferred between the furidamenta1 fiequencies and the various harmonics. As

a result, it is concluded that the energy losses due to momentum transfer and shear within the

fluid as energy is transferred between harmonic frequencies is small in relation to that lost due to

wave breaking and interaction between the fiuid and the breakwater. It is more likely that the

spatial effects of this energy transfer on the breaker location and local setup have more effect on

K, than the actual energy transfer mechanisms themselves. This variable has not been considered

further in the statistical analysis section.

2-D Submerged Breakwater Tests Kt vs. ds/Hi and ABS(Sin(WBeat2))

Figure 7.1 1 Effect of Dimensionlessi Beat Length

7.1.5 Frictional Effects

Given the typically shallow areas over the crest of submerged breakwaters and the often visibly

turbulent conditions, it i s possible that fiction losses in this area are an important factor in

detemining Kt. In order to consider the effects of Wction on K, it i s necessary to develop a

representative dimensionless variable. A variable analogous to the fnction loss expression for

pipes was developed. Head loss in pipes is defined as:

Considering the individual terms, the Wction coefficient # is related to a Reynolds Number and

relative roughness, L is a characteristic length, D is the pipe diameter and v is the flow velocity.

Considering only fully rough turbulent conditions, the fnction coefficient becomes a function of

the relative roughness (EX)) only. In the region over a submerged breakwater, the velocity term

is assumed to be based on the wave celerity given the shallow water conditions and it has been

made dimensionless by dividing by Hi. Therefore. the following substitutions are made.

For convenience sake, this term will be caIled the friction term throughout the remainder of this

thesis. The inverse of the fnction term was plotted against K, to assess its effect on the

transmission process. This plot is shown in Figure 7.12. The figure shows a relatively strong

relationship between K, and the dirnensionless friction tenn and therefore, this parameter is

considered to be important in defining Kr

- - - - -

2-D Submerged Breakwater Tests Kt vs. dsHilBD50

Figure 7.12 Effect of Friction Term

7.1.6 Interna1 Flow (Drag) Effects

As the incident wave is transmitted across the breakwater, there is some flow within the

breakwater material and as the depth of submergence is reduced, the relative portion of the flow

within the material is expected to increase. Therefore, it is important to consider a variable

representing the potential for transmission through the structure.

As previously noted, Seelig (1980) postdated that the transmission through a submerged

breakwater is negiigible when h/h, r 1 -2. For cases when h/h, s 1.2, it seems reasonable to

assume that the majority of the transmission through the breakwater would be through the upper

portion of the structure. For conventional submerged nibblemound structures, the core material

is armoured with two layers of relatively large Stone, providing a relatively porous matenal for

transmission of wave energy. Therefore, only the potential for flow within the amour layer wiIl

be considered here.

Given the increased transmission through conventional breakwaters with increasing wavelength

(L) (U.S. A m y Corps, 1984), the increased permeability with increased D,, and the increased

resistance with increasing B, a dimensionless t m was forrned on the basis of these ttiree

variables This term is not derived fiom any theoretical or analyticaI expression of porous media

flow but is simply assumed to be proportional to the resistance of transmission through the

armow on the basis of the variables involved. The term can be expressed as:

Again, the effect of this variable will be dependent on the depth of submergence. The

transmission coefficient is plotted, in Figure 7.13, as a function of d/TI, and P/LD,. The figure

shows a strong correlation between the variables when d, is small, as expected, with reduced

influence under higher submergence. Further analysis of this variable is presented in Section

7.2.

2-D Subrnerged Breakwater Tests Kt vs. dslHi and BA21LD50a

7.2 Statisticai Development of Design Equations

7.2.1 Analysis Techniques

On the basis of the graphical analysis of dimensionless variables, a number of these variables

have been considered statistically with respect to their ability to predict the transmission

coefficient for submerged breakwaters. These variabIes have been selected because they are

thought to represent the most important physical processes, directly affecting the transmission

mechanism. Although the transmitted wave height is a function of al1 of the variables presented

in [2.18], some of these variabIes have only a minor effect which is ofien not discernable in

random waves where non-linear wave/wave interactions and wavdstnicture interactions generate

a very dynamic condition at the breakwater. Therefore, it is important f?om a statistical

perspective, to limit the variables in the design equation to those with the most dominant and

consistent effects on the transmission process.

Statistical analysis of selected dimensionless variables was perfomed using the SYSTAT

software package. A database of dimensional and dimensionless variables was generated from

the test data and the associated wave properties (computed based on Iinear theory techniques).

This &tabase was subjected to a simple linear (Pearson) correlation test to check the relative

dependence of the transmission and reflection coefficients on the independent variables. While

the linear correlation coefficient is not conclusive for a non-linear relationship, it provides a

general confirmation of the variables assurned to be important in the transmission process.

A scatterplot matrix was aIso developed for a subset of the 2-D test data (structures GI - GS) to

graphically show the relationship between variables. These plots (included in Appendix H) are

more usehl in assessing general relationships between K, and the various dirnensionless

variables. The plots confirm the importance of those variables noted in the previous sections.

Subsequently, those dimensionless variables presented in Section 7.1 were used to develop a

number of alternative linear and non-linear design equations for transmission at submerged

breakwaters.

Although the measure of statistical fit (R2) is oAen used as the general measure of an equation's

suitability, it is important to consider the physical implications of the equation and the quality of

the input data. As a result, the adequacy of the alternative design equations was detemined on

the basis of a number of statistical and practical considerations. The best equation has been

considered to be one which combines the following features:

the equation fits the data well as measured by the R2 statistic;

the equation approaches appropriate limiting values in its approximation

(ie. O.OsKts 1 .O);

the residuals of the equation are disbibuted normally, indicating that there are no

significant trends unaccounted for, and

the equation incorporates physically relevant variables while minimizing the number of

statistically fitted parameters.

Development of alternative design equations proceeded in a trial and error manner given the

cornplexity of the interaction between the various factors influencing the transmission

phenornenon, Therefore, although numerous combinations of variables were assessed, only the

most relevant alternatives will be presented here.

7.2.2 Development of Alternative Design Equations.

The fxst consideration in developing an appropriate design equation for K, is the f o m of the

function. Linear and non-linear equations were considered. A simple linear equation would

have the forrn:

where C,, C and C, are fined parameters and X,, X, and X, are dimensionless variables. Non-

linear equations are much more complex in nature and can involve complex power functions,

logarithmic functions or derivative functions. A typical non-linear function may take the form:

The form of the equation should be defined by the nature of the processes taking place and it is

116

oftm evident fkom visual inspection of the data or consideration of the variables involved. A

number of arternpts were made to linearize the data by rescaling the dependent and independent

variables in various non-linear forms. Ai1 attempts were unsuccessful, indicating that process is

not linear with respect to any one or two independent variables, but is the combined effect of a

number of physical processes.

Therefore, the development of the design equation in this case has been undertaken assuming

that the transmission process is due to the combined effect of a number of individual

mechanisrns. As noted in Section 7.1, some of the potential mechanisrns which have been

considered include transmission of unbroken waves, transmission by waves breaking into the lee

of the breakwater, transmission h o u g h the breakwater rnaterial and the effects of friction losses

and losses through harmonic interactions. It was previously noted, however, that the harmonic

interactions did not appear to have any significant influence on the transmission process and

therefore have been dropped fiom the investigation.

Given the predominant effect of the relative submergence depth on the transmission coefficient

and the general consensus in the existing literature that this variable is the most important

influencing factor at submerged breakwaters, the wave breaking influence was assumed as the

basis for al1 alternative design equations. Other variables were added in an iterative fashion to

represent the effect of other physical processes. Where more than one dimensionless variable

was thought to represent the same physical process, they were evaluated individually.

Although some of the dimensionless variables discussed in Section 7.1 were observed to be

relatively well correlated with K,, the correlation could, in some cases, be due to the effect of one

of the dimensional variables making up the dimensionless variable, rather than the effect of the

entire dimensionless terrn. Therefore, the apparent correlation between K, and two or more

dimension~ess variables incorporating a comrnon dimensional variable does not necessarily

indicate that al1 of those dimensionless variables, or the processes which they are supposed to

represent, are physically or statistically important.

For example, a number of the dimensionless variables are based on the crest width (B). It is

speculative to develop numerous dimensionless variables on the basis of B, to be representative

of physical processes at submerged breakwaters, and suggest that they are al1 physically

important on the basis of correlation tests when in fact, it may simply be the influence of B that

results in the observed correlation. Therefore, the arguments made in this study to include or

preclude certain variables fiom the final equation development have been made on the basis of

statistical indicators and qualitative physical observations made during testing. In short, more

detailed investigations of the transformation of the various hydrodynamic variables across the

width of the breakwater would be required to conclusively determine the most important

physical processes taking place.

Nevertheless, the arguments made herein are consistent with the observed data and are in

accordance with the objectives of this study (ie. to develop a design equahon which is valid over

a wide range of design conditions). The following discussion outlines the rnost significant

aspects of the design equation development. As previously noted, the design equations were

developed through non-linear regression analysis using a statistical analysis package. A number

of alternative design equations involving different variables and different mathematical forms

were investigated in order to maximize the fit of the equation while providing some stability

with respect to the fitted parameters and the practical limits of the estimated Kt. A partial list of

the more promising alternative design equations considered and the basic statistical results for

each equation are presented in Appendix 1.

Considering only the relative subrnergence term in both a linear and non-linear fashion. a design

equation suff5ciently fitting the test data could not be acheved. The maximum R' value attained

with the single variable equation was 0.672 and a plot of the residuals against the relative

submergence tenn showed very poor estimates of Kr at d, /H, = 0.0. The modified surf-similan3,y

parameter (5,") incorporates a relative depth term and, therefore, was considered as an alternative

to d/H,. Substitution of &" for d, Hi as the single independent variable resulted in an RL of only

0.375 and although the estimation of K, at d, H, = 0.0 was improved, overall, the estimate was

degraded.

Building upon the relative subrnergence term, an overtopping term was added to the equation to

represent the transmission which is generated by waves breaking over the structure. This is

especially important where d/H, = 0.0. Both H, /B and X,/B were assumed to be representative

of this transmission mechanism and therefore, both were considered as additional tenns for the

equation. The H,/B term provided slightly better R2 values (0.894) as a non-linear term. This

seerns reasonable given that the number of waves breakmg into the lee of the breakwater would

decrease rapidly (at a non-Iinear rate) as the crest width is increased.

The general fonn of this equation is:

where the fitted coefficients C 1, C2, C3 and C4 are 0.35,0.58,0.63 and 0.64 respectively. The

approximate error associated with the coefficients was used to define the number of significant

figures. Ploning the estimate against the measured K, (Figure 7.14) shows a relahvely good fit.

There does appear to be a slight oscillation of the estirnate about the perfect fit line, with a

general overestimation of the lower Kr vaIues and perhaps a slight underestimation of the larger

K, values. However, the equation appears to be welI bounded and in general provides a good

visual fit.

--

Figure 7.14 K, vs Estimate - Eqn. 7.12

Inspection of the residuals of the statistical non-linear fitting procedure (Fig. 7.1 S ) , confirms the

oscillatory trend, with negative residuals for lower Kr values and positive residuals for higher Kr

values. Ploning these residuals against the relative submergence term (Fig. 7.16) shows that the

oscillation is consistent with the d/H, term and therefore, inclusion of other transmission

mechanisms in the equation, reducing the transmission at low submergence and increasing it at

moderate submergence rnay improve the fit. Terms representing £iictional processes and intemal

flow processes were therefore introduced to the equation for evaluation.

Figure 7.15 Residuals vs. K., - Eqn. 7.12

Figure 7.16 Residuals vs. d/H, - Eqn. 7.12

The frictional resistance term (dfij BD50a> and the intemal flow term (B-' /a) were introduced

to the equation altemately and simultaneously. The best fit was attained with only the intemal

flow term in a non-linear manner (R2 = 0.946). Equations incorporating both the intemal flow

temn and friction term provided estimates which were comparable (R2 = 0.94*) but there is a

general benefit associated witb the equation with fewer parameters. The fitted equation is:

The fit of this equation is shown in Figure 7.17. The residuals are plotted as a function of d,/H,

and as a normal distribution in Figures 7.18 and 7.19 respectively. As indicated in the figures,

the equation fits most of the data relatively well, with considerably less scatter than that depicted

in Figure 7.14. However, there are a number of estimates below Kt=û.O and a few above K,=1 .O

indicating that the equation underestimates transmission whm transmission is small (perhaps for

srna11 d/H, and large B) and ovemtimates transmission whm transmission is large (perhaps for

large dmi and maIl B). The normal probability plot of residuals does produce a relativeIy

straight line indicating that the residuals are nearly normally distributed. Figure 7.18 on the

other hand shows a definite trend in the residuals as a function of d/H,.

- -

Figure 7.17 Estimate vs. - Eqn. 7.13

Figure 7.18 Residuals vs. d/H, - Eqn. 7.13

Figure 7.19 Normal Probabifity of Residuais

In an effort to improve this condition and minimize the potential for estimates outside the

practical limits of K, the internal flow term was modified to include the influence of the

subrnerged depth. The revised variable is therefore of the form:

Assuming an internal flow term incorporating d,, the following fitted equation was found to

provide a good estimate of K,, while maintaining the estimate between 0.0 and 1.0.

This equation which now also includes the fnctional term provides an R' of 0.914. Visual

inspection of the equation's fit (Figure 7.20) shows a relatively tight chstenng of the estirnate

about the perfect fit line. The residuals are well distributed with respect to djH, (Fig. 7.21) and

appear to fit the normal distribution relatively well. Although a power exponent on the intemal

flow term does improve the R' rnarginally (0.927), it tends to degrade the normal distribution of

the residuals. Therefore, the additional parameter was not employed.

Figure 7.20 Estimate vs. K, - Eqn. 7.14

Figure 7.2 1 Residuals vs. d/H, - Eqn. 7.14

Figure 7.22 Normal ProbabiIity of Residuals

Given the factors noted above. the equations represented by [7.13] and [7.M] both have distinct

advantages. Although it is tempting to base the equation selecrion on the best R' value. the intent

of this exercise is to produce a robust equation based on the physical processes. Thmefore.

m e r comparison of these equations on the basis of sensitivity to input variables and

comparison with 3-D test results is provided in the folIowing section.

7.23 Evaiuation of Alternative Design Equations

The equations presented in [7.13] and [7.14] were compared in more detail on the basis of the

sensitivity of the estimate to changes in the equation variables, to changes in the equation

parameters and on the basis of their modelling of the 3-D test results.

Detailed statistical output for [7.I3] and [7.14] is provided in Appendix 1. The non-linear

regression analysis for the equation provides a number of statistical measures of the equation and

its parameters. An approximate standard error associated with each parameter estimate and the

associated 95% upper and lower confidence intervals for the parameters are provided.

Consideration was given to the size of the standard m o r of the esrimate, and its potential impact

on the equation's performance, by plotting the Kt estimates for the proposed parameten and for

two other cases: with al1 parameters set at the 95% interval that provided the lowest K, estimate

and the 95% intemal that provided the largest Kr estirnate. Thus, the stability of the equation to

changes in the parameter estimates was evaluated. The resulr of this analysis for [7.13] and

[7.14] are shown in Figures 7.23 and 7.24 respectively.

The results show that [7.14] is more stable with respect to changes in the estimated equation

parameters. This is desirable given the uncertainty associated with any statistical equation

development. This analysis also suggests that the strength of the equation fit lies more in the

equation's variables than in the fitted parameters.

Equation 7.13 : Predicted vs. Observed Sensitivity to Pararneters (95 % C.I.)

O 0.2 0.4 O. 6 0.8 7 Observed Kt

Est Parrim. v

+OS% Pamm. *

45% Parum.

Figure 7.23 Eqn. (7.131 Sensitivity to Parameter Estimates

l Equation 7.14 : Predicted vs. Observed Sensitivity to Pararneters (95 % C.I.)

O O. 2 0.4 0.6 0.8 1 Obsewed Kt

Est. Pamm. 7

+SS% Param. 0

-95% Param.

Figure 734 Eqn.ff.141 Sensitivity to Parameter Estimates

The suitability of the parameter estimates c m also be shown by comparison of the relative size

of the approximate standard error. The statistical output in Appendix 1 shows that parameters C5

and C6 of [7.13] have approximite standard mors (A.S.E.) of about 40% and 30% respectively

and they are strongly conelated with each other and with C3. The output for [7.24] shows

smaller A.S.E.'s and weaker correlation, indicating a more robust equation.

The sensitivity of the proposed equations to changes in design variables was tested by plotting

the estimates of 17-13] and [7.14] with the estimates of van der Meer's equation and Ahrens' reef

breakwater equation for the hypothetical design conditions considered in Chapter 6. The results

of the cornparisons for breakwater widths of 0.3 m and 3.5 rn are shown in Figures 7.25 and 7.26

respectively.

The comparison shows that both equations predict similar Kt values and they are comparable to

the predictions of van der Meer's equation for the narrow crest width. This is desirable since van

der Meer's equation predicted test data well under these conditions. The un-bounded nature of

[7.13] and van der Meer's equation is evident at the extreme values of djH, where predicted

values creep above 1 .O. Considering the wider crest (Fig. 7.26) the problems with [7.13] and

van der Meer's equation are quite evident The equation expressed by l7.141, however, is well

controlled and is consistent with the upper limits of the estimates of Ahrens' equation.

There fore, [7.14] appears to be the superior equation.

It is important to note that the estimates of [7.14] will not approach the correct limiting values

under al1 input design conditions. Much wider crest widths will begin to produce negative

iransmission coefficient estimates. However, given that the range of variables tested for this

thesis is approaching the limits of feasible design conditions for most situations, this limitation is

not considered to be crucial.

-- -- - -- -

Submerged Breakwater Transmission Corn parison of Methods (B=0.3 m)

n

Ahnns w

van der Mee 9

Eqn 7.13 A

Eqn 7.f4

- -

Figure 7.25 Sensitivity to Input Variables (B43 m)

Submerged Breakwater Transmission Cornparison of Methods (B=3.5 m)

I

Ahrrns v

van der Msei 0

Eqn 7.13 A

Eqn 7.14

- - - - - -- - -

Figure 7.26 Sensitivity to Input Variables (B=3S m)

The final comparison of the proposed equations is with respect to the ability to predict the 3-D

modelling results. It has been noted that the 3-D modelling conditions introduce many site

specific compIexities and in reality, accurate prediction of the hydrodynamic regime for a

specific design condition is best served by a physical hydraulic model. However, it is the

objective of this thesis to provide an equation which will provide a good approximation of the

transmission at a submerged breakwater, and in order for the equation to be practical, it must be

suitable for 3-0 appiications.

The comparison of predicted and observed K, values or the 3-D tests is shown in Figure 7.27.

The figure shows that visually, there is no clear indication of the better equation on this basis.

Both equations predict better when the transmission is low to moderate, while under conditions

where the observed transmission approaches 1 .O, both equations underestimate Kt.

Although this is not a desirable situation, consideration should be given to the fact that the 3-D

testing produced sorne observed transmission coefficients larger than 1 .O. W l e this is not

impossible at site specific locations under directional spectral transformations and complex

reflection and diffraction conditions, it is more likely that the 3-D testing apparatus and analysis

procedures have introduced some m o r into the measured K, values. Thmefore, ths comparison

serves a warning that the estimates for 3-D conditions are suspect, especially under higher

transmission conditions but on the whole, the estimates are not bad.

3-D Testing : Submerged Breakwaters Est. vs Obs. @ PZ (Eqns. 7.13 8 7.14)

I

Eqn. 7.14 *

Eqn. 7-13 - Est. - Ob..

Figure 7.27 Equation Cornparison with 3-D Test Results

Considering the cornparisons and evaluations noted above, the equation represented by [7.14] is

considered to be the superior equation and is suggested for application in the preliminary design

of submerged breakwaters. It is important to consider the following points regarding the

proposed equation.

The variables in the equation are considered to be representative of physical processes

related to wave breaking, overtopping and losses due to flow over and within the amour

of conventional submerged rubblemound breakwaters.

There is no term in the equation to account for the relative height of the breakwater.

This variable will affect the reflection characteristics of the structure. which in tum, will

affect the transmission characteristics. Application of the equation outside the range of

relative depths tested (0.56shfii 1 .O) may therefore result in some error in the K,

estimation. It is expected, however, that the tests presented herein represent the practical

range of design applications for conventional submerged nibblemound breakwaters.

The equation is based on testing variables representative of storm-type conditions but it

does not specifically account for secon- hydrodynamic effects related to wave

groupiness, local water level setup, resonance within the sheltered area, wave-current

interactions or tidal effects. Unique phenomena such as s d beats within the protected

zone (Nakaza et. al., 199 1) have been associated with such secondary effects and should

be considered with respect to the site specific conditions.

The equation approaches the appropriate limits for K, for the range of variables tested.

Given the relahvely large range of test variables, the results are considered to be

appropriate for most feasible design conditions.

The equation is generally consistent with the 3-D testing results, but tends to

underestimate Kr when the transmission is high (approaching 1 .O). The e ffec ts of the

mode1 scaling and the poorer representation of the Reynolds processes in the 3-D tests

(as discussed in Section 4.2.2) should be considered when comparing 2-D and 3-D test

results.

Although the equation provides good agreement with the 2-D test results and fair

agreement with the 3-D test results, the physicai processes affecting the transmission at

submerged breakwaters are cornplex. The proposed design equation should be used to

provide a preIiminary estimate of the structure requirements. Location specific

transmission coefficients and hydrodynamic characteristics should be evaluated using a

physical mode1 and the detailed structure requirements determined on the basis of these

physical rnodelling results.

Although numerous alternative design equations were considered over the course of this

study and the performance of the proposed equation is relatively good for the 2-D tests,

there is still some uncertainty as to the exact nature of the physical processes affecting

the transmission. The persistent trend observed in the plot of residuals vs. H, /B in

Appendix I suggests that al1 contributing processes are not fully accounted for.

8.0 Conclusions and Recommendations

8.1 Conclusions

increased environmental awareness and the need for rehabilitation measures for existing coastal

defence structures have resulted ÜI more frequent consideration of submerged breakwater

structures as coastal protection alternatives. Existing design equations, however. are unable to

accurately predict Kt ovn a wide range of design conditions. This is especially mie for wide

crested structures.

Based on an extensive set of 2-D tests of transmission at submerged breakwaters, a number of

general observations were made with respect to the effect of various independent variables on

the transmission process. In general, observations show that the transmission coefficient (K,) is

most sensitive to the relative submergence (dm). The second most important independent

variable affecting K, was obsemed tu be the crest width. A nurnber of dimensionless variables

related to the crest width can be assumed to be physically relevant. The present study proposes

that variables representing an overtopping terni (HP), a fnctional term (d,H/BD5&nd an

intemal flow term (BdJLD,) are the most appropriate. AIthough the individual effects of wave

penod (or related wavelength) and armour size on K, were not significant, they have been

incorporated into the dimensionless variables related to crest width.

On the basis of the 2-D test data and observations, alternative design equations were developed.

The dimensionless variables used in the equations are assumed to be representative of those

physical processes which have the most influence on the transmission phenornenon. The

recommended transmission equanon, selected on the basis of statistical measures and physical

characteristics can be expressed as follows:

The followîng specific conclusions can be made regarding the 2-D modelling results and

subsequent design equation development.

Relative subrnergence and crest width were found to be the most important factors and

formed the basis for the dimensionless design equation variables. Their importance to

the transmission process has been s h o w graphically and statistically.

The equation provides good agreement with the results of 2-D t e s h g performed for this

thesis, resulting in an R2 of 0.914. The equation is also retatively stable with respect to

the parameter estimates.

The equation approaches the appropriate limits for K, for the range of variables tested.

Given the relatively large range of test variables, the resuIts are considered to be

appropriate for most feasïble design conditions.

The equation is generally consistent with the 3-D testing results, but tends to

underestimate Kt when the transmission is high (ie. approaching 1 .O).

Results of 3-D tests show transmission trends sirnilar to those observed during 2-D testing, but

with considerably more variability. Observed Kt values above 1 .O are amibuted in pari to the test

setup which results in considerable reflection at times and some seepage of energy into the test

area. Complex directional spectral transformations, characteristic of m e 3-D conditions. will

also contribute to local increases in the measured transmission.

Transmission at submerged breakwaters under 3-D conditions appears to be locally specific with

minimum transmission coeficients achieved immediately behind the breakwater where

difhcted energy is relatively low. It is suggested that the use of a 2-D transmission equation in

conjunction with a coefficient to account for energy contributions f?om diffraction may be

appropnate for determining transmission under 3-D conditions. The nahire of the d i h c t i o n has

not been determined here but is suspected to be qualitatively consistent with that observed at

surface piercing breakwaters.

Beach development in the lee of submerged breakwaters appears to be similar to that observed at

detached offshore breakwaters as the initiation of tombolo-type beach forms was observed.

However, the beach development appears to be sensitive to the submerged depth and under

fluctuating water level conditions, a more dynamically stable condition may be achievable. Test

results are qualitative only, however, and are not conclusive in this regard.

8.2 Recommendations

The following recommendations are put forth on the basis of the present fmdings.

1. Although the proposed design equation provides good agreement with the 2-D test results and

fair agreement with the 3-D test results, the physical processes affecting the transmission at

submerged breakwaters are cornplex. The proposed design equation should be used to provide a

preliminary estimate of the structure requirements. Location specific transmission coefficients

and hydrodynamic charactenstics should be evaluated using a physical mode1 and the detailed

structure requirements determined on the basis of these physical modelling results.

2. Although numerous alternative design equations were considered over the course of this study

and the performance of the proposed equation is relatively good for the 2-D tests, there is still

some uncertainty as to the exact nature of the physical processes affecting the transmission. The

persistent trend observed in the plot of residuals vs. H, /B in Appendix 1 suggests that al1

contributing processes are not fully accounted for. Therefore, M e r investigation of the effect

of crest width on submerged breakwater performance is necessary to confirm the importance of

the physical processes suggested herein.

3. Tests have reportedly been undertaken with a number of narrow crested submerged

breakwaters in senes but results were not available at the time of this thesis preparation. The

ability of the proposed equation to predict transmission under such conditions should be

assessed.

4. More detailed investigations of the transmission process under 3-D conditions are necessary to

assess the effect of ditfraction on local transmission coefficients and to provide

recomrnendations regarding a suitable approach to accounting for this effect.

5. The effect of the breakwater location (ie. distance offshore) should be considered in regard to

transmission coefficients and secondary hydrodynamic impacts such as resonant long wave

effects given the potential for local setup and the sensitivity of the submerged breakwater to

relative submergence.

Adams, C B and Sonu, C.J. "Wave Transmission Across Submerged Near Suface Breakwaten". Coastal Engineering '86, American Society of Civil Enginem, New York 1986, pp. 1729 - 1737.

Ahrens, J.P. "Reef Breakwater Response to Wave Attack", Berm Breakwaters: Jnconventional Rllbble-Mound. Arnerican Society of Civil Engineers, 1988, pp 2 1-40.

Ahrens, J.P. "Characteristics of Reef Breakwaters". CERC Technical Report 87- 17, 1987.

Beji, S. and Battjes, J . k "Experimental Investigation of Wave Ropagation ûver a Bar", Coastal Engineering, 19: 1-2, 1993, pp. 15 1 - 162.

Beji, S. and Battjes. J.A. "Numerical Simulation of Nonlinear Wave Propagation Over a Bar", Coastal Engineering, 23: 1-2, 1994, pp. 1-16.

Cooker, M. and Peregrine, H. "Solitary Waves Passing Over Submerged Breakwaters", 21' International Conference on Coastal Engineering, American Society of Civil Engineers, New York, 1989, pp. 624-631.

Comett, A., Mansard, E. and Funlce, E. "Wave Transformation and Load Reduction Using a Small Tandem Reef Rreakwater - Physical Mode1 Tests", Ocean Wave Measurement and Analysis, Ammcan Society of Civil Engineers, New York, 1 994, pp. 1 008- 1023.

Cruz, E.C.. isobe, M. and Watanabe, A. "Nonlinear Wave Transformation Over a Submerged Permeable Breakwater", 23" International Conference on Coastal Engineering, American Society of Civil Engineers , New York, Vol. 1, 1993, pp. 1 10 1- 1 1 14.

Dai, Y.B. and Kamel, A.M. "Scale Effects Tests for Rubblemound Breakwaters", U.S. h n y Waterways Experimental Station Research Repo~t H-69-2, 1969.

Dalxymple, R.A. "Physical Modelling in Coastal Engineering", AA. Balkema, Rotterdam. 1985, pp. 3 - 9.

Dattaûi, J., Raman, H. and Shankar, N.J. "Performance Charactmstics of Submerged Breakwaters", 1 6 International Conference on Coastal Engineering, American Society of Civil Engineers, New York, 1979, pp. 2 152-2 17 1.

Dick, T.M. and Brebner, A. "Solid and Permeable Submerged Breakwaters", 1 1" International Conference on Coastal Engineering, American Society of Civil Engineers, New York, 1969.

Driscoll, A.M., Dalrymple, R.A. and Gdli, S.T. "Haxmonic Generation and Transmission Past a Submerged Rectangular Obstacle", 23" International Conference in Coastal Engineering, Amencan Society of Civil Engineers, New York, Vol. 1, 1 993, pp. 1 1 42- 1 1 60.

Fu1 ford, E.T. "Reef Type Breakwaters for Shore S tabilization", Coastal Zone '85, American Society of Civil Engineers, New York, 1985.

Gilver L.D. and Sorensen, R.M. "An Investigation of the Stability of Submerged Homogmeous Rubble Mound Structures Under Wave Attack", IMBT Hydraulics Report #IHL-110-86, 1986.

Goda, Y., Takada, N. And Monya, Y. "Laboratory Ivestigation of Wave Transmission over Breakwaters", Report of the Port and Harbour Research Institute, No. 13, 1967.

Goda, Y. "Re-analysis of Laboratory Data on Wave Transmission over Breakwaters", Report of the Port and Harbour Research Institute, Vol. 1 8, No. 3, 1969.

Goda, Y. and Suzuki, Y. "Estimation of Incident and Reflected Waves in Random Wave Experiments", 1 5' Intemational Conference on Coastal Engineering, American Society of Civil Engineers, New York, 1977, pp. 828-845.

Grilli. S.T. and Svendsen, I.A. "Long Wave Interaction with Steeply Sloping Structures", 22" International Conference on Coastal Engineering, American Society of Civil Engineers, New York, Vol. 2, 199 1, pp. 1200- 12 13.

Grilli, S.T., Losada, M.A. and Martin, F. "Characteristics of Solitary Wave Breaking Induced by Breakwaters", Journal of Waterway, Port and Coastal Engineering, American Society of Civil Engineers, New York, 120: 1, 1994, pp. 74 -92.

Hm, M., Yasuda, T. and Sakakibara, Y. "Characteristics of a Solitary Wave Breaking Over a Submerged Obstacle", 23d International Conference on Coastal Engineering, American Society of Civil Engineers , New York, Vol. 1, 1993, pp. 253-266.

Horikawa, K. and Kuo, C.T. "A Study on Wave Transformation Inside Surf Zone", 10th International Conference on Coastal Engineering, American Society of Civil Engmeers, New York, Vol. 1, 1966, pp217 - 233.

Kamphuis, J.W. "Incipient Wave Brealang", Coastal Engineering, Vol. 15, No. 3, 199 1, pp. 1 85 - 203.

Kamphuis, J.W. "Advanced Lectures in Coastal Engineering", Lecture Notes in Civil Engineering 853, Queen's University, 1995.

Kobayashi, N. and Wurjanto, A. *'Wave Transmission Over Submerged Breakwaters", Journal of Waterway, Port and Coastal Engineering, American Society of Civil Engineers, New York, 1 155, 1989, pp. 662-680.

Lee, J.J., Chang, C. and Zhrrang, F. "Interaction of Nonlinear Waves and Coastal Structures", 23* International Confmence on Coastal Engineering, Arnerican Society of Civil Engineers , New York, Vol. 2, 1993, pp. 2327-1340.

Liberatore, G. and Petti, M. "Wave Transformations Over a Submerged Bar : Experiments and Theoretical interpretations", 23" International Conférence on Coastal Engineering, Amencan Society of Civil Engineers , New York, Vol. 1, 1993. pp. 447-459.

Losada, I.J., Silva, R. and Losada, M.A. "3-D Non-Breaking Regular Wave Interaction with Submerged Breakwaters", Coastal Engineering, 28, 1996, pp. 229-248.

Madsen, O. S. and White, S .M. "Re flection and Transmission Charac teristics of Porous Rubblemound Breakwaters", MR-76-5, Coastal Engineering Research Centre, US Amy Corps of Engineers, Fort Belvoir, Va., 1976.

Mansard,E.P.D. and Funke, E.R. "On the Reflection Analysis of Irreguiar Waves", Technical Report No. TR-HY-O 1 7, National Research Council of Canada, Ottawa, 1 987.

Mansard, E.P.D., Funke, ER., Readshaw, J.S. and Girard, R K "On the Transfomation of Wave Statistics due to Shoaling", 2 l4 International Conference on Coastal Engineering, American Society of Civil Engineers, New York, 1989, pp. 106-120.

Massel, S.R "Harmonie Generation by Waves Ropagating over a Subrnerged Step", Coastal Engneering, 7, 1983, pp. 357-380.

Mei, C C and Ùnliiata, Ü. "Harmonie Generation in Shdlow Water Waves", Waves on m, Resultineted by RE. Meyer, Academic, New York, 1972, pp. 181-202.

Miles, M.D., "User Guide for GEDAP Version 2.0 Wave Generation Software", Technical Memorandum LM-HY-034, National Research Council of Canada, Ottawa, 1990.

Nakaza. E., Tsukayama, S. and Hino, M. "Bore-Like Surf Beat on Reef Coasts", 2zd International Conference Coastal Engineering, American Society of Civil Engineers, New York, Vol. 1, 199 1, pp. 743-757.

Ohyama, T. and Nadaoka, K. "Modelling the Transformation of Nonlinear Waves Passing over a Submerged Dyke", 23" International Conference on Coastal Engineering, American Society of Civil Engmeers , New York, Vol. 1, 1993, pp. 526-539.

Petti, M. and Ruol, P. "Expenmental Study on the Behaviour of Submerged Detached Breakwaters", 3rd International Conference on Coastal and Port Engineering in Developing Countries, Vol. 1, 199 1, pp. 167 - 179.

Petti, M. and Ruol, P. "Laboratory Tests on the Interaction Between Nonlinear Long Waves and Submerged Breakwaters", 23' International Conference on Coartal Engineering, American Society of Civil Engineers , New York, Vol. 1, 1993, pp. 792-803.

PelIetier, D. "Real Time Control System Users Manual", Technical Report TR-KY-035, National Research Council Of Canada, Ottawa, 1990.

Rojanakarnthom, S., Isober, M. and Watanabe, A. "Modeling of Wave Transformation on Submerged Breakwater", 22"" Intemational Conference on Coastal Engineering, American Society of CiviI Engineers , New York, Vol. 1, 1991, pp. 1060-1073.

Saville, T.Jr. bbHydraulic Mode1 Study of Transmission of Wave Energy by Low-Crested Breakwater: Ship Mooring Study, West Coast of of Point Loma, Sandiego, California",

Unpublished Memorandum for Record, U.S. -y Corps of Engineers. Beach Erosion Board. Washington, D.C., 1963.

Seelig, W.N. "Estimation of Wave Transmission Coefficients for ûvertopping of Impermeable Breakwaters", CERC Coastal Engineering Technical Aid 80-7, 1980.

Seelig, WN. "Effect of Breakwatm on Waves : Laboratory Tests of Wave Transmission by Overtopping", Coastal Structures '79, American Society of Engineers, New York, 1979, pp. 941-961.

Smith, E.R. and Kraus, N.C. "Laboratory Study on Macro-Features of Wave Breaking Over Bars and Artificial Reefs", CERC Technical Report 90-12, 1990.

Tanaka, N. "Effects of Submerged Rubblemound Bnakwater on Wave Attenuation and Shoreline Stabilization", Roceedings of 23" Japanese Coastal Engineering Conference, 1976.

Ting, F.C.K. and Kim, Y.K. "Vortex Generation in Water Waves Propagating Over a Submerged Obstacle", Journal of Coastal Engineering, American Society of Civil Engineers. New York, 24: 1-2,1994, pp. 23-49.

U.S. Army Corps of Engineers. " Shore Protection Manual", Coastal Engineering Research Centre, Vicksburg, Mississippi, 1984,

van der Meer, J.W. "Rock Slopes and Grave1 Beaches Under Wave Attack", Delft Hydraulics Communication No. 396, 1988.

van der Meer, J.W. and Pilarczyk, K.W. "Stability of Low-Crested and Reef Breakwaters", Coastal Engineering '90, Arnerican Society of Civil Engineers, New York, t 990, pp. 1375- 13 88.

Van der Meer, J.W. "Stability and Transmission at Low-Crested Stnictures", Delft HydrauIics Publication 453, 199 1.

Van der Meer, J. and Daemen, F.R. "Stability and Wave Transmission at Low-Crested Rubblemound Structures", Journal of Waterway, Port and CoastaI Engineering, Arnerican Society of Civil Engineers, New York, 120: 1, 1994, pp. 1 - 19.

Van der Meer, LW. and Angremond, K. "Wave Transmission at Low-Crested Structures", Coastal, Institution of Civil Engineers, London, 1992.

Vidal, C., Losanda, M.A., Medina, R., Mansard, E.P.D. and Gomez-Pina, G. "Universal Analysis for the Stability of Both Low Crested and Submerged Breakwaters", 23" International Conference on Coastal Engineering, American Society of Civil Engmeers , New York, Vol. 2, 1993, pp. 1679- 1692.

Weigert, A.W. and Edwards, E.P. "Capacitative Water Level Gauges for Hydraulic Models: Laboratory Memorandum", National Research Council, Canada, 198 1,2ûpp.

Yasuda, T. and Hara, M. "Breakmg and Reflection of a Steep Solitary Wave Caused by a Submerged Obstacle", 22* International Conference on Coastal Engineering, American Society of Civil Engmem, New York, Vol. 1, 199 1, pp. 923-934.

Appendix A Hypothetical Input Design Conditions

For Preliminary Evaluation of Existing K, Models

Hypothetical Design Conditions for Prelirninary Cornparison of Existi ng Design Equations

Parameter alues Tested

All possible combinations of the variables in the table above were input to the existing design equations for cornparison.

Appendix B Test Breakwater Configurations

and Stability Calculations

SUMMARY OF BREAKWATER CHARACTERISTICS 2-D and 3-0 Testing

-Er (mm)

'ET 17 17 17 17 17 17 17 17 17 17 17 17 4 4 4 4 4 4 -

Breakwater Material Distri bution Curves - By Mass

, Bnakunter Material D1.Mbutfon Cum ! m u r ~.teriat - 3-û ~.itr

BREAKWATER STABILITY CALCULATIONS

1. By Method of Vidal et al, 1994.

9

(Source :Vidal et. al., 1994)

Pr where A = (- -1) p.

Cons ide~g Iribarren's Darnage level and F, < 0.0: N, min -2.1 for crest area. Therefore for:

' Range of armour assumed to be 0.75 W, 5 W i 1.25 W,

2. Method of van der Meer, 1991.

1 H.

- - r D,=- - 4 s~ where sp - -

(A N;) =P

Where L, is local (at structure) wavelength based on linear wave theory.

Considering Darnage Level = S2 and 4 1 h = 1 : N,' - 6.0. Therefore for:

&=20cm p, = 2650 kg/m3 pl = 1000 kg/m3

s,, - 0.10

Appendix C Wave Generation Routines and Characteristics

$ ! Filename: WAVETRAIN.COM $! Develops wavetra in from a s p e c i f i e d i r r e g u i a r wave spectrum $! ~ n i t i a t e d be and receiving input parameters from SRSTRAIN-COM $ PARSPEC

! number of program c y c l e s (11 # p l r ! name of f i r s t wave spectrum output file [.O011

! type of wave spectrum ( 1 - 5 ) (11 ! water depth for wave power c a l c u l a t i o n s ( m ) [10000 .0]

l p2* ! frequency at which spectrum peaks (Hz) ! va lue of P h i l l i p s Alpha constant [ 0 . 0 0 8 1 ]

4 ! JONSWAP spectrum option ( 1 - 4 ) [l] ! Gamma [ 3 . 3 ]

' P 3 ' ! s i g n i f i c a n t wave height (m) $ RWSYN

! number o f program c y c l e s [lj ! mode1 scale factor [ 1 . 0 ]

? 5 ' ! durat ion i n full-scale minutes ( 0 . 1 - 1 0 9 . 2 2 7 ) [ 2 0 . 0 ] ! seed opt ion [1] ! wave s y n t h e s i s option (11

' Pl1 ! name of wave spectrum input file [.001) ~ 4 ' ! name of f i r s t wave record output file [ .O01 1

$! filename: WAVESIGNAL.COM $! Develops a voltage wave signal for wave machine from input wave t r a i n $! file (previously developed by SRSTRAIN-COM and WAVETRAIN.COM). $ ! Initiated by and receiving input parameters £rom SRSSIGNAL.COM 5 RWREP2

! number of program cycles (11 ! wave generation option ( O to 5 ) [ l ]

UIRcal-3ft-flume ! name of wave machine calibration file [.O011 'Pl' ! DEPTH (mode1 m)

! wave propagation distance D (model m) [0.01 ! lower cut-off frequency Pl (model Hz) [0,03] ! upper cut-off frequency F2 (model H z ) [4-01 ! Scale Factor [1.0]

e 2 ' ! wave amplification factor (1.0 J ' e 3 ' ! name of wave record input file [ .O01 1

! Apply spectral matching transfer function? [No]: ' e 4 ' ! name of wave machine control signal output file [.O011

O

CI - 3 0 - 0

O 0 .. "O,

V1 6 a --- w E x b u < w se= Z C m '

b - x * L 7 > < œ - < x < X h 2 a

G E N E R A T E D I N P U T S I C N A L S

Z C A P A R A M E T E R S

H 1 3 - A V (m) : 0 . 0 4 7 TAV ( m ) : 1 . 0 0 1 H M A X - M A X ( m ) : 0 . 0 8 b

S P E C T R A L P A R A M E T E R S

H M 0 ( m ) a 0 . 0 6 0 S P E C . TP : 2 .000 9P : 3 .027

S u b m c r g e d Q r c a k w a t e r S t u d l c s q u e a n ' i U n l v e r i l t y S i u B a - b r e a k WT.I~OCU-rreoo

- - --f -- - - ----- - ._I----

C E N E R A T E D I N P U T S I C N A L S

2 C A PARAMETERS

H 1 3 - A v (m) : 0 .094 T A V ( a ) : 0 .901 H Y A X - M A X (m) ; 0 . 1 7 3

SPECTRAL PARAMETERS

C E N E R A T E D I N P U T S I G N A L S

2 C A P A R A M E T E R S

H l 3 - A V ( m ) : 0 . 0 9 5 T A V ( 8 ) r 1 . 1 3 1 H M A X - M A X ( m ) : 0 . 1 7 3

S P E C T R A L P A R A M E T E R S

HM0 (m) t 0 . 1 0 0 8PBC. TP r 1 . 6 0 0 QP : 3 . 0 6 6

C1

;= r: q o o 0 -

6 ,

+-C I A

VI O g A"-

W E x + - < b - x

4 m,* œ - < x ( TC.% a

Appendix D Velocity Probe Calibration

CAUBRAlïON OF VELOCiTY PROBES

j Velocity Probe Calibration t probe NO. 1 (Chmnel O : x V ~ L )

l 3

Pmbe No. 1 - (Channel 9 = X )

Velocity Probe Catibation Ptok No. 1 (Ciunml 10 : Y VeL)

l J

Probe No. 1 - (Channel 10 = Y )

Voltage Oiatance v (ml

Velocity Probe Caltbration Pmk No. 2 (Clunml 31 : X VeL)

Pmbe No. 2 - (Channel 11 = X )

Voltage Diatance Time Velocity Linear Fit v (ml (SI (WS) v

Veloctty Pmbe Calibration Probe No. 2 (Chtnrwll2 : Y VOL)

Probe No. 2 - (Channel 12 = Y )

Voltage Diatance v (ml

Appendix E Selected 2-D Transmission Results

SELECTED 2-0 TESTING TRANSMISSION RESULTS

1. Kt vs. ds ( Varled Hs)

1 Submerged Breakwater Transmlsslon

1 Submerged Breakwater Transmlsslon Ktnd8(B=Z+Sm,Tp - 1 1 8)

Submerged Breakwater Transmission Kt v8 d8 (8.2.5 m, Tp - 1.5 S)

Submerged Breakwater Transmlsslon Kt n ds (B4.6 m, Tp - 2.0 8)

- - -- Submerged Breakwater Transmlsslon

Kt va d i ( B a s ml Tp - 2.0 a)

SELECTEO 2-0 TE STlNG TRANSMISSION RESULTS (Cont.)

6. Kt vs. D50a at 8 ~ 2 . 5 m ( Varled Tp) -a-p- --- *

Breakwater Transmission

0.055 0.04 0.045 0.05 0.055 0.06 DM (8-2.5 rn)

w h

Su bmerged Breakwater Transrnlssion Kt vs Dm (dsS).lS m, H i - 0.05 m)

Su bmerged Breakwater Transmission

Su bmerged Breakwater Transmlssion Kt vs DSO (dsrO.15 m, Hs - 0.10 m)

- .. -

Submerged Breakwater Transrnlssion Kt va Dm (dso0.15 m, Hs - 0.45 m)

1 0.0 1 I

SELECTED 2-D TESTING TRANSMISSION RESULTS (Cont.)

7. Kt vs. Tan f heta at B ~ 0 . 6 m ( Varled Tp)

Su bmerged Breakwater Transmission Kt vs Slope (ds=O.OS m, Hm - 0.05 m)

0.2 0.3 0.4 0.5 0.6 0.7 Tan Thrtr (04 .6 m)

--- œ

I p - II.

I p - 1 6 i

r p - 2 0 . -

Submerged Breakwater Transmlsslon Kt vr Slop (ds.0.15 ml Hs - 0.05 m)

0.1 1 O - . . . I 0 2 0.3 0.4 0.5 OS 0.7

Tan Thrh (BiO.6 m)

Submerged Breakwater Trsnsmlsslon

Submerged Breakwater Transmlsslon Kt vs Slopr (dsrO.15 ml H6 - 0.15 m)

0.1 1 O . . 1 0.2 0.3 0.4 0.5 0.6 0.7

Tan T b t r (bO.6 rn)

- . - .. . - - . - - . . - - & - - -. - - . . . - . . .

Submerged flreakwater Transmtssion Kt vs Slope (ds*O.OS m, Hs - 0,lS m)

0.2 0.3 0.4 0.5 0.6 0.7 Tan Thta (84 .6 m)

Submerged Breakwater Transmlsslon

Appendix F Selected 2-D RefIection Results

SELECTED 2-D TESTING REFLECTION RESULTS (Cont,)

2. Kr vs. ds ( Varied Tp) - A -. . - - -

Refiectlon

-- P

I p - 1 l r

Ip- 1 K i *

r p - 2 0 8 -

1 Submerged Breakwater Refiectlon Kr vs ds (B=Zd m. Hm - 0.05 m ) I

T p - 1 2 8

Tp- l E i

Tp -20 .

- - - -- -- - - --- - - - -- - - - --

Submerged Breakwater Reflectlon

Submergeci Breakwater Refiectlon Kr w ds (Bm2.1 m, Hs - 0.10 m ) l

T p - $ 2 8

Tp- l o i

T p - 2 O l

-W.--- - - - - . - - -

Submerged Breakwater Refiection

Submerged Breakwater Reffectlon Kr vs ds (6~2.5 m, Hs - 0.11 rn )

60

* ' 40

Tp- I Z r

Tp- 1 E r

- -. - I p - 2 0 a 20 @ . -!- .- U

(Y-

. = 4 . 9*

C

ru- . é

. I r Y?m F

' ( Y -

I - E I * m m I F

g f t l 1 z - i ' t

4 * @

i . ,

i f ! / n o. a * * * C

-. 2 1 : .

1: : * r 2

( Y -

. E - *. y

SELECTEO 2-D TEST ING REFLECTION RESULTS (Cont.)

7. Kr vs. Tan Theta et B=0.6 m ( Varied Tp)

Submerged Breakwater Reflection Kr vs Slope (ds=O.OQ m, Hs - 0.05 m)

Subrnerged Bmakwater Reflectlon Kr w Slow (ds-0.15 m. Hs - 0.05 m)

Submerged Breakwater Reflectlon Kr vs Slopr (dsr0.05 rn, Hm - 0.10 m)

0 2 0.3 0.4 0.5 0.6 0.7 Tan Theta (BnO.6 m)

-

Submerged Bmakwater Reflectlon Kr vr Slop. (dr=O.lb m. Hs - 0.10 m)

0.3 0.4 0.1 0.6 Tan Theta (BrO.0 m)

Tp- 1 6 s

T p - 2 0 s

0.7

Subrnerged Breakwater Reflection

Subrnerged Breakwater Reflectlon Kr va Slopo (dsmO.lB m, Hs - 0.16 m) l

O ' l

0 2 0.3 0.4 0.5 0.6 0.7 Tan Thot. (64.0 m)

1 a. a * a C C C u

Appendix G 3-D Testing Transmission Results

3-0 TESTING at 60 Deg.

3-D Testing : Submerged Breakwalers Effecl of Cresl W i h (BO Deg ; P2)

1 4

1 X

3-D Testlng : Submerged Breakwaters Effecl of Cresl W l h (60 Deg ; Pl )

1 4

3-0 Testlng : Submerged Breakwaters Eiiect of Crest Wdth (60 Oeg ; PB)

1 4

3-0 Tesllng : Submerged Brsikwilsn Effect of Cmst Wldth (60 Deg ; P4)

1 4

3-D Tesling : Submerged Breakwalers Efl813 of Clesl Width (80 Dw ; P3)

! 4

3-D Tesling : Submerged Breakwaters EfleCl d Cmst Wldth (80 Deg ; Pr)

1 4

3-D Tesllng : Submerged Breakwakrs Efiad of Cm81 Wdlh (&Y 089 ; PB)

Appendix H Scatterplot Correlation of Independent Variables

-- .- .&.\-.. . a..

Notes: 1 : KT = K,

B 2 : BHI = - H f

3 : DJOHi = 5 H,

d , 4 : DSHI = - Hl

h 5 : HHI = - H, L 6 : L i ï I = -

H ,

Scatterplot Matrix - 2-D Test Results (Cont.)

Notes: 1 : KT = K,

B 2 : BZLDSO = - D ,

dl 3 : BDSLD50 = - = D,, d B

4 : DSBHIDSO = - 4

4 5 : DSHI = - H, H, 6 : HIE = - B

Notes:

*:;--.* . - - &*

&. .y : . - k&-:.-... : . . . y>.; - 5 . . . p;.. & ! ; fi.?.. %Y .:-. -- il! i 9; :. - r if &'. 1 i ! ' ? - 1 - ..- - . -- -

1 - --. . . - -. - - --. . - .- - I -*

=.. -.- - L - -- - - - 15 .Y..- L - 1 L. - - -- 1

& . .- V . . .Y- E . .-- -.- / ...a . . -- L- - :1:--3f. -y... .- r . 1 . . { i ; -. . , 2 - 1

1 . 1 .,. l

. . . .- - i - * . - - - . ........ i . . . --:. i

H128L - - - 1 . . :-.. . . . . . . 5 ; ; . .. . . . . . . . . . . . . . . . . ! . ,

*...: i 1 : . ? * - . , . . . . ; ; ! ! i , < 9 . . , a<::-, . -,ï ! . ! ; . . . . . .

1 I ! 1

. - . 9

5 -- ;.*: - .- - b . ,&i. lii , , i L-* ;/ .: -.-. [ 1 *. . . . . . . S.. - - 1

. . . . . . . . - - +- I

....... - - - .- - -.. -- - - -. - - . IMtLOSS -n-. - - - -.-- - . -. - . . - - - * - - - .*. l -. . . p-2 m+w:.-k .* . -- +

fl- . . . . - . - . . I

\ - I :

. . . . - LEHI

.. I

....... . C l . . . . # . 0 2 . * * : # - . &:$-'-.. ... h::.. - . . . - - .at ; ;! ; ! - .. S . . .- _ _* -

-. . . . . 5- ! :.- . . . . . . I f ! * - - - , . I f . -

*... -.* 1 Xrc - - . . . . . . * I 0 - t

:. - - X88 -.-. - . # 1 .. y- - - . . . . v- -.:- . t - . . . . ai..- &&. p. p:.

Notes: L : C R l V = K ,

B 2 : BHI = - Hl

4 : DSHI = - *, h 5 : H H I = - *, L 6 : LHI = - Hl

7 : REHI = p rn 189 P

Notes: 1 : CRAV = K ,

d, 2 : DSB = - B

h 3 : HG27 = -

g T' h

4 : HHI = - ff ,

Notes: 1 : CRAV = Kr

8' 2 : BILDSO = - = Q,, 8 4 3 : BDSLDSO = - = 4,

d B 4 : DSBHIDSO = -

H, DID,

d. 5 : DSHI = - H, H, 6 : H I B = - B

h. - B h H t 0 4

7 : SuRFSm = [- --]O1 ( f ) 191 h 3 . 5 U o O

Appendix 1 Alternative Mathematical ModeIs

Table 1-1 AIternative Mathematical Models

Mode1 Form

Table 1-1 Alternative Mathematical Models (Con

I

eve-

ITERATION LOSS PARAMETER VALUES

[TERATION LOSS PARAMETER VALUES

DEPENDENT VARIABLE IS KT

SOURCE SUM-OF-SQUARES DF MEAN-SQUARE

REGRESSION 185.047 6 30.841 RESIDUAL 2.116 621 0.003

TOTAL 187.163 627 CORRECTED 38.925 626

RA W R-SQUARED ( 1 -RESIDUAL/TOTAL) = 0.989 CORRECTED R-SQUARED (1-RESIDUAUCORELECTED) = 0.946

PARAMETER ESTIMATE A.S.E. LOWER 4 5 % UPPER Cl 0.392 0.006 0.380 0.404 C2 0.504 0.01 1 0.483 0.526 C3 0.486 0.048 0.392 0.580 C4 0.173 0.035 0.104 0.243 CS -0.120 0.052 -0.222 -0.019 C6 0.219 0.061 0.099 0.339

ASYMPTOTIC CORRELATION MATRIX OF PARAMETERS

Eqn [7.13] Predicted vs. Obla-ved - 2-D Tests

- -

Eqn. [7.131 Normal Plot of Residuals

Eqn. (7,131 Residuals vs. d/H,

Eqn. (7.131 Residuals vs. Observed K,

Eqn. 17.13) Residuals vs. HJB

p.p- - .

Eqn (7.141 Predicted vs. Observed - 2-D Tes&

Eqn. I7.141 Normal Plot of Residuals

Eqn. [7.14J Residuals vs. Observed K,

I C3 O 1 0.2 03 04 05 36 3 7

use

Eqn. (7.141 Residuals vs. H,/i3

Eqn. 17.141 Residuals vs. d/H,

S Y S W N N 7 = 14.141

ITERATION LOSS PARAMETER VALUES

DEPENDENT VARIABLE IS KT

SOURCE SUM-OF-SQUARES DF MEAN-SQUARE

REGRESSION 187.338 4 46.835 RE S IDUAL 3.343 623 0.005

TOTAL 187.163 627 CORRECTED 38.925 626

RAW R-SQUARED (1-RESIDUA.L/TOTAL) = 0.982 CORRECTED R-SQUARED ( 1 -RESIDUAL/CORRECTED) = 0.9 14

P W T E R ESTIMATE A.S.E. LOWER 4 5 % UPPER Cl -0.646 0.016 -0.677 -0.614 C2 -1.083 0.049 -1.180 -0.986 C3 0.047 0.003 0.041 0.053 C4 -0.067 0.010 -0.086 -0.048

ASYMPTOTIC CORRELATION MAT= OF PARAMETERS