~ Hydraulics Research Wallingford

LOW-CRE ST BREAKWATERS , HYDRAULIC PERFORMANCE AND STABILITY

K A POWELL & N W H ALLSOP

Re port SR 57 July 1985

Registered Office: Hydraulics Resea; .:h Limited, Wallingford, Oxfordshire OXlO 8BA. Telephone: 0491 35381. Telex: 848552

This report describes work carried out by members of the Coastal Structures Section in the Maritime Engineering Dept of Hydraulics Research, under three contracts concerned with research on the stability and performance of rubble mound structures:

(a) PECD 7/7/130, funded by the Department of the Environment (Water Directorate), nominated officer Mr R B Bussell;

(b) DGR 465/30, funded by the Department of Transport from April 1 982 to March 1984 and thereafter by the Department of the Environment , nominated officer Mr A J M Harrison;

(c) Commission B, funded by the Ministry of Agriculture, Fisheries and Food, nominated officer Mr A Allison.

At the time of reporting this project, Hydraulics Research's nominated project officer was Dr S W Huntington.

This report is published on behalf of the Department of the Environment and the Ministry of Agriculture, Fisheries and Food, but any opinions expressed are not necessarily those of those ministries.

(c) Crown Copyright 1 985

Published by permission of the Controller of Her Majesty's Stationary Office

ABSTRACT

The research reported advances design methods for low crest and submerged breakwaters. A low-crest breakwater may be defined as a breakwater that is

frequently overtopped by wave action, but that still performs a significant function in dissipating wave energy. Such breakwaters are considerably

simpler and cheaper to construct than conventional harbour breakwaters, but do allow higher levels of wave activity in their lee. Low-crest breakwaters may be of considerable benefit in shoreline protection, and their use in this context is discussed separately by Brampton and Smallman(lS).

This report reviews techniques available to predict wave transmission and overtopping effects and armour displacement. Hydraulic model test results obtained both at Hydraulics Research and in other labs are presented graphically in terms of appropriate non-dimensional parameters. The degree of attenuation of wave effects may be estimated from these graphs for a wide range of structure crest levels and/or incident wave conditions. Test

results for structures of approximately 40% porosity allow the estimation of the wave transmission and overtopping performance with sufficient accuracy for preliminary design purposes. The attenuation provided by breakwaters of other porosities is less certain, but a basis for estimating it is suggested. More accurate prediction will depend on an ability to define the permeability of the breakwater to oscillatory flow. At the moment this is limited to estimating the porosity of the prototype breakwater and testing an appropriate scale model.

Measurements of armour movement under random waves have also been analysed. The results have been presented graphically in terms of non-dimensional stability numbers. This provides the basis of a method for estimating armour sizes needed for breakwaters subject to frequent overtopping. A number of anomalies are discussed and it is concluded that all such designs should be finalised by appropriate physical model testing.

NOTATION

B c d db ds D Ed Ei E r Eto Ett g ii Hb Hi H r Hs H to Htt Kd K r Kt Kto Ktt L p L s ra

N N a p

R R c

Breakwater crest width Wave celerity Nominal water depth Depth at wave break point Depth of water at structure Representative stone diameter Wave energy dissipated Incident wave energy Reflected wave energy Wave energy transmitted by overtopping Wave energy transmitted through structure Acceleration due to gravity Mean wave height Breaking wave height Incident wave height Reflected wave height Significant wave height Wave height corresponding to E to Wave height corresponding to Ett Coefficient of dissipation Coefficient of reflection Coefficient of total transmission Coefficient of transmission by overtopping Coefficient of transmission through structure Wave length corresponding to T p Wave length at structure toe Sea bed slope (= tan 0) Number of waves overtopping Number of armour units in structure Porosity Notional run-up level Breakwater freeboard Sea steepness (= H/L )

0 Nominal wave period Period of peak spectral energy Zero crossing wave period Median weight of armour stone Number of armour stones displaced

Dens i ty of rock

Dens i ty of wa te r

CONTENTS

1

2

3

4

5

6

7

6

LOW-CREST BREAKWATERS

1.1 Introduction 1.2 Object of study 1.3 Outline of report

PREVIOUS WORK

2.1 Definitions 2.2 Wave reflections 2.3 Wave transformations 2.4 Wave transmission by overtopping 2.5 Wave transmission through porous structures 2.6 Rock armour stability

WAVE TRANSMISSION

Page

1

1 2 2

3

3 4 5 6 6 9

10

3.1 General 10 3.2 Transmission characteristics of low-crest structures 10 3.3 Effect of wave period 12 3.4 Effect of breakwater porosity 14 3.5 Effect of breakwater crest width 16

BREAKWATER STABILITY

4.1 General 4.2 Hudson stability number 4.3 Spectral stability number

DESIGN GUIDELINES

5.1 Wave transmission 5.2 Stability

CONCLUSIONS AND RECOMMENDATIONS

ACKNOWLEDGEMENTS

REFERENCES

FIGURES:

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

3.9

3.10

Breakwater cross-sections - Reference 5 Breakwater cross-sections - Reference 5 Breakwater cross-section - References 30 and 39 Wave transmission data - Reference 5

Wave transmission data - References 30 and 39 Wave transmission data - Reference 5 Wave transmission data - References 30 and 39 Variation of wave transmission with wave period -References 30 and 39 Variation of wave transmission with wave period -Reference 5 Effect of porQg ity on wave transmission coefficient (after Seelig{ ))

17

17 17 19

2 1

21 22

22

23

24

CONTENTS (CONT'D)

FIGURES (cont'd)

3.11

3.12

3.13

4.1

4.2

4.3

4.4 4.5

4.6

5.1

5 .2

Effect of porosity on wave transmission coefficient -data from Reference 27

Effect of breakwater crest width on wave transmission -Reference 39

Effect of breakwater crest width on wave transmission -Reference 27.

Front and ��8� face damage as a function of N8 -long waves Front and b9l�)face damage as a function of N8 -short waves\. Damage as a function of Hudson's stability number , Ns Damage as a function of Hudson's stability number , Ns * Damage as a function of the spectral stability number, N Damage as a function of the spectral stability number , N! Wave transmission envelope for low crest breakwaters as a function of Rc/Hs Wave transmission envelope for low crest breakwaters as a function of R�

1 LOW-CREST BREAKWATERS

1.1 Int roduction The hyd raul i c performance of a breakwater wi t h a low c res t subjected to f requent wave overtopping may be t reated as resulting from the combination of four d i f f erent mechanisms : -

(a) Where the s ea bed topograp hy changes, and part icularly in the form of a s t ructure that p ierces the water surface, s ome propo r t i on of t he inc ident wave energy will be ref lec t ed. In general t he longer t he inc ident wave leng t h, the higher the proport i on that may be ref lect ed by a breakwa t er. S horter or s t eeper waves may d i s s ipate more of their energy at t he s t ructure ref lect ing a lower proport ion.

(b) Where the cres t of the s t ructure is always immersed, some of the energy not reflected will then be d i s s i pated in turbulent f ri c t ion in the flow over the s t ructure cres t. S hor t per iod waves may pass over such a breakwater rel at ively unchanged, but longer per iod waves will tend to diss ipate some energy.

(c) Under some combinat i ons of c res t f reeboard and i nc ident wave cond i t ions , waves may s hoal and break on the s t ructure, or run up and overtop, d i s s ipating further energy. In breaking, or overt op ping, each wave will give rise to a number of smaller and s horter waves , in the lee of the s t ructure, causing a shift in t he wave peri od, or f r equency, cor responding to t he energy peak, as wel l as a reduct i on in the to tal energy. Again t he degree and f orm of energy d i s s ipat ion or change wi l l depend s t rongly on the inc ident wave period, or wave length, as well as the wave height. Generally longer shallow waves will give r i se to g reater levels of overtopping, and hence wave t ransmi s s i on over the s t ruc ture, than s hor ter steep waves. S t eep waves may break on the breakwater d i s s ipat i ng much of their energy.

(d) Wave energy may also be t ransmitted by wave-induced osci llatory f low through the vo ids in a porous rubble, or other, breakwater. In rubble s t ructures such flow occurs princ i pally in the outer and upper layers, especially if a layered cons t ruction i s used. Wave t ra nsmi s s ion may usually be regarded as negligible for s hort period waves , but may be of importance for porous s t ruc tures subject ed to long waves.

The t ransmi tted wave may t herefore be viewed as t he res ul t of t he combined ef fec t s of each of these four mechanisms. As, however, t heir interac t i on is complex and ill-def ined mos t researchers have chosen to concentrate on only one, or two, mechanisms at a t ime. The technical literature available may therefore be cons idered under t he following general head ings:-

1

1.2 Object of study

1.3 Outline of report

(a) Reflection at a structure {b) Transformations over submerged structures (c) Transmission by overtopping {d) Transmission through a porous structure

As indicated above, the relative effect of each of these mechanisms will differ with both wave conditions and structure geometry. In considering each mechanism , researchers have therefore often considered different aspects of wave behaviour, usually with widely differing analytical or experimental techniques. Their results may therefore not be fully compatible. The rest of this report does however seek to identify the principal effects of each mechanism on the overall performance of low-crest breakwaters.

The primary aim of this study is to provide design methods to allow the estimation of the hydraulic performance and stability characteristics of rock armoured breakwaters. Using these estimates, it is suggested that preliminary cross-sections may be produced for initial costing and feasibility purposes.

The data needed to provide this information has been obtained from recent site specific and fundamental research studies conducted at Hydraulics Research and from published work principally by the US Army Corps of Engineers. Most of the data considered in detail was derived from random wave model studies.

Before considering the data available in detail, a summary of previous work in this field is given in chapter 2. The main transmission characteristics of low-crest breakwaters are considered in chapter 3. Special attention is paid to the mechanisms of wave transmission as well as to the effects of wave period , breakwater porosity and breakwater crest width on that transmission. Dimensionless design graphs are given , from which the wave transmission coefficient may be estimated for various breakwater freeboards and incident wave conditions. Chapter 4 considers the stability of low crest breakwaters in terms of the long established Hudson's stability number and the relatively recent spectral stability number. Qualitative comparisons of the two parameters are made. In chapter 5 the accuracy and applicability of the results is assessed , particularly with regard to their use as design guidelines. Further discussion is offered concerning the recurrent anomalies observed within the data scatter. A tentative explanation of these anomalies is given. Chapter 6 draws together the conclusions arising from this report and makes recommendations for further work in this field.

2

2 PREVIOUS WORK

2.1 Definitions In considering the reflection, dissipation and transmission of wave energy at a breakwater or similar coastal structure, a number of simple coefficients and terms may be useful. Each of these may most easily be described by considering a simple structure in water of constant depth, d, subjected to waves of incident energy, Ei, and ha�ing a wave height, Hi, generally proportional to Ei (For the purposes of definition regular waves will be considered, however for random waves energy may be divided into frequency bands).

At the structure part of the incident energy is reflected, Er, equivalent to a reflected wave height, Hr· The coefficient of reflection, Kr may then be defined: -

(2 .1)

Some of the remaining energy may be transmitted by overtopping, Et0, equivalent to a wave height, Hto• Similarly energy may be transmitted throug h a porous structure, Ett• equivalent to H tt• Wave transmission coefficients may be defined for each of these two cases, using the same terminology as above: -

H (

E to)� Kto -�-i E"f (2.2)

and

Ktt == Htt _ (

Ett)� � - Ef i (2. 3)

T hat proportion of the energy incident on the structure that is neither reflected nor transmitted must necessarily be dissipated in the various processes at the structure. The energy dissipated, Ed, may be used to define a coefficient of energy dissipation, �:-

Kd • (

Ed)� Ei

The energy balance may be written:-

Ei = Ed + Er+ E to + E tt and the coefficient of dissipation:-

Kd = (1- Kto2 - Ktt2 - Kr2)�

(2.4)

(2. 5 )

(2.6 )

In some circumstances no distinction is drawn between energy transmitted throug h the structure and that transmitted by overtopping, a single coefficient may

3

2.2 Wave reflections

then be defined:-

(2. 7) As mentioned earlier, in the process of wave reflection and transmission by overtopping, a shift of energy from low frequencies to high frequencies may occur as single large waves break and reform as a number of small short waves. Care must be taken in using the definitions above to ensure such a frequency shift is allowed for.

The prediction of the level of reflected wave energy is addressed by various researchers using different approaches. Both analytical and experimental techniques are reported. In general, however, most methods available rely on model tests to determine values of the empirical coefficients used, and many of these tests are reported in the literature. The case of breakwaters that both transmit and reflect is relatively lightly covered.

Much of the recent work is summarised by See�ig{l), and, in a longer version, by Seelig & Ahrens<2 J. Both present simple prediction methods for reflections from beaches, seawalls and breakwaters. The special case of both transmission and reflection is c�3�red briefly citing the model work of Sollitt & Cross J and the prediction method of Madsen & White<4). Seelig(l,5) presents a simple prediction equation for the reflection coefficient at a rubble breakwater in terms of the surf similarity parameter, or Iribarren number, Ir, and empirical coefficients a and �:-

K = a rr2 (2.8)

r � + Ir2

where Ir = tan 9/s\ and s = H/L 0

Using a= 0.6 and � = 6.6 this is likely to give conservative results. It should be noted however that it is only directly applicable to regular waves, and moreover it does not account for any significant transmission of wave energy.

Seelig & Ahrens<2) also discuss the influence of layers of armour over an impermeable core or embankment. They suggest that equation 2.8 should be used with � = 5. 5 and a defined by:-

a= exp [-1.7 <k>0•5 cote- 0.5 (�)1•3] b

(2.9)

where Ls is the wavelength at the struciy�e toe, D is the representative stone diameter {W/p) and Hb is a representative breaking wave height. Values of a correction factor a' to be applied to a are tabulated for 2-4 layers of armour, and for ranges of relative

4

2.3 Wave trans­formations

armour size, D/Hi.

Other methods for the p�gdiction of wave reflections are presented by Moraes� )� Batt�es<7>, Madsen & White (4) and Madsen and others< 4,25, 6).

Methods for the measurement and analysis of incident and r�{ lected waves have beeo

8discussed

cf!)Madsen &

White� ) Thornton & Calhoun� ) Kajima , Gilbert & Thompson{13), Gaillard et a1<14,, and Goda & Suzuki (15). Measurements of reflections from breakwaters in model studies are presented by Seelig & Ahrens(2), Madsen & White(4), Seelig(5), CERc(9J, Kondo et al (10), and Ijima et al(11J. Apparently the sole analysis of wave reflections from a prototyp� structure is presented by Thornton & Calhoun�8).

For convenience, wave reflections and transmission a t wholly submerged structures are considered separately as elements of wave transformations in the following section.

When waves encounter a change in sea bed level a number of transformations may occur to the waves. These may be simply summarised as:-

a) refraction b) diffraction c) friction d) shoaling

and e) breaking.

Of these, refraction and diffraction effects are not considered further in this report.

(9Jhe interested reader is refer�ed to work by C�RC , Longuet-Hi§gins\16), Brampton<1 ) and Brampton & Smallman<1 J.

The effects of significant changes in sea bed level on wave transformations, such as produced

(f9)a submet!Bj breakwater, are conside��d by Brampton , Miles

and in passing, by Lamb� l). The effect of turbulent friction at the sea bed boundary i�yer has been discussed by Treloar & Abernethy( J and by Hydraulics Research(22J. Both laminar and rough turbulent friction laws are considered, and simplified calculation methods are presented.

The influence of sea bed slope and water depth on wave height and length, shoaling, has been discussed by very many authors, and is summarised by CERc(9). Many authors have also addressed wave breaking, and this remains an active topic for research. For waves passing over a simple sea bed of slope m ( = tan 8), the relationship between the wave height at br{��ing, Hb, and the depth at breaking, db, is given by :-

5

2.4 Wave trans­mission by over topping

(2.10)

where a and b are functions of the beach slope, m:

a a 43.75 (1 - exp (-19m)) b a 1.56/(1 + exp (-19.5m)).

Research into the performance of low-crest breakwaters has concentrated primarily on the level and form of wave transmission. As no fully satisfactory formulation for the hydrodynamic processes of wave overtopping has been produced, researchers have generally concentrated on physical model studies, from which they have derived empirical expressions for the wave transmission coefficient. These model tests have generally studied one or more of three principal structure types:-

(a) wholly impermeable, solid; (b) rubble mound with impermeable layer, barrier or

core; (c) permeable rubble mound.

The general form of such structures is trapezoidal, although some experimenters have used rectangular sections, principally to study transmission through porous structures.

In a sig§le instance, reported by Thornton and Calhount ), measurements were made of both wave reflection and transmission at a prototype structure, a rubble mound breakwater at Monterey, California.

The results of model studies of wave transmission, principts}Y by or§}topping, are prefz?}ed by Seelig , ��RC , Dattatri et al , Bade &

Kaldenh�!} ( ), Ouellet & Eubanks(29) and Allsop<30). Seelig presents wave transmission coefficients for a wide variety of breakwater sections, principally in terms of a wave steepness parameter, H/gT2. A major conclusion of that study is that the transmission by overtopping may be estimated from the crest freeboard, Re, and a notion run-up level, R, by:-

R Kto = c (1 - ___£) (2 .11)

R

where

c = 0.51 - O.ll B , Re + ds

and B is the structure crest width and ds is the water depth at the structure. For submerged breakwaters with approach sea bed slopes around 1:15 an extended expression is suggested:-

6

Kto • C(1 -R

C) - (1 - 2C) Rc

R R (2 .12)

For both equation 2.11 and 2.12, Seelig suggests that the value of the notional run-up level, R, may be calculated using normal prediction methods for run-up levels. A number of analytical and empirical methods are discussed. For stable rock breakwaters a single expression is suggested: R "" __.:;;;__.:;;;.;;;.__

H 1 + b Ir where

Ir ,. tan 0

s\

(2.13)

and a and b are empirical coefficients having values a • 0.692 an4 b = 0.504. Recent work by Allsop et al(31,32) has discussed the prediction of wave run-up levels under both regular and random waves, and a number of empirical expressions have been identified for various armour units.

CERc<9> use Seelig's method for predicting Kto• Graphs are presented allowing the prediction of the overtopping coefficient under random as well as regular waves. It is however noted that this method may overpredict the value of Kto•

Dattatri et al<27) present results of regular wave tests on submerged breakwater sections of rectangular, triangular and trapezoidal section. The effects of the relative breakwater crest width, B/L, and relative freeboard, Rc/ds, on the wave transmission are presented, but no general prediction method is derived. The authors conclude that the permeability of a submerged breakwater has relatively little effect on the wave transmission char��g�ristics of such a structure. Bade & Kaldenhoff\ J studied the transmission of short sequences of irregular waves over a cube armoured trapezoidal breakwater of crest freeboard, Rc/ds from -0.1 to +0.1. The authors present graphs of the transmission coefficient � against a dimensionless freeboard, 1-Rc/Hs, but concentrate primarily on the different transmission charact{2g�tics of the wave groups used. Ouellet & Eubanks also used irregular waves against a trapezoidal rubble breakwater, armoured with dolos on the seaward face. A range of water levels were used, all below the structure crest level. Values of the transmission coefficient, �, are plotted against frequency. In general, however, the results of this study are presented in an unconventional manner, and it is not possible simply to generalise the results.

Allsop<30> considers a series of multi-layer rock armoured breakwaters of conventional form with crest

7

freeboards in the range, Rc/ds a 0.23 to 0.56, subjected to random wave attack. Measurements of the number of waves overtopping, N, and the transmission coefficient, Kt, are presented against values of dimensionless freeboard RciRs and R*, where:-

R R* =�<h)\ s it

(2.14)

Allsop concludes that N may be described by a function of Rc/Rs, but that � is better described by R*. Expressions suggested by regular wave tests are tried, but are found to underpredict N and Kt at low values of relative freeboard.

2.5 Wave trans­mission through porous structures

The transmission of wave energy through the voids of rubble mound breakwaters has been studied both experimentally and analytically. Some researchers have sought to derive mathematical models using linearized formulae for viscous drag, calibrating the empirical expressions against physical model results, and then re-running the mathematical model for prototype conditions. Mathematical models have been presented by Madsen and co-authors(4,24,25,26), Kondo et al(lO) Ijima et al(ll), Seelig(33) and Uassel & Butowski(J4). The results of physical model tests are presented by Madsen & White<4>, Seelig(33) and Kondo et al(lO). Measurements of transmission through a prototyge breakwater are presented by Thornton & Calhoun\8) .

Madsen & White(4) describe the derivation of, and Seelig(5) documents and lists, a computer program to estimate wave transmission, and reflection, at a porous rubble breakwater. The method models transmission of long waves through and reflection from a porous structure in two stages. Energy dissipated and reflected at the seaward face is considered in the first stage. The second considers the energy dissipated in viscous drag in the flow through the voids of the structure. Simple empirical relationships tested against model test results are used to estimate energy losses. In their derivation, Madsen & White only consider reg�!,r waves, having height, R, and period T. Seelig however suggests that the method may be used to estimate the performance of structures subjected to random waves by setting H=H and T=Tp.

In later work, Madsen et al<26) attempt to analyse the flow at the seaward face of a trapezoidal breakwater in a less artificial manner in order to refine the estimation of wave reflections. T��G conclude however that neither the new method >, nor the previous method<4>, estimate the level of reflections well, although both give relatively good estimates of

8

2.6 Rock armour stability

transmission.

Ijima et al(ll) describe an analytical method for the estimation of wave reflection and transmission at various structures. Work with triangular and rectangular section porous breakwaters is presented very briefly. The prediction method is claimed to work well for rectangular sections, but less well for triangular, and tb§!efore trapezoidal, structures. Massel & Butowski( J use arbitrary wave spectra and rectangular porous breakwaters. Using a similar argument to that presented by Hydraulics Research(22), random waves are treated as the sum of small amplitude periodic waves, each of which may be treated linearly. The fluid damping in flow through the porous structure is represented by a linear term, approximating turbulent friction. Values for transmission and reflection are determined by integrating over the spectrum. The method is tested against the results measured at Monter�y

)breakwater, California by

Thornton & Calhounltl • Agreement with the total level of energy disipation is reasonable, but the division between reflection and transmission is not well described.

Kondo et al(lO) present model test results demonstrating the influence of core permeability, and position within the structure, on the reflection and transmission performance. The results are again compared with those calculated by an analytical approach using a linearized friction loss method.

Low-crest breakwaters, designed to allow some overtopping, may have a stricter design criterion, with regard to the stability of the primary armour layers than breakwaters which do not

(��jrtop. A gymber of authors (Lgrding and Scott , Raichlen lJ6) and Lillevang(J/)) have noted that the armour on the back face of a low-crest breakwater is more likely to be displaced by heavy ovef§B�ping than the armour on the seaward face. Allsop concludes that the total damage to a low-crest rock armoured breakwater attacked by waves of steepness Hs/Lp>0.03 is dependent upon the stability number Ns, but not upon the freeboard. Furthermore damage to the back face of such a breakwater is best described by a dimensionless freeboard parameter, R*.

Ahrens(JS) relates low-crest breakwater damage to the stability number, Ns• He concludes that there is a wave period effect with damage increasing with increasing Tp· Furthermore he suggests that this wave period effect might be accounted for by the use of a modified stability number, Ns*, which includes a measure of the wave length.

Little other stability testing of low-crest breakwaters in random waves has been documented.

9

3 WAVE TRANS­MISSION

3.1 General

3.2 Wave trans­mission characteristics of low-crest breakwaters

Indeed, most designers seem to rely on stability formulae and coefficients derived from regular wave tests only.

As discussed previously there are two basic modes of wave transmission for surface piercing, permeable low-crest breakwaters, that is, transmission through and transmission by overtopping. In the case of submerged permeable structures shorter waves will propagate above the breakwater while longer waves pass partly above the structure and partly through it. Thus there is a third mode of wave transmission, that is, transmission above a submerged structure. The relative proportions of any of these modes of transmission are dependant upon the relative freeboard, the breakwater permeability, the water depth and the wave period.

The coefficients of transmission by overtopping, Kt0, and through the structure, Ktt• have been defined in Section 2.1. However, Kto and Ktt' as defined, have a basic failing in that they are determined solely by the ratio of transmitted to incident wave height. They cannot therefore account for any frequency shifts which may occur as waves are transmitted through or over the structure. To incorporate such information in a simple transmission coefficient may well require the coefficient to be defined in terms of frequency, as are the incident and transmitted wave spectra. This may however be over-complex given the present state of understanding of the hydrodynamics.

The data used in this study was obtained from a number of different sources and consequently a wide range of breakwater constructions have been investigated. The cross-sections of these breakwaters are shown in Figures 3.1 to 3.3. All are of the low-crest type and both surface piercing and submerged structures are included.

The transmission performance data from reference 5 and references 30 and 39 is presented in Figures 3.4 and 3.5 respectively, in terms of Kt and the dimensionless parameter Rc/Hs; Re is the breakwater freeboard and may be either positive (surface piercing) or negative (submerged). As might be expected the overall trend is one of decreasing wave

10

transmission with increasing freeboard. For values of Rc/Hs < 1.0, Figure 3.5 suggests slightly higher levels of wave transmission than those given by Figure 3.4. However this discrepancy may be ascribed to the different permeabilities of the breakwaters tested. Where the breakwaters are more permeable, as in the case of Figure 3.5, the increased transmission through the structure will result in an increase in the overall value of�· It might also be expected that this effect would account for much of the data scatter apparent in Figure 3.4.

The Shore Protection Manual(9 ) contains a curve for the prediction of wave transmission by overtopping only. This curve, when re-worked and plotted in Figures 3.4 and 3.5, provides qualitative agreement with the general trend. It also serves to emphasise that for 0.0 < Rc/Hs < 1.5, overtopping is the major mode of wave transmission. The upper limit to this range will however vary from breakwater to breakwater depending on the attenuation to wave run-up afforded by the primary armour layers. For further details of run-up performance the interested reader is rt��}red to work by Allsop, Hawkes, Jackson and Franco •

For surface piercing structures with Rc/Hs ) 1.5, the qualitative agreement

(gjtween Figures 3.4 and 3.5

ceases, with Seeltj'� data showing an increase in Kt while Allsop's ( 0 J data suggests a small, but constant, level of wave transmission, presumably through the structure. The problem is that for relatively high breakwaters transmission is no longer dominated by wave overtopping but by energy transmission through the structure, which is a function of, amongst other things, wave steepness. It is this changing role in the dependence of � on Hs which causes the paradoxical trend in Seelig1s data. Thus, Figure 3.4 does not imply that for a fixed incident wave height the transmitted height will increase if the freeboard is increased, but rather that with a fixed freeboard the transmission coefficient will increase if the incident wave period is increased (or if the wave height is reduced and the waves become less steep). A similar trend should be expected in Figure 3.5 but, owing to the lower permeability of the breakwater modelled by Allsop, its onset may be considerably delayed.

Owen<40) suggests that the overtopping performance of seawalls can be expressed in terms of a dimensionless discharge, Q*, and a dimensionless freeboard, R* , where

(3.1)

11

3.3 Effect of wave period

and Tz is the zero crossing wave period.

Tz is however a less desirable measure of periodicity than Tp, the period of maximum spectral energy, due to it being difficult to assess from wave spectra, and because it may vary with the frequency shifts occurring as a result of wave transformations at the breakwater. Tp, on the other hand, is relatively unaffected by wave transformations. Furthermore its use allows compa�� �ons to be drawn with published American results l J.

The dimensionless freeboard has therefore been re-defined as:-

(3. 2)

The physical significance of which is perhaps best appreciated if equation 3. 2 is re-written in the form:-

R*' =

Rc (-S )� (3. 3) Hs 2n:

where S is the wave steepness corresponding to TP.

Equation 3.3 implies that for waves of constant steepness, R*' is simply related to the ratio of structure freeboard to significant wave height.

Figures 3.6 and 3.7 present tg) transm{�§}on performance data from Seelig and HR , in terms of Rt and R*'· Both figures exhibit a trend very similar to that given by previous plots of Kt against Rc/Hs• However the upward trend of Kt for large values of Rc/Hs is nullified in Figures 3.6 and 3.7 by the inclusion of the wave steepness in the parameter, R*'· Furthermore, it appears that the use of R*' as the freeboard parameter may result in a slight reduction in the scatter of data, perhaps due to the inclusion of wave period effects within the parameter.

It may be assumed that the influence of wave period will differ for both submerged and surface piercing breakwaters. For permeable surface piercing structures Kt will increase with increasing wave period due to the longer waves propagating more freely through and over the structure. For submerged breakwaters, short period waves should pass almost unhindered over the structure (depending on the

12

freeboard), while longer period waves, which propagate deeper in the water, will be partially attenuated. However, no matter how permeable the breakwater is, the efficiency of wave transmission through it will never be 100%.

The changing influence of wave period on the transmission characterisitics of low crested breakwaters is illustrated in Figures 3.8 and 3.9. Both figures are purely qualitative and no reliance should be placed on the absolute values they suggest. However the reversal of the wave period effect is clear, occurring at Rc/ds � 0.1 for data from reference 39 and Rc/ds � -0. 35 for data from reference 5. The relative depth of submergence, Rc/ds, at which this reversal occurs will probably depend upon the relative permeability of the breakwater to the wave periods being considered.

One further point that arises from Figures 3.8 and 3.9 is the apparent discontinuity of the general trend at Rc/ds = -0.05, ie 5% submergence. Had such an effect occurred in just one of the figures it might well have been discounted as an anomaly, but for the same effect to occur at exactly the same point in both figures seems to be more than sheer coincidence. Indeed, retrospective analysis of Figures 3.4 to 3.7 reveals a similar trend within the scatter. The explanation for this phenomenon possibly lies with the mechanisms responsible for wave transmission, or more precisely with the interaction of the mechanism of wave transmission above a submerged structure and the mechanism of transmission over a surface piercing structure. It may be postulated that transmission by overtopping commences not at the still water level, but at the level at which the breakwater crest is first exposed. This level is determined by the incident wave conditions. Thus, the discontinuity in the overall trend may result from the superimposition, between the still water level and some critical level determined by the wave conditions, of the rapidly declining coefficient of transmission (wave transformation effect) above the structure and the comparatively large coefficient of transmission by overtopping.

This argument is lent {�i1ence by the theoretical considerations of Lamb concerning the reflection and transmission characteristics of shallow water waves (L>)d) propagating over a submerged step. From these considerations Lamb derived reflection and transmission coefficients in terms of the wave celerity, C, where,

13

3.4 Effect of

and

c2+ cl K = 2C2

t c2+ cl For the shallow water approximation:-

(3.4)

(3.5)

(3 .6)

The subscripts 1 and 2 refer to the wave celerities and water depths above and beyond the step respectively. Combining equations 3.5 and 3.6 yields an expression for the transmission coefficient in terms of the water depth only:-

K • t 2d l:i 2

d2�ll:i (3. 7)

This expression has been plotted in Figures 3.8 and 3.9. Equation 3.7 effectively represents the transmission of wave energy above a submerged breakwater. The resulting curve agrees remarkably well with the general trend but, more importantly, it also passes almost directly through the point of discontinuity. As such Lamb's curve would appear to confirm the hypothesis previously suggested.

Further research into this phenomenon is however still required in order to;

(a) Confirm the phenomenon (b) Substantiate the explanation given above (c) Assess the effect of this small 'window' in the

general transmission trend on the optimum cost/benefit design of submerged breakwaters.

breakwater porosity The porosity of a permeable breakwater may be defined as the ratio of the volume of voids within the breakwater to the total breakwater volume. As such, the porosity, together with the incident wave conditions, will determine the level of wave transmission through the structure. This scenario may however be complicated, particularly for low permeability structures, by the additional effects of void shape and the tortuosity of the flow path through the breakwater. There is however virtually no data available for these latter two effects. Similarly, there is very little published data concerning the effects of breakwater porosity on the transmission of wave energy.

14

Seelig(S) presents a plot of wave transmission through a rubble mound breakwater as a function of wave steepness for different breakwater porosities. This graph is reproduced in Figure 3. 10 and is based on transmission coefficients Pf�1icted by the computer program of Madsen and White , for regular waves. The most important features of the graph are that:

1. The predicted transmission coefficient increases with decreasing wave steepness, and

2. The change in the predicted value of Ktt for a given change in porosity is greatest for waves of small steepness, i. e. long waves.

These two effects are likely to be true for both submerged and surface piercing structures.

Dattatri, Raman and Shankar<27 ) studied the effect of breakwater porosity on the transmission of wave energy past a submerged breakwater. Unfortunately the only data readily available relates to rectangular structures under regular waves. Nevertheless this data for breakwaters of porosity 0%, 41% and 42%, can be presented in terms of Kt and Rc/H - Figure 3. 11.

The general trend agrees favourably with that of Figures 3. 4 and 3.5. Dattatri et al's data also suggest however that there may be a non-linear increase in Kt with porosity, such that Kt is far more sensitive to porosity over a range of, say, 0.4 < P < 0. 5 than it is over a range of 0 < P < 0. 2. This sensitivity would appear to be heightened as Rc/H decreases.

The results presented by HR(39 ) (Figure 3. 5) also relate to a breakwater of porosity 0. 4. However, transmission coefficients for this breakwater are considerably higher than those given by Dattatri et al for breakwaters of comparable porosity. This is probably almost entirely due to the differing ranges of breakwater crest widths, B, relative to wave length, L, used in the two studies. Dattatri et al used 0.08 <B/L <0. 32 while HR used 0. 03 <B/L<0. 07. Naturally, as the value of B/L increases the transmission through the structure will decrease, hence the lower values of Kt for Dattatri et al's results. The different breakwater shapes and wave conditions used in the two studies may also in part contribute towards the different values obtained for Kt•

For surface piercing structures, porosity may also act to restrict transmission by overtopping through the attenuation of wave run-up. The greater the porosity and roughness of the breakwater armour layer, the greater the attenuation of this run-up, and hence the

15

3.S Effect of breakwater crest width

reduction of wave transmission by overtopping.

Due to the paucity of detailed information, regarding the effect of structure porosity on the transmission performance of low crest breakwaters, it is almost inevitable that further research will be required if realistic design guidelines are to be produced. However, it appears that a short, but well planned, series of model tests may provide sufficient information to allow the confident prediction of this transmission performance.

Results obtained in the previous section have indicated that the transmission through a porous breakwater may be reduced by increasing the crest width. It is logical to suppose that the transmission by overtopping may similarly be reduced by an increase in crest width. The problem arises in determining the size of any subsequent reduction in Kt for a given increase in crest width. Moreover, such an increase may not be cost effective.

At present there is little or no data available on the quantitative effect of increasing the breakwater crest width for surface piercing structures. However some, limited, data is av����ble from recent studies at Hydraulics Research� for submerged structures. This data is presented, in terms of Kt and the dimensionless freeboard R*', in Figure 3. 12. Surprisingly the results indicate that increasing the breakwater crest width by SS% {equivalent to a change in crest width from 4. 5 m to 7.0 m prototype) yields at best only a 10% reduction in wave transmission. This reduction comes about by reduced transmission through the structure and increased friction losses across the crest.

Dattatri, Raman and Shakar<27) also present data for crest width effects on wave transmission passed sumnerged structures. These results are plotted in terms of Kt and Rc/d in Figure 3.13. Unfortunately there is insufficient information availabl�3g � allow a direct comparison with the results from HR� •

However Dattatri et al's results do appear to confirm the previous findings, with a 400% increase in crest width resulting in a reduction in Kt by a factor of only 0. 2. This reduction factor appears to be fairly constant over the range of structure heights considered. Dattatri et al also suggest that the crest width influences the transformation of wave energy by prompting wave breaking above the structure. They conclude that any increase in the crest width over the minimum necessary to trigger breaking, is

16

4 BREAKWATER STABILITY

4.1 General

4.2 Hudson stability number

unlikely to have any significant influence on the transmission characteristics. However, it should be emphasised that Dattatri et al considered a totally impractical range of relative crest widths, B/L, (0.08 < B/L < 0.32) from the point of view of prototype constructions.

This report has so far been concerned mainly with the wave transmission characteristics of low crest breakwaters. It is however, worth remembering that these breakwaters will only continue to function as required whilst they are relatively undamaged by wave action. If the breakwater is unstable under the design wave conditions its performance in respect of, amongst other things, wave transmission, will be impaired. Consequently higher levels of wave activity may occur in the lee of the breakwater than allowed for. The stability of the breakwater crest is therefore of particular importance.

The stability data used in this study W�q ta�en principally from Allsop(30) and Ahrens(JH,41 ). Although the stability of a rubble mound breakwater is usually described by the "zero damage" wave height or sea state, it is clear that in random waves some small armour movement is possible at comparatively low sea states. It is, therefore, more useful to determine the damage behaviour of the structure over a range of wave heights, or sea states. In this study damage has been defined as the number of units extracted from their original position, �, expressed as a percentage of the total number of armour units, Na.

Hudson and Davidson<42) concluded from the results of tests in regular waves with no overtopping that the stability of rubble mound breakwaters is a function of a dimensionless stability number, Ns,

Ns = ____ H..::..s ___ _

where

(W5o//3

(Pr - 1) Pr Pw w50 is the median Pr is the density Pw is the density

armour stone weight of armour stone of water.

(4.1)

In effect Ns is a dimensionless wave height and as such does not contain a wave period or sea steepness term. Furthermore its derivation in tests that

17

allowed no overtopping suggest that it may overestimate the median armour weight required for front face stability on low-crest breakwaters.

Back face damage is considered by Allsop<30) to be dependent more upon the overtopping discharge and hence upon R*, than on any stability number. Howeve r a comparison of the data for back and front face damage, as presented by Allsop, reveals that for long waves (Figure 4.1) damage to both faces is equally well described by Ns. Conversely, for short waves (Figure 4.2), Ns is a poor descriptor of the damage to both faces. Figures 4.1 and 4.2 also show very similar damage levels for both face s as a function of Ns. This suggests that one particular design value for Ns may be applicable to both the front and back face of a breakwater.

Figures 4.3 and 4.4 present damage, t:JNa, in terms of Ns for various relative freeboards Re/d. The overall trend is that of an increasing number of stones being extracted as the wave attack becomes more severe. Although the stability number, Ns, would seem to offer a reasonable explanation of the damage to the structures, closer inspection of the data in each of the figures suggest that there is a wave period effect; damage increasing with increasing values of TP' all other factors being equal.

An exponential regression analysis may be performed to fit curves of the following form to the data:-

� = A exp(B.Ns) (4.2)

where A and B are empirically derived coefficients. The results of such an analysis are summarised be low:-

Coefficients Correlation DATA Rc/d A B r2

Ref (38) o.o 0.23 0.86 0.90 Ref (41) 0.2 0.32 0.89 0.88 Ref (38) 0.4 0.21 1.19 0.62 Ref (30)' Long Wave 0.29 Ref (30), Long Wave 0.39 0.028 2.25 0.74 Ref (30), Long Wave 0.57 Ref (30), Short Wave 0.23 Ref (30), Short Wave 0.38 0.008 2.31 0.56 Ref (30), Short Wave 0.57

Compari�gn yt the resulti�� curves for the data of Ahrenst •4 ) and Allsopt ) is not strictly valid owing to the diffe rent breakwater constructions tested by the two authors. Ahrens used homogeneous surface

18

4.3 Spectral stability number

piercing rubble breakwaters constructed from a rock size sufficiently small that the structure would collapse under wave action, eventually forming a stable submerged mound. Allsop, on the other hand, used a multilayered surface piercing rubble mound breakwater that was designed so as to minimise damage. The use of the damage parameter A/Na, as defined, implies that damage for Ahrens homogeneous breakwaters is a volumetric measure whilst that for Allsop's breakwaters is effectively a surface area measure. Thus Allsop's damage level will, by definition, be much lower. Indeed the damage levels measured by Allsop are so small that the extrapolated trends may not be significant.

In conclusion the trend of increasing damage with increasing stability number, Ns, is reasonably described by curves of the form A/Na = A exp (BNs), however it is difficult to envisage such curves being universally applicable to all types of breakwater construction. Furthermore, use of the Hudson stability number tends to introduce a wave period effect, which increases the separation of the data.

In an attempt to account for the wave period effect appareq§ in plots of the Hudson stability number, Ahrensl 8 ) suggested a modified stability parameter the spectral stability number, Ns*, where,

Ns* = (H 2 L )1/3 s p w 1/3 p

(�) (.....!'.- 1) Pr Pw

and LP is the wavelength corresponding to Tp.

(4. 3)

A similarly modified par�meter had previously been mooted by Gravesen et all43J, based on model studies of breakwaters which, similar to Hudson•s(42), did not overtop. Again, therefore, this new stability number, Ns*, may overestimate the weight of armour stones required for front face stability, on breakwaters that are designed to overtop.

Figures 4. 5 and 4. 6 present plots of damage, A/Na, against Ns* for the various sets of data. The trends are very similar to those observed with the Hudson stability number but there is now no apparent dependence on Tp for Ahrens data. Allsop's data, however, still exhibits a wave period effect; damage occurring more rapidly under wave spectra with a greater T • It is interesting to note that Allsop (38) distinguishes between front and back face damage, though total combined damage has been plotted

19

in this study. Moreover Allsop's results indicate that back face damage is generally greater than front face damage, for the longer wave periods, while the trend is reversed for the shorter wave periods. This implies that the apparent wave period effect is in reality a result of the increased overtopping, which occurs under the longer waves. This leads to significantly increased back face damage and hence to disproportionately higher levels of total damage. It seems unlikely therefore that a stability number, specifically derived for the case of no overtopping, could be used to adequately account for the stability of low crest breakwaters over a realistic range of wave periods. By virtue of their design, Ahrens breakwaters were subjected to only a very short duration of overtopping before they retreated below_ the water level. It is not surprising therefore that the wave period effect is not apparent in Ahrens results when plotted in terms of Ns*.

An exponential regression analysis to fit curves of the form given by equation 4.2 has been carried out. The results are summarised below:-

Coefficients Correlation DATA Rc/d A B r2

Ref (38) o.o 0.152 0.31 o. 75 Ref (41) 0.2 0.168 0.33 0.88 Ref (38) 0.4 0.048 0.05 0.77 Ref (30), Long Wave 0.29 0.0007 1.66 0.98

Ref (30 ), Long Wave 0.39 0.0018 1.58 0.96 Ref (JO), Long Wave 0.57 0.0009 1.92 0.95 Ref (30 ), Short Wave 0.23 Ref (30), Short Wave 0.38 0.0059 1.07 0.57 Ref (30)' Short Wave 0.57

On average there is little difference between the correlation coefficients for Hudson's stability number and the spectral stability number.

It is interesting to note however that the trends for Rc/d dependency in Figures 4.5 and 4.6 flO qualitatively similar for both Allsop's ) long wave regime and Ahrens data. A comparison of the range of wave steepnesses used in both studies reveals that,

1. 2.

3.

for for for

Allsop 's long wave regime, Hs/L < 0.03 Allsop's short wave regime, Hs/£ 1> ,. 0.03 Ahrens data, 0.0012 < Hs/Lp < 0.036

In other words, Ahrens wave conditions bear closest resemblance to Allsop's long wave regime. This suggests that the apparent dependence on relative freeboard, is characteristic of long waves (Hs/Lp< 0.03). The trend of increasing damage with increasing

20

values of Rc/d may therefore be partly due to the effect of the water depth, d, on the shoaling of the longer waves.

5 DESIGN GUIDELINES

5 .1 Estimation of wave transmission

It is clear from the preceding discussions that the wave transmission characteristics of low-crest breakwaters are primarily determined by:-

(a) the breakwater freeboard, (b) the incident wave conditions, and (c) the breakwater porosity.

The above factors may be adequately represented by a graph of Kt against dimensionless freeboard (Rc/Hs or R*'), for various values of breakwater porosity. In its ultimate form such a graph would comprise a series of curves each relating to a particular value of porosity. Unfortunately, there is at present only sufficient random wave data available to enable one curve, that for structures with a porosity of 40% (P =

0. 4), to be plotted. This curve is presented in Figures 5 .1 and 5.2 together with the envelope of wave transmission coefficients covering the range of realistic breakwater porosities (say 0 < P < 0.5). The upper bound of this envelope represents relatively permeable structures and the lower bound relatively impermeable structures.

Given the 40% porosity curve and the boundary conditions it may be possible to estimate from Figures 5.1 and 5.2 the overall transmission coefficient, Kt, for any particular breakwater porosity. However it should be recognised that for a particular value of the dimensionless freeboard, Kt may not increase linearly with increasing porosity. Thus, the accuracy of any estimate must be uncertain, until such time as sufficient data is available to allow more porosity curves to be plotted. For surface piercing structures it is recommended that Figure 5 .2 be used to obtain the transmission coefficients owing to the reduced range of Kt over that of Figure 5.1.

It is clear that the determination of the porosity of a breakwater is of particular importance if accurate values of Kt are to be obtained. This may lead to problems particularly if the breakwater is of multilayered construction, with an armour layer, filter layers and core. In such a case, if Ktt is predominant, then the required porosity ought to be that of the least permeable material, probably the core. If transmission by overtopping, Kt0, dominates then the porosity chosen may be that of the primary

21

5.2 Estimation of stability

6 CONCLUSIONS AND RECOMMENDATIONS

armour layers. Ultimately the choice of porosity used must lie with the designer.

The present study has demonstrated that the stability of the primary armour layers of low-crest breakwaters cannot, as yet, be adequately represented by a single simple design graph. However it is clear that if the structure has been designed to provide a certain level of wave attenuation, then the level of damage which that structure can sustain whilst still providing the required degree of wave attenuation will be very limited. This is particularly true if the damage should occur at the crest.

For st��gtures similar to those of Allsop()O) and Ahrens\ J , Figures 4 . 3 and 4 . 6 may be used to obtain a rough estimate of Ns or Ns* for the permissible level of damage selected. For other structures Ns or Ns* may be estimated by approximating that structure to those used by Allsop and Ahrens. It should however be noted that both Ns and Ns* may overestimate the weight of armour required for front face stability. By the same token, they may underestimate the weight of crest and back slope armour required to resist overtopping forces .

Clearly there are still considerable uncertainties involved in selecting a suitable armour weight to ensure stability of a low crest breakwater under the design wave conditions. At present these uncertainties may only be satisfactorily resolved by physical model testing of the stability aspects of the breakwater design. It is therefore strongly recommended that all designs be finalised by physical model testing.

The aim of this study has been the production of a design methodology for estimating the wave transmission coefficients of low crest breakwaters. This methodology is graphically illustrated in Figures 5.1 and 5.2. Further research is however required to improve the accuracy and applicability of these design graphs. In particular it is recommended that research be conducted into the influence of breakwater porosity on the wave transmission coefficient.

On the basis of this study it may be concluded that the wave transmission characteristics of low crest breakwaters are primarily dependent upon the incident wave conditions, the breakwater freeboard and the breakwater porosity. Other factors have, however, been considered. These factors include the breakwater

22

7 ACKNOWLEDGEMENTS

crest wid th and the effect of wave period . For submerged structures, increasing the crest width, above that needed for stability, has little effect on the wave transmission. It is therefore unlikely to be a cost effective method of improving the wave attenuation capability of such a breakwater. For surface piercing structures an increase in crest width may have a significant effect on the level of wave transmission.

The influence of wave period has proved difficult to quantify, with several anomalies apparent within the general trend. These anomalies suggest that there may be an optimum freeboard for a submerged breakwater at which maximum wave attenuation is achieved for minimum cost. However, further research is required to substantiate this hypothesis.

The stability of low crest breakwaters has been considered in relation to bo th the Hudson's stability number and the spectral s tability number. It has been concluded that , although the spectral stability number may adequately account for wave period, neither of these two parameters is likely to form a satisfactory basis for a comprehensive design method. Further research into the mechanisms responsible for front and back face damage is required, and until these mechanisms are fully understood low-crest breakwater designs can only be finalised by physical model tests.

The authors are grateful for permission from Southern Water Authority to use data derived in a study performed for them, and for data and preliminary analysis from John Ahrens of the Coastal Engineering Research Centre, now at Vicksburg, Mississippi.

23

8 REFERENCES 1. Seelig WN "Wave reflection from coastal

structures" Proc Coastal Structures 83, ASCE, Arlington, 1983 .

2. Seelig WN & Ahrens JP "Estimation of wave reflection and energy dissipation coefficients for beaches, revetments and breakwaters" CERC Technical Paper 81-1, Fort Belvoir, February 1981.

3 . Sollitt CK and Cross RH "Wave relection and transmission at permeable breakwaters" CERC, TP 76-8, Fort Belvoir, July 1976.

4 . t1adsen OS & White SM "Reflection and transmission characteristics of porous rubble-mound breakwaters", CERC, MR. 7 6-5, Fort Belvoir, March 1976.

s . Seelig WN "Two-dimensional test of wave transmission and reflection characteristics of laboratory breakwaters", CERC TR 80-1, Fort Belvoir, June 1980 .

6 . l-foraes CD "Experiments of wave reflection on impermeable shores" Proc 12th Coastal Eng Conf 1970 .

7. Battjes JA " Surf similarity" Proc 14th Coastal Eng Conf Copenhagen, 1974 .

8. Thornton EB and Calhoun RJ "Spectral resolution of breakwater reflected waves" Proc ASCE, Jour W'ways, Harbours and Coastal Eng Din WW4, November 1972 .

9 . Coastal Engineering Research Centre "Shore Protection Manual" 2 vols, 4th edition, CERC, Vicksburg, 1984 .

10 . Kondo H, Toma s , Yano K "Laboratory study on pervious core breakwaters" Proc 15th Coastal Eng. Conf. Honolulu, 19 76.

11. Ijima T, Chou CR, Yoshida A "Method for analyses for two-dimensional water wave problems " Proc 15th Coastal Eng. Conf, Honolulu, 19 76.

12 . Kajima R "Estimation of an incident wave spectrum under the influence of reflection" Coastal Eng in Japan Vol 12, 1969 .

13. Gilbert G & Thompson DM "Reflection in random waves: the frequency response function method" HRS Report IT 173, Hydraulics Research,

Wallingford, March 1978.

14. Gaillard P, Cauthien M & Holly F "Method of analysis of random wave experiments with reflecting coastal structures" Proc 17 Coastal Eng Conf Sydney 1980.

1 5. Goda Y & Suzuki Y "Estimation of incident and reflected waves in random wave experiments " Proc 1 5th Coastal Eng Conf Honolulu, 1976.

16. Longuet-Higgins MS "On the transformation of a continuous spectrum by refraction " Proc Camb Phi! Soc. V53, 1957.

17. Brampton AH "A computer method for wave refraction " Hydraulics Research Report IT 172, Wallingford September 1981.

18. Brampton AH & Smallman JV "Shore protection by offshore breakwaters " Hydraulics Research, Report SR8, July 1985.

1 9. Brampton AH "Surface waves over a step, interim report" Hydraulics Research, Report IT 170, Wallingford, Nov 1977.

20. Miles JW "Surface-wave scattering matrix for a shelf" J Fluid Mech V28 , 1967.

21. Lamb H "Hydrodynamics" Cambridge University Press, 1932.

22. Hydraulics Research "Wave attenuation over uneven sea bed topography" HR Report EX 1143 , Wallingford, Sept 1983.

23. Treloar PD & Abernethy CL "Determination of a bed friction facto� for Botany Bay, Australia", Coastal Engineering 2, 1978.

24. Madsen OS "Wave transmission through porous structures " Proc ASCE Vol 1 00, WW3, 1974.

25. Madsen OS & White SM "Wave transmission through trapezoidal breakwaters" Proc 15th Coastal Eng Conf. Honolulu, 1976.

26. Madsen OS, Shusang P and Hanson SA "Wave transmission through trapezoidal breakwaters" Proc 1 6th Coastal Eng Conf, Hamburg, 1978.

27 . Da t ta t ri J, Raman H & Shankar NJ "Perf ormance characteristics of submerged breakwaters" Proc 16th Coastal Eng Conf Hamburg, 1978.

28 . Bade P and Kaldenhof f H " Energy t ransmi s s i on over b reakwat e r - a des ign criterion" Proc 1 7 th Coas tal Eng Conf Sydney, 1 9 8 0 .

2 9 . Oue llet Y & Eubanks P "Ove r topping o f rubble-mound breakwaters b y irregular waves " Proc 1 5 th C oastal Eng Conf, Honolulu, 1 9 7 6 .

30 . All sop NWH " Low-cres t breakwa ters, s tud ies i n random wave s " Proc Coas tal St ructures 83, ASCE Arl ingt on, 1 983 .

3 1 . Allsop NWH , F ranco L & Hawkes PJ "Wave run-up o n s teep slopes, a literature review" HR Repor t SR1 Wal lingford, March 1985.

3 2 . All s op NWH , Hawkes PJ, Jackson FA & F ranco L "Wave run-up on steep slopes - mode l tests und e r random wave s " . Hydraulics Research Report No SR2, Wal l ingford, June 1 985 .

33 . S eelig WN "Est ima t i on of wave t ransmi s s ion coef f i cients for permeable breakwaters " CERC, CETA 7 9-6, Fort Belvoir Oct 1 9 7 9 .

3 4 . Mas s e l S R and But owski P, "Wind wave t ransmi s s ion through porous breakwa t e r " Proc 1 7 th Coas tal Eng Conf Sydney, 1 980 .

3 5 . Lordi ng P T and Scott JR, "Armour s t ab i l ity o f ove rtopped breakwa t e r " , Jour o f Waterways, Har bours and Coas tal Eng . Div ASCE WW2, 1 9 7 1 .

3 6 . Rai chlen F , " Armour s tabi l i ty of ove rtoppe d breakwa t e r " , Jour of Wa t e rways, Harbours and Coas tal Eng Div ASCE WW2 D i s cus s ion of paper 8 1 3 8, 1 97 2 .

3 7 . Lil levang OJ, " A breakwater subject t o heavy ove r topping : Concept, Des ign, Cons t ruc t ion and Experience " , Proc ASCE Speciality Conf Ports 7 7 , Long Beach, Ca l i f . , 1 97 7 .

3 8 . Ahrens JP, " Reef type breakwaters " , Proc 1 9 t h Coa s t al Eng Conf, Hous ton, 1 9 84 .

3 9 . Hydraulics Re search , " Seaford f rontage - mode l tes t s o f low cres t breakwaters, perf ormance and s ta bi l i ty ", HR Report EX 1346, Wallingford, Sept 1 985

40 . Owen MW "Design of s e awal ls a llowing f or wave ove r t op ping " HR Report EX 924, Wa llingford, June 1 9 8 0 .

4 1 . Ahrens J P , Viggos s on G , Z i rkle KP , " S tabi l i ty and

wave transmission characteristics of reef breakwaters" CERC Interim report , Fort Belvoir , 1982 .

4 2 . Hudson RY and Davidson DD, "Reliability of rubble mound breakwater stability models" , Proc ASCE Symposium on modelling techniques, San Francisco , 1975 .

43 . Gravesen H, Jensen OJ and Sorensen T, "Stability of rubble mound breakwater II" , Danish Hydraulic Inst. , 1980.

D D B Dd 650449 1 1/85

Figures

, l.O w ........

CD ..., (t) OJ ::::r\ � OJ � (t) ..., n ..., 0 VI VI I VI (t) n � 0 :::l VI

:::0 (t) � (t) ..., (t) :::l n (t) Vl

68g Angular s t one

Breakwater sect ion 1

68g Angular stone

0.33m

Breakwater sect ion 3

3690g Angular stone

0.66m

0 0.5 1.0m

68g Angular stone

0 .54m high plate

3690g Angular stone

3690g Angular stone

0.7m

0.66m

Breakwater sect ion 2

68g Angular stone

390g Dolos

Breakwater section 4

11 LCJ

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Breakwater sect ion 5

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p o15 1 fm

0.33m

Breakw ater sect ion 7

68g Angular s t one 3690g Angular stone

0.33m

15

Breakwater sect ion 6 & 8

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15

0 .22m High p l a t e

Breakwater sect ion 9

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0.32m

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0 0.1 0.2m

0.43m

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!" "!

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0 0.28m

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0 0

0 � <x A 0 '<1 X +

X •• A 0 t O A 0 0

· onl� SPM, overt opptng y

- 2 0 -1.0

I 0

0 0 0 �0 0 A �� AA V fiA '<1 '<;1 '<;1 • '<;1

'<;1 go. V

1 0

0 A '<;1 0 0

0 o·

2 0 3.0

Dimensionless freeboard , Rc/Hs

11 lO w Ul

� (lJ < ru --, OJ ::J Vl 3 Vl Vl 0 ::J 0... (lJ -(lJ

::::0 ro .....,.. ru ' ru ::J rl ro w 0 QJ ::J 0... w -.a

� c .� � .... OJ 0 V c::: 0 ·v; Ill ·e Ill c "' '-,_

1.0 •

•

0 8

0.6

0 .4

0.2

• • • •

I Submerg ..

ed J �urface p1ercing l • • •

•••

•

•

• •

..

•• •

• •

• • •

• •

• I I

I 0

• • • • •

• • SPM. overt opping only - 1 -.\.

•

0 • •

• Ref 3 9 o Ref 3 0 Short wavE. reg 1me x Ref 30 Long wave reg1me Rc;d = 0 39

Ref 30 Long wave regime Re; d = 0 57

X 0 0

0 0 X

X X ••

0 0 0 0

•

0

0 L-----------�------------L-__________ _L ____________ L_ ________ ��------------L-� -3 0 -2.C -1.0 0 U 2.0 3.0

Dimensionless freeboard.R�/Hs

F i g 3 . 6

en c

u '­w

0.

w l.J ru

"­'­:;:)

(/)

"0 w CJ'\ '­QJ E .0 ::J

(/)

0

0 C> •

J> •

0 0 0

0

<l

0 o o

--- --- --- ----- - - - ------��---------- - - - -<] ll<l 0 ��' .t �

0 0 0 0

'Se � 0 0 )(

<II> >< e • •

• •

� 0

0 0(/1) (!!) 0

0

� <l

0

0

O e •

"' 0

<l

[> {>. � [>t:t>

• +0 0

)( + 0

)(

N

3 3 ' ' (]) (])

+ )(

0

0 0

(Y) -4

3 3 ' ' (]) (])

0 [>

"" ..cJ

3 3 ' ' (]) (])

0 •

l)l �UC:IIJIJ!<IOJ UOISSIWSUeJl

r- CXl Cl'

3 3 3 , , , (]) (]) (])

"" <I •

W a v :c.:. t r a n s m i s s i o n d a t a - R e f e r e n c e 5

0

0

6 I

N 6 I

"" 6 I

-J' 0

o '

' * er: u· '-"' 0 .D w w '-�

Ill Ill w c 0 'V. c w E i5

"'T1 - ·

lD w I 1 .0 I -·---

-.l •

• Submerged I Surf ace piercrng "' I •

� • QJ • • < • • • ro • • -t-

0.8 ·� ..., QJ :::1 • Vl • • 3 • •

Vl

J Vl • •• 0 :::1 •

• 0.. QJ ::.::: • -t-QJ � I • c QJ • I � I ..__ ::::0 ..__ J QJ •• ro 0 • � u • ro c T ..., 0 ro Vl ·. I :::1 Vl 0 .4 ,.., E I 0 ro Vl c • Ref 39 I ••• w ro . • · L..

I • 0 I-

0 Ref 30 Short wave regime I -QO X Ref 3 0 Long wave regime I 0 w

I •

'-0 • I X

0 .2 x 0° I & I X X 0 I X 0 I X � % I Cl> q, 0 0 I

0 -0 .4 - 0 3 -0 .2 - 0. 1 0 0 . 1 0 2

Oimensionless freeboard, R :

0.8

..... c: Q) Ll 0.6 ..... ..... Q) 0 ...., c 0 "' Vl ·E 0 .4 "' c ro L

1-

F i g 3 .8

...._ c: Q) u

....._ ....._ Q) 0 u c: 0 V) V)

0 .2

1.0

0.8

0.6

.E 0 . 4 V) c: ro L 1-

0.2

- 0 . 5 -0 .3

Submerged,. Surf ace piercing .... ----+--••

- - -x----o-

T P1 < T P2

-0 1 0 . 1 0 .3 0. 5

Oimensiontess freeboard Re/d. 0.7

Var iat ion o f w a v e t r ansm iss ion w i t h w av e· per iod - Re f e r e n c e 30 & 3 9

Submerged Surface piercing ---r---�:

0 X

e; Tp, X T P2 0 T P3

T P1 < T P2 < T P3

/

0 - 0 7 -0.5 -0.3 - 0. 1 0 1 O J 0.5 0.7

Relat ive freeboard Rc/d

Fi g 3 . 9 Var iat ion o f w a v e t r an smission w i t h w a v e per iod - Reference 5

0.5

c � 0 4 � ..._ "­QJ 0 � 0 3 0 Vl Vl ·e � 0 2 ro '­

f-

0 1

O L_ ________ L_ ____ L_ __ ��--����L---------�----�--�--� 1 x 10-4 3x10-4 6x10- 4 1x10-3 3x10�3 6x10-3

H/g T2

F i g 3 . 1 0 E f f e d o f p o r o s i t y o n t h e w a v e t r a n s m i s s i ,J n c o e f f i c i e n t ( a f t e r S e e l ig

!Sl )

1 . 0

0 .8

::.:::: ...... c QJ u

0.6 ..._ ..._ QJ 0 u c 0 Vl VI

·e Vl 0 4 c ro '-

f-

0.2

----� -------- '\ "- '

' ...... , \ ' '

',\ '\ '

\\ \ '

'\ \ '

\\ \ . ', \

\ \

\

p = 0 .42

' p = 0 .4 1

p = 0.0

0�----�------�------�------�------�------�----� -5.0 -4 .0 - 3 . 0 -2 .0 -1 .0 0 1 .0

Rc/H

Fig 3 . 1 1 E f f e c t o f p o r o s i t y o n t h e w a v e t r a nsmiss ion coeff i c ient -

D a t a from R e f e r e n c e 2 7

2.0

1 0

0 8

'::£ ..... c QJ '-'

"-"-QJ 0 u c 0 Vl Vl E Vl c to '-- -o- B , . r-

---·--- 82

0.2 8 1 > B2

0 -0 4 -0 3 -0. 2

Submerged I .. I

I

�� I I l I

',\ I I 1 I I ' 0

-0. 1

\ \ I 'x I

\� : � I I I I I I I I I I

0 .0

Surface piercing ..

0 . 1

Oimensionless freeboard 0 2 '

R.

F 1 c 3 . 1 2 E f f e c t o f b r e a k w a t e r c r e st w i d t h o n w a v e t r ansmiss ion - R e f e r e n c e 3 9

1 0

0 . 8 X

� .._

0 . 6 c QJ u

..._ ..._ QJ 0 u c I 0 0 .4 Vl Vl

"§ Vl c -�>- B1 to '--

r- --··-o --- Bz - x - B3

0 2

0�--------�--------�----------L----------L--------�--------� - 0 5 -0 .4 -0.3 -0 .2 - 0 . 1 0 0 0 . 1

Relat ive freeboard Rc/d

Fig 3. 1 3 E f f e c t o f b r e a k w a t e r c re s r w i d t h on w a v e transmiss ion - R e f e r e n ce 2 7

0 V)

-c (1) 0 w <.... ro I.L ......

F i g 4 . 1

>(

)(

)(

)(

)(

•

<l

0 0 C> 0 � ,..-; N

• )( -

)(

.. z

0 N

)(

<l • •

� )( )(

)( )(

0

• •

V) 0

0

(1) 0'1 1'1) 1'1) z E " � m 0 Cl <l

)(

)(

>(

)(

• • )(

• •

• <l

a-N 0

1 1 u " w a::

)(

:3 0 0 N ,..-;

a-rT'1 6

1 1 u " w a::

•

r-lfl 0

" -o " w 0::

<l

<l

0 0 � ..,..;

X QJ u u ro ro ro ....._

F r o n t and back f a c e damage a s a f u n c t i o n o f N5 - l o n g w a v e s 130)

( T p = 1 9 . 0 s )

I C> "'

.._ c Q/ 0 u '- ro LL "-

Fig 4 . 2

0

<l O

0

I I I o. 0 0 � rri C'i

O <l

"' z

0 - ,....

<l 0

t> t>

0 8 �

<l <l «< 0 t> 0

<l O

0

0 - -"

I 0 0 -"

OJ I'D 01 z � ...... � 0 ro <l Cl

0

l 0 .-'

_l_ 0 ....;

F r o n t a n d b a c k f a c e d a m age as a f u n c t i o n o f ( 7 . 0 s < T p < 1 3 . 0 s )

r"f"l N 0

" "0 '-u �

<l

I

a:> 0'-r"f"l rr\ 0 0

11 " "0 "0 '- '-u u � �

0 0

c--Lf"l 0

1 1 "0 '-LJ �

t>

I I 0 lf1

N 5-short w a v e s 130l

11 1.0 +-w rr

0 ClJ 2 5 3 (lJ

1.0 ro (lJ Vl (lJ

20 -c :::J n � - · 0 :::J 0 � 15 - 0 -I 1'0 c z

' a. <J Vl 0 C)J

0"1 :::J 1'0 E Vl ro 1 0 Vl Cl

� ClJ rr -- · �

'<

:J 5 c 3 er ro '

�

z VI

X Tp = 1.45 secs

0 Tp = 2.25 secs

t> Tp = 2. 86 secs • T p = 3 58 secs

L> / N a = 0. 23 e0 86 Ns

•

• <9

X

0! Q (1. X

)J/. n wlt � .

2. 0 4.0

X

Q

0

Stabil ity No. Ns

6 .0

Data from Reference 38 (Rc/d = 0 0)

11 \0

+-w ClJ

0 OJ 3 ClJ

\0 ro ClJ Vl QJ -c :::J n �

0 :::J 0 � 0 -

ro :r: z c ' a. <l Vl <U 0 0"1

1'0 :::J E - 1'0 Vl C) VI � OJ er - · -- · �

'< :::J c 3 0"' ro '

z VI

25 I

20 r-

15

10

5

X Tp = 1 .45 s e c s o T o = 2 . 2 4 secs

<J Tp = 2.86 secs • Tp = 159 secs

L> / N a = 0.323 e0.89 Ns

0 Q 0

X

• I • •

� · Q

Q •

•

X Ox

X

0 ���--�----L----L--�--__J 0 2.0 4.0 6.0

S t abil i t y No. Ns

Data from Reference 4 1 ( R c / d = 0 . 2 )

11 lCl +-

+-cr

0 25 OJ 3 a 0 Long wave reg1me OJ

LO ro

b • Short wave regime

OJ a b Vl

OJ 20

-c ::J n -- · 0 ::J 0 � 0 15 -I � z c: " Cl. <I (/) 0 Q) :J en ro (/) E 1 0 ro Vl 0

� /Na = 0.028 e 2 25N, - a QJ CT 0 b t>/Na = 0.008 e 2 31N, ... . ,__

........ "< :J 5 c: 3 CT rD -,

z 11'1 0 0

I I I I

Stab i l i ty No. Ns

O a t a from Re f 3 0

11 - ·

lCl

+-

+-OJ

0 25 OJ I 3 OJ

LO ro

OJ Vl I OJ 20 __,., c ::J n --0 ::J 0 � 15 0 -

� I z c " 0... <I Vl 0 OJ

en ::J Vl

11) E 10 �

0 Vl -OJ CT

-t-'< :.:J s c 3 o-ro -,

�

z 11'1

X 0 Q •

T p = 1 . 4 3 s e c s Tp = 2 23 s e c s Tp = 2 .86 secs Tp = 3 56 secs

•

Q 0

0

Q

•

Q

0 0

�

� /Na = 0 213 e 1. 19Ns

S t abil i t y No . N5

Data from Ref 38 (Rc/d = 0.4)

11 lO +-V1 CT 0 OJ 3 25 QJ

lO ro QJ

x Tp = 1 .45 secs VI OJ o Tp = 2.25 secs _,., 2 0 'V Tp = 2.86 secs c Tp = 3.58 secs :::J • ,.., • _,... - · 0 :::J

0 � 15 .

__,., .6/Na :: 0 . 1 5 2 e (O.JlNs 1 0

_,... � ro ro

z VI "

-o <J (t) QJ • ,.., 0'> ___,... E 10 ..., • QJ ro - 0 VI

I . J ___,... OJ CT 0 0

___,... 5 '< X :::J c 3 rr ro -,

�z * I 0 0 5 10 15 20

. Spedral s t abil i t y No. N s

D a t a from Ref 38 !Rc/d = 0 . 0 )

11 lD +-Lf1 OJ 0 OJ 3 2 5 OJ

lD ro OJ VI OJ __,., 2 0 r-c: :::J ,.., ___,... - · 0 :::J

0 __,., � 15 L 0

_,... ;::::;-ro

ro z "

{/1 "'0 ro ,.., ___,... -, w

<J

J QJ 0'> ro E ro 0

'l/1 -w CT - · -'< 5 ::J c 3 CT ro -, -

:z Cl\ *

00

x Tp = 1.45 secs 0 Tp = 2 24 secs <:� Tp = 2 82 secs • Tp = 3 59 secs

(0 33N• l .6/N a = 0 . 168 e · s

0 •

s 10

• ' •

I Q

Q

I <(> <J/ Q

15 *

Spectral st abil i t y No. Ns

Data from Ref 41 (Rc/d :: 0.2)

20

11 I lO +-a-. er 0 QJ 3 QJ 25 lO m QJ (/l QJ __,., c 20 =:J n -.... - · 0 =:J 0 __,., -.... 15 -=r m (/l Damage

-o c. IN a m n (%) -;---, QJ 10 L/1 -QJ er � - · -;-

'< =:J 5 c 3 er m --,

:z. "' * 0

0

x T p = 1 .43 secs 0 Tp = 2 .23 s e c s Q Tp = 2 .86 s e c s • Tp = 3 .56 s e c s

X •

X Q

5 10

0

0

I

Qo

.

c. / N a � 0.048 el0·53 N. 1

1 5 •

Spect ral s tabi l ity No. Ns

Data from Ref 38 (Rc/d = 0. 4 ) \

11 lO +-er-QJ 0 QJ 3 QJ 25 lO Long wave /Short wave m regime reg1me QJ ---- -(/l c b a d QJ

a c./Na = 0. 0007 e (1.65 N; l __,., c . =:J 20 b c./Na = 0.0018 e (1.56 Ns l n

c: c. /Na = 0.0009 e (1.92 Ns.l -;-0

d: c. /Na = 0.0059 e 1107 Ns" l =:J 0 __,., --.- I 15 -=r m (/l Damage

-o c. /Na m n (%) Rc/d ---, X a 0 .29 QJ 10 L/1 0 b: 0.39 -;- ('. 0.57 • QJ • er I X

Q d· 0 23,038,0.57 - · � - · -

'< =:J I 5 c 3 er ro --, -:z.

V> * 0 0 5 10 15 •

Spec tral s t ability No. Ns

Data from Ref 30

C> rvi

C> N

C> ....c

C> ,.., '

L_ __________ _L ____________ L_ __________ _L ____________ L_ __________ _J �

F i g 5 . 1

-t_ <""! 0 I 0 C>

W a v e t r a n s m i s s i o n e n v e l o p e f o r l o w c r e s t b r e a k w a t e r s a s a

f u n c t i o n o f R c / H s

VI :X: "

u a::: -a· <... ro 0 ..0 <11 <11 <... -"' VI <11

c 0 ·;:;; c <11 E 6

/�/ 01 c '-' '-<11

·a. <11 � I '-' ru ...... '-:::J

V1

-o <11 01 '-ClJ E ..0 :::J V1

N 0

- · 0:: "C.

� , 0 .0 "' "' �

.._

VI VI � c 0

� ·v; 0 c

0

0 I

N 0 I

,....., 0 I

"' E i5

-4-�--------�--------�------��---------L--------� o

F i g 5 . 2

CD 0

1>1 �U<lOilJ<lOJ U01551WSUI?J!

N 0

W a v e t r a n s m i s s i o n e n v el ope a s a f u n c t i o n o f R �

0 •

11 11 - · - · LO LO

+- +-V1 V1 o- QJ 0 0 QJ QJ I 25 3 2 5 3 QJ QJ

LO LO ro ro

QJ QJ

20 � I • • Tp = 1 .45 secs • {/l {/l x Tp = 1.45 s e c s

QJ o Tp = 2.25 s e c s QJ 0 Tp = 2 24 secs - 2 0 " T p = 2.86 s e c s - <v Tp = 2 82 secs c

Tp = 3.58 s e c s c Tp = 3 .59 secs V/ '0 ::J • ::J • n • n -r -.... - · - · 0 0 I I '0 ::J ::J 0

� 15 . 0 � 15 r - 0 AjN a = 0.168 e !O.BN: l -

A/Na = 0 . 152 e (O.JlNs I � 0

-r -'V -r

;r ::r 10

:z: ro

ru ro " z

{/l " {/l <l -o <l "'Cl f 10 L � ro QJ • ro

<v/ n en n -....

E 10 -r '0 ' • '

Cl QJ 10 QJ - Cl {/l

I . I Vl I I X X -r 0 -r

QJ QJ

o- 0 0 o-- ·

I 5 � X -

-r 5 --+-0

'< '< • 1:1 . ::J ::J

c c 3 3

L�·�" o- o- 3- • ro ro ' 0 '

• -..,z

* I 0 z 0 0

5

10 15 20 0 5 10 15 20 VI * * *

Spectral stabili t y No. N s Spectral s t- abil ity No. N s

D a t a f r o m Ref 38 (Rc/d = 0.0 ) Data from Ref 41 !Rc/d = 0 .2 }

11 - · LO +-er-o-0 OJ 3 OJ 2 5 LO ro (l) (/) OJ --... c 20 ::I n ........ - · 0 ::J

0 --... ........ 15 ::::r ro l/1 Damage

""0 £) / N a ro n (%) -t-J cu 10

l/1 ........ (l) Cl � - · ........

'<

::J 5 c 3 Cl ro J

z "' * 0

0

X ·r p = 1 4 3 secs 0 Tp = 2.23 s e c s 17 ·�· p = 2 8 6 secs "' T p = 3 .56 s e c s

•

X

5

0

X

<7

1 0

<70

.

£1 / N a = 0.048 e t053 N, 1

15 *

S p e c t r a l s tab i l i ty No N5

D a t a f r o m Ref 38 \Rc/d = 0 4)

ll - · LD +-er-OJ 0 OJ ::3 OJ

LD ro OJ (/) OJ -;-, c ::J n -t-- · 0 ::I

0 -... ........ ::::r ro (/)

-o ro n ........ J OJ �

l/1 _,... OJ o-- · � - · ........

'<

::I c ::3 Cl ro J

�

z VI

25

20

I 15

Damage £1 /Na (%)

10

I 5

* 0 0

long w a v e regime

-----..

c b a

•

I )(

5

/Short wave reg1me

-

d

10

a: £�/Na = 0.0007 e t1.66 N; 1 .

b li / N a = 0.0018 e (1.56 N5 l

c: £� /Na = 0.0009 e (1.92 N5• l

d: £1 /Na = 0 0059 e ttoJ N; 1

Rc/d X a 0 29

0 b: 0.39 • (. 057 <7 d. 0 23,0.38,0 57

15 *

Spect ral s t abi l 1 ty No. N5

Data from Ref 3 0

0 rri

0 N

0

0

� N I

0 m I

0 L_ ____________ L_ ____________ L_ ____________ L-------------�----------�. �

F i g 5 . 1

CO C>

oo4' � 0 I 0 C>

l )l jUC!IJ IJjC!OJ IJOISSIWSUeJl

W a v e � r a n s m i s s i o n e n v e l o p e f o r l o w c r e s t b r e a k w a t e r s a s a C. . , _ _ !. : _ _ _ C. ll � I LJ -

"' J: '-u a:: -o' '-

"' 0 .0 ClJ ClJ '--Vl "' ClJ

c: � Vl c ClJ E

Ci

C1l s w '--QJ 0-QJ w /t)

0'; ...... '--::J l/)

/

"0 QJ 0' '-QJ E ..0 ::J l/)

� �I N 0

- · 0: -c' '-tO 0 ..0 Ql Ql '--"' "' Ql c 0

� v; 0 c:

0

N 6 I

,.., 0 I

Ql E (5

-.1' L---------�---------L------��----------�--------� 0

F i g 5 . 2

CX) 6 �. 0

1>1 jUCIIliJJCIOJ UOISSIWSUeJ!

N 6

W a v e t r a n sm is s ion enve lope a s a funct ion o f R !

C> '