A Breakwater Design for Wilson InletSupervisor: Jorg Imberger

(Source WRC)

Environmental Engineering Honours Project

By: Laurence Andrew Huizinga

Student Number: 0011116

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Contents Page

A BREAKWATER DESIGN FOR WILSON INLET 1

Contents Page 3

Introduction 4

Acknowledgments 6

Literature Review 7Wilson Inlet 7Breakwaters and coastal engineering 12Wave propagation in coastal engineering 29

Methods 37Wave Analysis 37Storm Surges 41Tides 42Longshore transport 42Breaking Waves 42Wave Propagation 43Interaction with structures 45Breakwater Properties 46

Results 48Bathymetry 48Wave Analysis 48Refraction diagrams 50Refraction/Diffraction diagrams 51

Synthesis 53Sizing of Armour Units and Breakwater Specifications 53

Conclusions 55

Recommendations 56

Appendices 57Appendix A 58Appendix B 62Appendix C 64

References 70

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Introduction

A major issue for the town of Denmark, on the Wilson Inlet, nestled in the corner of

Western Australia’s South West, is the opening and closing of the inlet to the ocean.

The opening of the inlet has occurred through a large portion of the last century as

remaining closed the banks of the Wilson Inlet will flood the adjacent farms.

However, during summer, a sandbar which runs perpendicular to the channel running

from the Inlet to the sea is formed. During the summer months the Wilson Inlet

becomes stratified and a large influx of nutrients causes eutrophication. Worse, the

ruppia which grows in the Inlet has a foul smell which becomes offensive to tourists

and local residents near the Inlet. Because of the eutrophication of the inlet, some of

the residents desire to have a permanent opening. The effect of a permanent opening

is studied elsewhere, but it is assumed that an opening will change the resume to

semi-salt water and will decrease the residence time for water and nutrients entering

the inlet. One drawback to a permanent opening is that release of nutrients from

sediments due to increased mixing may occur. However, the pros and cons are not

further discussed in this project. This project assumes that an opening of some kind

will occur and thus focuses on the processes on the ocean side of the inlet and

essentially a breakwater design to maintain both ecological and hydrodynamic

equilibrium as much as possible. However, the breakwater design is in essence

coupled with the environment it protects and if it fails then the environment is not

safe. The environment necessary to protect is both the foreshore around the inlet and

Wilson Head, as well as the marine environment of Ratcliffe Bay and Wilson Inlet.

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Figure 1 – Lichens at the western extent of Ratcliffe Bay and Ocean Beach, as a

part of the natural environment (Photo: Alex Bond).

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Acknowledgments

I would like to take this opportunity to acknowledge the assistance of many people

involved in this project. Firstly, I would like to thank Jorg Imberger for the idea of a

breakwater for Wilson Inlet, as well as the encouragement to make it a fun project.

Next, I would like to thank the staff at the Department of Planning and Infrastructure

of WA for all their efforts to provide accurate data for this project, namely Rodney

Hoath and Steven Hearn.

Personnel at CALM should also be thanked for providing information on data

collection namely Lawrie Ray and Nick D’Adamo.

The Centre for Water Research has provided excellent facilities and staff and students

there also aided in this project providing moral and technical support, including Phil

Bussemaker (for helping out with data manipulation) and Charitha Pattiaratchi (for his

advice on ray tracing).

Further, Annie Mose and Leona Lim were exceedingly patient with my failure to

attend meetings on time.

Finally, the support of my family never failed to tide me over in stormy sections of

the project.

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Literature Review

The literature review undertaken encompassed all types of breakwaters previously

designed. Found within the discourse are rubble mound breakwaters, permeable

breakwaters and semi-permeable or composite breakwaters. It also encompasses

papers on the propagation of waves into shoaling waters and the site Wilson Inlet.

Wilson Inlet

The Wilson Inlet is located on the south coast of Western Australia. It has been part of

a study by the fourth years at the Centre for Water Research at the University of

Western Australia, Nedlands, in the year 2003. This study focussed on providing a

Sustainable Future for Denmark, which included the Wilson Inlet as a local icon for

tourism and trade.

Figure 2 – Aerial Photo of the Wilson Inlet ocean entrance, looking towards the

Nullaki Peninsula (Photo: S. Neville)

The Wilson Inlet has a surface area of 48 km2, is 14 km long from east to west and is

4 km wide approximately. The average depth of the Wilson Inlet is 1.8m below sea

level where sea level is 0m AHD (Australian Height Datum). The Wilson Inlet has a

volume of about 90GL at 0m AHD and 130GL at 1m AHD. There are five rivers

which discharge into the inlet and the catchment area is 2300 km2. The Wilson Inlet

opens to the ocean at the west side of Ratcliffe bay, east of the Wilson Head.

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Figure 3 – Major features of the Wilson Inlet at the opening (source: WRC 2002)

The Wilson Inlet incorporates a shallow delta at the opening through which a channel

is cut (almost) every winter. The swell from the sea is generally from the south/south

east and the Wilson Head provides significant shelter from this predominant wave

action (WRC 2002; Ranasinghe et al 1999). However, Ratcliffe Bay is not at all

protected from the south east and easterly winds (WRC 2002). The mean annual

stream flow in the Wilson Inlet is 207 x 106 m3.

The field study done by Ranasinghe et al (1999) was done at a station shown in the

figure 2 below. The peak period and wave height is given in figure 3. This data shows

a significant wave height at the depth of 17m.

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Figure 4 – Data acquisition and position of wave recorder. The Wave recorder

was deployed at 17 meters depth, just off Wilson Head (Source: WRC 2002)

The peak period value was 13 seconds as determined by the peak spectral analysis

described in Sherman et al (1986) and determined by Ranasinghe et al (1999). The

offshore 50% exceedance wave height was 1.0m.

The mouth of the Wilson Inlet is completely blocked by a sand berm for about one

half of the year (from February to July) which is on average 150m in width and 500m

in length (WRC 2002). This is known as a ‘seasonally open tidal inlet’ (Ranasinghe et

al 1999). They usually occur in micro-tidal, wave dominated coastal regions with

strongly seasonal stream flow and wave climate experienced (Ranasinghe et al 1999).

The processes which govern the closure of the Wilson Inlet are not conclusively

identified, although hypotheses are present (Ranasinghe et al 1999). A combination of

factors could be reduced river discharge, growth of a sub-tidal sill and build up of an

ocean bar due to onshore transport as described by Hodgkin and Clark in 1988. A

second hypothesis by Marshall in 1993 is that the sand deposited offshore during

winter scour, is deposited as a bar in the summer when weakening stream flow occurs.

On the other hand, Todd in 1995 describes the closure as due to longshore transport,

due to south-easterly wind waves. However, these wind waves would not

refract/diffract as much as the south-south-westerly swell waves resulting in oblique

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incidence. This hypothesis is unsubstantiated and is contradictory of other studies on

the closure mechanism for Wilson Inlet (Ranasinghe et al 1999).

Limestone cliffs occur at the western reaches of the mouth, and the Nullaki Peninsula

exists on the eastern reaches. It is noted that the sands at the eastern end of this

closing bar are being stabilised as vegetation is becoming more established there.

Thus, the tip of the Nullaki Peninsula seems to be growing westwards, probably due

to this stabilisation (WRC 2002). The bar is accrued by medium to coarse grained

marine sands. It builds up to a height of 1.8m above the AHD. This is due to scoured

sands that build an offshore berm during winter and are accreted as a sand berm

breaching the gap during summer. As seen in Figure 2, there is a large tidal delta

extending about 2km into Wilson Inlet, which, it is presumed, has been formed by

predominant wave energy on the coast opposite Wilson Inlet (WRC 2002). Several

channels have been scoured due to ‘after opening’ effects; after opening these

channels will either become unstable and fill in, or scour according to the pattern of

flow (WRC 2002).

The Wilson Inlet sand bar has been artificially opened since the 1920s to protect low

lying farmlands adjacent to the Wilson Inlet from becoming flooded. A breach in the

sand bar less than 200m from the cliffs at the western edge of the bar is called a

‘western opening’ and a breach in the sand bar more than 300m from the cliffs at the

western edge is called an ‘eastern opening’. It is found that the time for the bar to

remain open shows no correlation at all to being an ‘eastern opening’ or a ‘western

opening’ but rather depends on the annual rainfall, river discharge and Inlet water

level at time of opening (WRC 2002). The position of this opening has been

historically variable, from 50m to 450m from the western cliff. Further, the opening

times of the bar have varied considerably with openings being as early as June and as

late as October, but averaging around July/August. Closure of the bar varies between

November and May but averages around February. The bar is closed every year and

the length of time opened varies from 50 days to 334 days, with an average of 191

days (WRC 2002). A linkage to a rip cell which may be set up in Ratcliffe Bay will

reinforce the opening procedure enhancing the integrity of the opening as an exchange

mechanism (WRC 2002). The longshore currents in Ratcliffe Bay are found to be

small compared to net longshore transport at sites such as Dawesville (WRC 2002).

Due to both longshore transport and the curvature of channels, the channels through

the sand bar migrate in a south-westerly direction (WRC 2002). Importantly,

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computer models suggest that the closure of the bar is caused by incident summer

swell waves (WRC 2002). Sand deposition in the bar during the closure of the bar is

estimated at 300m3 of sand infill per day over the months of closure (WRC 2002).

Opening the bar at the same position in successive years results in increase flow rates

at the next opening of the sand bar (WRC 2002). The width of the berm beach

opposite the Surf Life Saving Club may be under threat after opening the bar. Even

with protection of the headland, this beach berm is at risk due to storm erosion, as

seen in other WA beaches in the south west (WRC 2002). This is perhaps another

reason for a breakwater for Wilson Inlet although the author does not put forth this

question. Two methods for closure are seen: one is longshore sediment transport and

the other is onshore sediment transport (Ranasinghe et al 1999). Isolated stream flow

events and storm events occur in this natural environment, which may have

significant impact of the closure of the Wilson Inlet channel (Ranasinghe et al 1999).

Figure 5 – A western opening shortly after a breach. The photo shows location of

the surf club building in the bottom left (Source: WRC 2002) (photo: T.

Carruthers)

The ocean storminess at Ocean Beach is a major driver of sand causing infill of the

channel and thus also stratification in the Wilson Inlet as a consequence. Further, it

was found that increasing marine exchange may result in an unexpected release of

more nutrients from the sediments of Wilson Inlet (WRC 2002). A field experiment

by Ranasinghe et al (1999) measured parameters including pressure, current speed

and direction This data was analysed using methods in Sherman and Greenwood

(1986) (Ranasinghe et al 1999). Within this data is no evidence of south-easterly wind

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waves even though the data collection occurred over 2 days from the 11th to 12th of

December in 1995. Tidal velocities were at a maximum ebb flow of 80 cm s-1 and

maximum flood flow of 100 cm s-1 (Ranasinghe et al 1999). However, flood/ebb

dominance can vary in inlets within a few days and above velocities do not mean that

the inlet is flood dominant. The maximum longshore current measured was

approximately 50 cm s-1 with a variable direction of currents (Ranasinghe et al 1999).

Thus, longshore currents are neither strong or have a dominant direction and can not

be responsible for closure of the channel (Ranasinghe et al 1999). Because the Wilson

Inlet is on an embayed coastline, the longshore currents are expected to be small.

Estimates of longshore transport from aerial photography and shoreline maps also

indicated very low longshore transport numbers (Ranasinghe et al 1999). Hence the

necessity of dealing with onshore sediment transport becomes imperative. Bed levels

along transects out to sea indicate an offshore bar at about 110 metres from the beach.

During the study period the bar moved inshore by 35 and 60 metres by the 3rd and 19th

of December (Ranasinghe et a 1999). Thus, onshore transport of sediment may be the

main mechanism for inlet closure.

The Wilson Headland is critical in defining the parabolic shape of waves taking on the

shape of the bay, and modelling domain incorporated the headland. With longshore

processes only in the modelling exercise done by Ranasinghe et al (1999), the inlet

channel did not close. This might be similar to the case when a breakwater protects

the inlet channel from onshore sediment transport. Margin bars are formed around the

edges of the Wilson Inlet channel to Ratcliffe bay in the model with longshore only,

and the inlet currents were sufficient to maintain this opening to equilibrium

(Ranasinghe et al 1999). It is hypothesised by Ranasinghe et al (1999) that during a

storm in an embayed coast, there is a higher possibility that more sediment would be

moved offshore than to the vicinity of the Wilson Inlet, thus keeping the inlet open

longer by eroding the bar at the opening.

Breakwaters and coastal engineering

Breakwaters have been in use in areas where protection of the coastline is essential.

However, Silvester in Coastal Engineering I notes that there have been many

instances of failure after the order of 5-10 years of installation as a result of poor

design.

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Wave interaction with breakwaters

The Offshore Design Manual from the US Army Corps of Engineers explains basic

concepts for breakwater designs. The manual looks at the modelling of propagation of

water around the breakwater and thus some assumptions need to be made for this:

1. Water is an ideal fluid, inviscid and incompressible.

2. Waves are of small amplitude and can be described by the linear wave theory.

3. Flow is irrotational and conforms to a potential function which satisfies the

Laplace equations.

4. Depth shoreward of breakwater is constant.

The equation for the height after diffraction is:

K' = H/Hi i.e. H = K' * Hi (1)

where H is the water height above the datum at any point, Hi is the initial height at the

toe of the breakwater and K' is a site specific constant, a diffraction coefficient (US

Army Corps of Engineers 1984).

The shore protection manual from the US Army Corps of Engineers explains the

interaction between a perforated breakwater and waves. It states that wave energy is

dissipated due to viscous effects and the linear diffraction theory. A rubble mound

breakwater is of this form, with large diameter rocks. Further, they state that density

and diameter of these rocks are keys to the design of a rubble mound breakwater.

However, it is found that at the time of publishing of this manual, there was little

work done on a rock filled core breakwater. The fluid motion around the breakwater

could be described according to the velocity potential which satisfies the Laplace

Equation within the fluid region.

Shoreline changes behind a detached breakwater are effects of waves interacting with

a breakwater. These changes are affected by sediment supply, sediment properties,

wave properties, topography and breakwater properties such and length and position.

A series of experiments scaled down were done measuring certain variables. These

were, A, the area of salient (accreted sand) behind the breakwater; B the breakwater

length and X, the distance of the breakwater from the shoreline.

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Kumagai and Foda (2002) have designed an analytical model for the response of

seabed to waves approaching a composite breakwater. There has been no analytical

modelling for the seabed response of a caisson type breakwater piled on top of a

rubble mound breakwater. The understanding from this analytical approach, is an

insight into the physical problems associated with the movement of breakwater

caisson and hence the response (Kumagai et al 2002). Assumptions for the model are:

1. Sinusoidal waves are acting on the caisson, and the response is analysed after the

waves are given

2. The caisson is impermeable and rigid with a no-slip condition at the interface of

caisson and rubble mound

3 . The rubble mound is homogeneous, isotropic, poro-elastic, saturated with

compressible pore water

4 . The seabed is also homogeneous, isotropic, poro-elastic, saturated with

compressible pore water, and semi-infinite;

5. The response of the rubble mound and the seabed is periodical over time

6. Hooke’s Law is used to describe relationship between the effective stress and the

velocity of the solid (Kumagai et al 2002).

These seem fairly reasonable assumptions considering the scale of a breakwater are

generally of 103 metres. Kumagai et al (2002) break the problem into scattering and

radiation modes. This means that in the scattering mode, the breakwaters’ position is

assumed fixed and the forcing comes from wave induced pressure along the soil

exposed. Radiation mode implies that there is no wave action and the forcing is the

motion of the caisson itself (Kumagai et al 2002). The motions of the caisson can be

thus broken into 4 modes: heave, pitch, surge and scattering. Using full poro elastic

equations in the horizontal direction, the response inside the mound is obtained, that

is, fluid momentum equations, solid momentum equations and mass conservation

(Kumagai et al 2002). They then consider motions in the outer regions, and the

boundary layer regions. The analytical model they have developed has solution to all

of the four modes above at the same time. The motion of the mound is broken into a

Fourier series and effective stress can be determined. The same approach is given for

the seabed solution. These two solutions are connected by then prescribing interfacial

conditions as previously described in the model outlined. Thus, the solution can be

determined by equating Fourier ‘harmonic’ coefficients separately (Kumagai et al

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2002). This process is called ‘matching conditions’, as it matches conditions at the

interface between two regions. A relaxation force is introduced at the mudline after

this matching of conditions. The radiation problem is approached based on Newton’s

law and the balancing of arbitrary caisson motions, in the horizontal, vertical and

rotational modes. The model is thus broken up into 4 steps:

1. Set calculation conditions, i.e. for a wave and mound and caisson.

2. Decompose the problem into heave, pitch, surge and scattering and then solve for

each mode. Matching conditions is then utilised to determine unknown Fourier

coefficients. Relaxation-surface solution is then imposed to satisfy the real

boundary condition outside the mound region.

3. Then, after solutions for arbitrary caisson motions are found, determine velocities

and phases of the caisson’s motion.

4 . Compose solutions to the simplified modes according to the velocities and

motions (Kumagai et al 2002).

The model is verified by comparing with previous models. It was found that the

mound thickness needs to be set to very thin to model the case of no rubble mound.

Pore pressure at specific depths is also in agreement with the other models based on

Mei and Foda’s approximation method (1981). Advantages of the model are that it

requires far less computational time than a numerical model and further, that it

provides insight into effects of properties of waves, caisson and the mound and the

response thereof (Kumagai et al 2002).

Use of a horizontal plate inside a breakwater has also been studied wherein the plate

is hoped to minimise the reflection and maximise energy dispersion (Yip et al 2002).

The construction of a breakwater to absorb the energy is one of the advancements in

port design (Yip et al 2002). Yip et al (2002) note that the use of a perforated wall

breakwater with a wave chamber close behind it has been used increasingly

worldwide. They also note the previous discovery that the incoming amplitude of

waves is minimised when and if the distance between the porous barrier and the

chamber end wall is equal to one quarter wavelength plus a multiple of a half-

wavelength of the incident wave. Yip et al (2002) theorise that a horizontal plate, as

an alternative to reducing water depth inside the chamber, to reduce overturning of the

front portion of the breakwater is a worthwhile installation. This is also founded on

previous suggestions for submerged horizontal plates which from a hydrodynamic

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performance perspective, suppresses vertical of motions strongly. A key outcome

from their study is that a small breakwater with a plate can have the same effect as a

large breakwater without a plate (Yip et al 2002). The ratio of reflected wave heights

to incident wave heights is Kr and is related to the ration of wave chamber length (b)

to incident wavelength (_). An important outcome that is verified through rigorous

variations of b is:

b/_ = _ + n/2 (n = 0, 1,2,…) for minimum reflections (2)

and is easily applied to engineering projects. However, the shortening of the wave

length does cause sharp changes in the wavelengths approaching the breakwater and

thus it is necessary to determine the actual wavelength within the chamber i.e. _1 (Yip

et al 2002). Further, an internal plate can minimize overturning moment and force on

the front of the breakwater (Yip et al 2002).

Shoaling occurring in a harbour is also an effect that is previously studied with

regards to breakwaters. As sand is accreted in front of a breakwater, it will also be

transported by currents towards the tip of the breakwater where it accrues. This sand

will cause waves to break easier and thus the sediment becomes suspended and is thus

transported into the bay (Yuksek 1995). Yuksek (1995) finds that the second part of

the main breakwater should thus be perpendicular so the dominant wave direction and

that the secondary breakwater (to minimise shoaling effects) should be located so that

the line of dominant waves occurring from the tip of the first breakwater is on or near

the second breakwater (Yuksek 1995).

Breakwater Constructions

Per Anders Hedar (1986) has developed improved formulas for rubble-mound

breakwaters and has been developed further to the present. He presents the Hudson

equation as the equation used the world over historically for stability of rubble-mound

breakwaters (Hedar 1986). Initially, the Iribarren formula (1938):

Q = (_s K Hb3)/((_s/_f)

3 (cos _ – sin _)3 (3)

was developed for weight of units, where K = 0.015 for ‘rock-fill’ and 0.019 for

concrete blocks. The Iribarren-Hudson formula in 1951 was then adapted to:

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Q = (_s K’ tan3_ H3toe)/(_s/_f – 1)(tan _ cos _ – sin _)3 (4)

where K’ is Iribarren-Hudson’s stability coefficient. In 1958 we see the Hudson

equation emerge:

Q = (_s H3

toe)/(KD(_s/_f – 1)3 cot _ (5)

Which is applicable to each structure slope ranging from 1 on 1.5 to 1 on 5 (Hedar

1986). KD values are found in the Shore Protection Manual (1984) for the US Army

Corps of Engineers. It may be quite difficult to correctly choose a KD value (Hedar,

1986). Hedar (1986) then outlines improved formulas for the stability of armour units

of a breakwater. The equation for the weight of an armour unit is:

Q = pi/6 * _s k3 (6)

where k is the diameter of a block sphere of an equivalent weight. For the uprush of

water into a breakwater, the pervious under layer unit diameter is given by:

k = (0.33(db + 0.7Hb).(tan_ + 2))/((_s/_f – 1)(3.6 – 1/(e4tan_))*cos_(tan_ +

tan_))

(7)

and for the impervious under layer of the breakwater:

k = (0.41(db + 0.7Hb).(tan_ + 2))/((_s/_f – 1)(3.3 – 1/(e4tan_))*cos_(tan_ +

tan_))

(8)

where _ = _ + (_ – 48o) (Hedar 1986). For the down rush of water from a breakwater,

the pervious under layer is given by:

k = ((db + 0.7Hb).(tan_ + 2))/((_s/_f – 1)(e4tan_ + 13.7)*cos_(tan_ + tan_)) (9)

and the impervious layer diameter for the down rush is:

k = (1.6(db + 0.7Hb).(tan_ + 2))/((_s/_f – 1)(e4tan_ + 16.5)*cos_(tan_ + tan_))

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(10)

where _ = _ – (_ – 48o) (Hedar 1986). The solutions for the above equations are also

shown graphically in Hedar (1986) for angle of reposes of 40o, 45o, and 48o. The

design wave height is also important in determining size of tetrapods or any other type

of breakwater component. The equations should incorporate the breaking wave height

as well as the depth at breaking (Hedar 1986). Thus, the armour units of the

breakwater shall be according to wave heights at least between hb33 and hb10 (Hedar,

1986). Because there is risk of sudden failure, caution must be taken by choosing a

high design wave. Further, if the breakwater toe is installed at a depth that is less than

the breaking wave depth, then the waves will break onto the armour slope (Hedar

1986). Hedar also reports that there is an increase necessity in unit weight when the

under layer is impermeable. A void ratio should be above 40% for the above

equations to be valid (Hedar 1986).

It is also recommended by Hedar that the wave characteristics should be calculated

for at least 50 years return period. This will achieve infrequent maintenance (Hedar

1986). Also, the breaking wave height should be calculated according to the SMB

method as outlined in the Shore Protection Manual on page 7-7.

The angle of repose for the breakwater armour units is given by _. It is a characteristic

of the armour layer (Hedar 1986). Once it becomes equal to the slope value _ then the

breakwater armouring units are unstable. The difference between the two angles is a

value of the strength of the breakwater to resist wave forcing (Hedar 1986). It is also a

value of the degree of interlocking. A high value of _ means a high value of

interlocking and thus, if the blocks are artificially interlocked carefully and not

haphazardly, then the value of _ may be increased. However, this may not be wise as

the breakwater would be under-designed when the units do break up a little. Rubble

mound breakwaters are more progressive in their breaking in comparison to concrete

armoured breakwaters, where sudden failure occurs (Hedar 1986). Hedar concludes

by noting that from diagrams it may be shrewd to design with the gradient of the slope

for the armouring layer of the breakwater between 1 on 2 to 1 on 3.

Cost effectiveness of breakwater cross sections (which is related to construction) is

discussed by Smith (1987). He argues that a structure that eliminated all likelihood of

damages would be far too expensive to build, and a ‘trade off’ between acceptable

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likelihood of damages and design criteria is better suited to engineers, especially in

the public works sector (Smith 1987). Smith goes on to say that one must minimise all

costs in a coastal design project, i.e. construction cost and user (total) cost. Thus one

reaches a maximum structure cost, which minimises the user cost with the structure

present (Smith 1987).

Figure 6 – Breakwater configurations with respect to cost and benefits. The

project feasibility area is the desired part of the figure (Source: Smith 1987).

Further, structural costs initially will decrease maintenance costs in the future up to a

point, as seen in figure 4. The project feasibility shaded area is the desired area for a

project to be within. The best possible project is where the severity of design criteria

is approximately “9” where the alternative has the maximum net benefits, as well as a

minimum total cost. It is noted that this position can be determined without

knowledge of the “without project” condition (Smith 1987). Smith then applies this

idealised diagram with a practical approach to rubble mound breakwaters and other

structures.

The first step is to define the site conditions. These include: water levels, tidal

currents, foundation characteristics and wave climate (Smith 1987).

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The second step is to identify an estimate of expected economic losses. This may

necessitate an estimate of a minimum wave height wherein there is no effect on

harbour and surrounds, and an upper limit for the worst possible wave attack to

completely destroy a harbour (Smith 1987). Thus:

$L(Hs) = $Lmax{1 – exp [A(Hs – HLo)]} (11)

where A is a coefficient determined by regression. This exponential function

incorporates the information conveyed in figure 5.

Figure 8 – Exponential function of economic losses in comparison with the

significant wave height (Smith 1987)

Smith then shows how this is useful for estimating the expected or long term average

annual economic losses as E ($L/yr):

E($L/yr) = _ ∫ $L (Hs) (_F (Hs))/(_Hs) * _Hs (12)

where F(Hs) is the cumulative probability distribution for a significant wave height as

derived by smith earlier in step 1 (Smith 1987). The quotient _F(Hs)/_(Hs) is the

associated probability density function. _ is the average number of storms per year as

determined through estimating for F(Hs) and this is necessary so as to present the

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expectation as an annual average (Smith 1987). The limits for this function are given

as a very high significant wave height and HLo.

The third step outlined by Smith is then to formulate the ensemble of all the variables,

which is described as highly subjective, depending a lot on the designer and his or her

wishes. Individual breakwater configurations need to be investigated (Smith 1987).

Too few alternatives will preclude the identification of an optimum design, and thus,

the designer must allow for some objectivity (Smith 1987). This step includes

conferring with a table which looks at functional performance per return period and

wave height exceedance (or x year storm) and rates each with a structural integrity

according to return periods for each wave height Hd. Thus, Smith (1987) recommends

that 50 year return periods are important to address since repairs of rubble mound

breakwaters generally require large mobilization and demobilization costs. Hence,

the 4th step involves identifying the optimum combination of armour size and type,

slope and crest elevation for each alternative breakwater (Smith 1987).

The 5th step involves detailing a cross section design for each alternative breakwater

which is very subjective (Smith 1987) and should utilize all engineering knowledge

available. Wave transmission characteristics for each alternative would then be

determined, as a function of incident wave conditions, as the 6th step (several can be

more severe than actual measured conditions) (Smith 1987). Step 7 is then to

measure the economic losses as previously outlined for each alternative.

Step 8 is to estimate the annual expected breakwater damage for each alternative,

which may be a function of armour layers and specifics (Smith 1987). Step 9 is to

sum the costs for each alternative. Verification of this model should then be done

using a wave model such as JONSWAP (Joint North Sea Wave Project) (Smith 1987).

Smith notes that a breakwater should be designed according to two major criteria. The

first criterion is to design the breakwater to function as a wave barrier depending on

its transmission characteristics. The second criterion is to design the breakwater with

its ability to withstand infrequent storms in mind, especially with maintenance in

mind (Smith 1987).

The Hudson Equation has also been found by van der Meer (1988) to be

unsatisfactory because it does not take into account the peak period or other important

wave characteristics. He notes that it can only be used as a very rough estimate and an

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error factor of about 8 in the stone mass results (van der Meer 1988). Van Der Meer

also describes stability formulas for rubble mound breakwaters. These are described

for both plunging and surging waves (i.e. for the conditions where the wave either

breaks before the breakwater, or after respectively). For plunging waves, the equation

is:

Hs/_Dn50 * √_z = 6.2P0.18(S/√N)0.2 (13)

and for surging waves:

Hs/_Dn50 = 1.0P – 0.13(S/√N)0.2√(cot _) _zP (14)

where Hs is the significant wave height at the toe of the structure, _z is the surf

similarity parameter (where _z = tan_/√(2�Hs/gTz2)), Tz is the average period of the

waves, _ is the slope, _ is the relative mass density of the stone (i.e. _ = _a/p-1), pa is

the mass density of the stone, _ is the mass density of the water, Dn50 is the nominal

diameter of the stone (i.e. Dn50 = (W50/_a)1/3, W50 = the 50% value of the mass

distribution curve, P = the permeability coefficient of the breakwater, S = the damage

level (i.e. S = A/Dn502), A is the erosion area in a cross section and N is the number of

waves in a storm (Van Der Meer 1988). The slope should be between 1.5 and 6 and

the significant wave height is used in these equations. The average of the highest 1/3

of waves can be used as the significant wave height (Van Der Meer 1988). He also

finds that for Hs in shallow water, we must use H2%/1.40 in formula (8) and (9)

instead of Hs. The wave steepness should be between 0.005 and 0.06. If the wave

steepness (2�Hs/gTz2) is greater than 0.06, the waves are unstable and break (Van Der

Meer 1988). Hence we can use this as an upper boundary. The wave period can be

determined from a wave signal defined from the zero up-crossings in the signal and is

given by: Tz = √(m0/m2), where m2 is the second moment of the energy density

spectrum. This gives the average period as in opposition to the peak period, although

both can be used. Values of permeability are given for various types of breakwaters. a

permeability of 0.1 is given for an impermeable core of either clay or sand. A

homogeneous structure consisting simply of armour stones will have a permeability of

approximately 0.6. The damage level S can be determined from the number of cubic

stones with sides of Dn50 eroded around the water level within a width of one Dn50 to

the water level (Van Der Meer 1988). The S values are given in a table for upper and

lower damage levels corresponding to a cot _ level (Van Der Meer 1988). The above

Page 23

formulas can be used when the storm duration or number of waves is between 1000

and 7000. The mass density of stones in tests done by Van Der Meer was between

2000 and 3000 kg. Thus mass densities (_) were between 1 and 3 (Van Der Meer

1988). Design graphs have been drawn by Van Der Meer for various parameters in

the two equations. Various parameters were assumed to demonstrate influence of a

certain parameter in the above equations. These were: Dn50 = 1.0; mass density stone

_a = 2600 kg/m3 i.e. W50 = 2600 kg; mass density water _ =1000 kg/m3 and thus

relative mass density (_) of 1.6; slope angle cot _ = 3.0; damage level S = 5 (which

means that there is tolerable damage in 50 years); permeability of P = 0.5 (with a

permeable core); and a storm duration of 3000 waves. It is found that for surging

waves, rundown is determining for stability and for plunging waves, runup is decisive

(Van Der Meer 1988). Important findings were that a steeper slope allows a smaller

wave height to cause instability of a breakwater. A larger permeability provides more

protection and stability and is also more stable for surging waves. The stability

increases as wave period increases. The mass of stone changes by a factor of 2.5 when

the permeability changes from 0.1 to 0.6. Graphs based on equations in Van Der Meer

make it easy for the designer to design the armour layer of a rubble mound and to

look into effects of various changes on the stability and thus to improve a design.

The stability of a rubble mound breakwater’s head and trunk is discussed by Vidal et

al (1991). They give new experimental information about the stability of head units as

functions of wavelength, and head shape. They also recall the rule of thumb enounced

by Iribarren and Nogales which states that size of head units should at least weigh 1.5-

2 times more than trunk units (Vidal et al 1991). Damage criteria are outlined in four

areas:

1. Initiation of Damage, which specifies damage conditions where a certain number

of units are displaced from their original position. In their study they used a value

of 2%.

2. Iribarren’s damage is the damage which occurs when the extent of failure on the

main layer is so large that waves may extricate armouring units from the lower

layer.

3. Initiation of destruction is where a small number of lower armour units are

extricated and waves will thus work on the secondary layer.

Page 24

4. Destruction is when pieces of the secondary layer are removed and thus, if the

wave height remains the same, the breakwater will be destroyed and the

functionality of the breakwater is lost (Vidal et al 1991).

They also introduced a scatter value for a breakwater unit which is:

√((1-p)/(pn)) (15)

where p is the probability of extraction of a unit and n is the number of units. Thus, if

p is small, the indicator takes on large values and vice versa (Vidal et al 1991).

Further, a stability function based on Losada and Gimenez-Curto (1979) was used to

evaluate the weight of an armour unit as a function of the incident wave height,

specific weights of water and armour units, Iribarren’s number, slope angle, damage

criterion, type of armour unit, unit placement, roughness and permeability. Thus we

get:

W = _w . Q.H3.Y (16)

Q = S/(Ss – 1)3 (17)

where

Ss = _s/_w (18)

and _w and _s are specific weights of water and armour units respectively and Y is a

function called the stability function which depends on the Iribarren number, slope

angle _, damage criterion, type of armour unit, unit placement and roughness and

permeability. The Iribarren number (or surf similarity parameter) is:

Ir = (tan _)/√(H/L) (19)

Where L is the wavelength and H is the wave height (Vidal et al 1991). The critical

stability sector was seen 60o from the normal to the wave ray tangent to the head toe.

It is less stable than other head sectors. As well as initiation of damage, destruction

also initiates at this sector (Vidal et al 1991).

Page 25

Figure 8 – Critical sector of head regarding its stability. This area needs to be

over designed in its size of rocks used to maintain its stability with respect to

incident waves (Source: Vidal et al 1991)

Thus, stability function values should be chosen that are 1.3 to 3.0 times higher than

the normal trunk values of unit weights are (Vidal et al 1991). The radius of the

breakwater head or slope length (S) can be divided by the wavelength (L) to produce

the ratio S/L. As S/L is small or moderate, the waves, whether they are breaking

waves or not, will pass the head with a forward motion. However, for large values of

S/L, we get a breaking of the waves where most of the energy is broken into the head

structure. As most breakwaters have a head slope value of about cotan _ > 2, test

breakwaters also used this value. In all of these tests, damage was produced by

forward breakers whereby units can be carried forward, minimising the protection

value of the breakwater and creating a hazard for navigation (Vidal et al 1991).

Vidal et al (1991) conclude to provide recommendations for the designing engineer.

They recommend that the least stable sector 60o as outlined previously should be

reinforced by units of weight at least 1.3 to 3.8 times higher than the weight of trunk

units (Vidal et al 1991). Further, it should be realised that the head is more brittle than

other areas on the breakwater. All wave directions should be considered to understand

Page 26

areas of the head that are susceptible to damage (Vidal et al 1991). Further, variations

in the trunk wave climate of the breakwater due to a lateral boundary such as a cliff

can cause standing longitudinal variation in the wave height along the breakwater.

Thus, this will cause standing longitudinal variation of damage along the trunk (Vidal

et al 1991).

Dickson et al (1995) show how reflection from a breakwater in Monterey Bay,

California was important in the breakdown of the breakwater and subsequent patching

up with extra armour units on the harbour side. A new method of estimating

breakwater reflection is also introduced (Dickson et al 1995). This is done using an

array of pressure sensors or of surface-height gauges, seawards of the breakwater.

However, lack of varying wave conditions prevented this study from determining how

reflection occurs under different wave conditions, the reflection and transmission

characteristics of a breakwater were not fully realised. However, it is generally

accepted that to obtain maximum protection for an area, a breakwater normal to the

wave orthogonal is the most efficient design (Dickson et al 1995).

Medina et al (1994) study the damage on armour units of a breakwater with an

experimental set up. They find that the equation stated by Van Der Meer and the

Shore Protection Manual (SPM) assumes a permeability that is not fully founded on

extensive experimental data. Further, the 0.15 power relationship between the SPM

and the approximation by a regression should not be considered. It is shown that the

Van Der Meer formulas calculations of damage for armour units could vary by up to

50% and these formulas should be taken with caution (Medina et al 1994).

Various types of armouring units are described by Bakker et al (2003). Often it is the

case that a breakwater will be designed without considering alternative concepts

(Bakker et al 2003). Large tetrapods were the first interlocking armour unit and were

developed in France. The armour units not based on weight but on friction for

stability are the Cob, Shed and Seabee and have an extremely high hydraulic stability.

Compact blocks as armouring units are stable mainly due to their own weight. The

average hydraulic stability is also very low. However, the structural stability is high

and the variation in hydraulic stability is relatively low. Hence armour layers are in a

‘parallel system’ and have a low risk of progressive failure (Bakker et al 2003).

Page 27

Blocks can either have a random or uniform placement pattern. Further, armour units

can be classified according to their shape and number of layers and the stability

factor. The stability factor is dependant on whether the units are stable by their own

weight, by interlocking, or by friction. Where uniform placement is necessary, the

cost of construction increases greatly (Bakker et al 2003). Tetrapod breakwaters will

typically have a void ratio of about 0.5. Tetrapod placement can be random, and

typically is double layered, although the second layer does not necessarily increase

the stability as it tends to create rocking (Bakker et al 2003). Double layered

randomly placed blocks are a sensible design only for compact blocks, according to

Bakker et al (2003). The Accropode is the first randomly placed single armour unit

and has been the world leader for twenty years since 1980. The Accropode is a

balance between interlocking and structural stability. Steep slopes (15/12) are

recommended for the Accropode as these increases the hydraulic stability. Only minor

incident to blocks occurs when being dropped from 3m and thus, are quite robust

(Bakker et al 2003). Conservative values are recommended for the design of

Accropode breakwaters as there are uncertainties related to interlocking properties

(Bakker et al 2003). This shall be more favourable than double layered armour units

(Bakker et al 2003). Core-Loc is a new introduction by the US Army Corps of

Engineers in 1994. It has single layer placement, high hydraulic stability and a reserve

stability if the design wave height is exceeded, has no tendency to rock and large

residual stability after breaking, has a high porosity and roughness of the armour units

(for maximum dissipation of energy), possible incorporation between other types of

armour units, has a large structural stability, is easy to cast, and has an easy

construction of armour layers (even in low visibility water), has a minimum casting

yard and barge space as well as employing conventional constructional materials and

techniques (Bakker et al 2003). However, a large risk of progressive failure related to

breakage of armour units justifies to preference to the safety margins of the

Accropode concept (Bakker et al 2003).

Torum et al (2002) find the breaking characteristics of berm breakwater rocks by

finding the velocity of a rock on the point of impact. The velocity is obtained by

filming through a window in a wave flume wherein a breakwater model is positioned

(Torum et al 2002). The incoming wave is determined by:

Hmo = Hmo, measured/√(1+Kr2) (20)

Page 28

Where Hmo, measured is the measured wave height and Kr is the reflection coefficient

which was assumed to be 0.25.

Diffraction

Diffraction occurs when waves pass objects which alter the bathymetry sharply such

as a breakwater, causing energy to pass along the wave front. Bowen and McIver

(2002) concern themselves with the diffraction by a gap in an infinite permeable

breakwater. Cartesian coordinates are generally used for the diffraction of waves as

they pass such obstacles and are also used by Bowen et al (2002). Their solution is for

a breakwater that is permeable with a gap. The breakwater is assumed a length b.

Further, the breakwater alignment is that of the x axis and has a gap of 2a. Polar

coordinates (r, theta) are used where the angle is derived from the x axis. Linear wave

theory is also assumed applicable due to small angular frequency (Bowen et al 2002).

The length of the incoming wave is assumed to be much greater than the thickness of

the breakwater, so the thickness may be neglected and thus modelled as a permeable

barrier at y=0 (Bowen et al 2002). The velocity potential for the flow as a function of

the Reynolds number, can be used where the complex-valued function at satisfies the

Helmholtz equation:

_2_T/_x2 + _2_T/dy2 + k2_T = 0 (21)

The depth of the discussion in this paper by Bowen et al (2002), is not necessarily

important for this project as it is for a gap breakwater whereas the project is primarily

involved in an attached breakwater. However, the approach for continuity in the

velocity potential and breakwater boundary conditions is duly noted.

Diffraction around a breakwater is also studied by Briggs et al (1995) where an

experiment was set up to produce a numerical model for wave diffraction around a

breakwater (Briggs et al 1995). Similar wave height, period, and mean directions were

input to decipher the difference in wave height patterns for regular and irregular

waves. It was found that the directional spreading is more important than frequency

spreading in determining the energy pattern in the lee of a breakwater. Because the

diffraction process is sensitive to directional spread, the actual incident data should

include incident spectra of directions on a breakwater in field studies (Briggs et al

1995).

Page 29

Diffraction diagrams included in the Shore Protection Manual are derived by

equations based on a velocity potential assumption. Exact solutions have been

presented for diffraction around a permeable breakwater, valid for all angles of

incidence on breakwaters (McIver 1999). However, if original assumptions in the

Shore Protection Manual are met then the accuracy of the diagrams therein is assumed

to be good for design purposes.

Wave propagation in coastal engineering

Sherman and Greenwood (1986) recognise that wave propagation in the near shore

area is not a simple method. They find in previous literature that the wave angle just

prior to breaking is essential understanding for longshore currents and transport of

sediment (Sherman et al 1986). A small error in a wave angle will result in about 5

times the error in the calculation for radiation stress (Sherman et al 1986). They also

realise that previously, the calculation has been done through simple refraction

analysis or by visual inspection and offshore wave parameters (Sherman et al 1986),

although the more complex models are still based on the same equations the simple

method employs.

Wave measurements including angle of approach can be found through a spectral

analysis of a wave measurements by remote sensing (Sherman et al 1986). These will

also include peak period, peak offshore wave height, and a peak or most frequent

direction. These peaks may include waves from different sources for example,

alongshore currents or smaller period waves and longer period or swell waves

(Sherman et al 1986). The method employed is cross-spectral analysis, to determine

what percentage of the wave comes from offshore forcing, or from local or tidal

forcing.

Models

Liu and Losada (2002) discuss the various models used for calculating wave

propagation into the surf zone from the deep water zone. They find that in general

wave climates are determined offshore and these are transferred inshore to determine

near shore or project site (Liu and Losada 2002). They note that in general, the trend

is for more accurate data for the design wave and for near shore circulation, especially

of sediment (Liu et al 2002). Further, it is noted that the incoming wave is refracted

by shoaling waters, diffracted around abrupt bathymetric features such as underwater

canyons and ridges. The wave can also be refracted by currents (Liu et al 2002). They

Page 30

note some simple understanding of wave propagation such that the long waves lead

the short waves into shoaling waters. Loss of energy is due to breaking in shoaling

waters. This is because the speed of a wave is proportional to the depth of water and

the front of the wave will travel slower than the crest causing overturning and

turbulence and loss of energy (Liu et al 2002). This turbulence is also responsible for

sediment movement.

They note also that unlike in the early 60s, the use of powerful computers and

numerical models are utilised by engineers to provide a wave environment assessment

(Liu et al 2002). However, they do note that taking into account all of the physical

processes involves varying temporal and spatial scales. Two types of numerical wave

models can be distinguished by Liu and Losada. These are phase-resolving models

which are based on vertically integrated, time dependant mass and momentum

balance equations and phase-averaged models, based on spectral energy balance

equations. Phase resolving models application requires from 10-100 time steps for

each wave period and is limited to spatial extent from order 1-10 km (Liu et al 2002).

However, neither of these models considers all physical processes. They have also

reviewed the solving of the Reynolds Averaged Navier Stokes equations to simulate

the wave-breaking process.

The Navier Stokes equation and free surface boundary conditions are nonlinear. Thus

the solving of the truly three dimensional wave propagation problem is simply too

large over the scales of up to 100 wavelengths in engineering practice (Liu et al

2002).

The ray approximation technique is discussed only in passing. The technique is to

approximate the propagation of an infinitesimal wave across bathymetry that varies

slowly over horizontal distance. However, because the wave rays are not allowed to

cross in this approach, the approach fails near focal regions where rays intersect (Liu

et al 2002).

Thus, there are more complex models to deal with three dimensional obstacles. One

improvement to the ray approximation was suggested by Eckart (1952) and later by

Berkhoff (1972, 1976). This improvement could deal with large areas of both

refraction and diffraction. It assumes that diffraction is only important in close

Page 31

proximity to drastic bathymetric changes. Thus for a monochromatic wave with a

frequency _ and a displacement _ the velocity can be expressed as:

_ = -g_/_ * cosh k(z+h)/cosh kh * e -i_t (22)

where the k(x, y) and h(x, y) vary slowly over the horizontal direction, according to

the linear frequency dispersion relationship which is:

_2 = gk tanh kh (23)

and g is the gravitational constant (Liu et al 2002). Further, this led to the mild slope

equation:

grad . (CCggrad_) + _ = 0 (24)

where

C = _/k, Cg = d_/dk = (C/2) * (1+(2kh/sinh2kh)), (25)

are the local phase and group velocities of a plane progressive wave (Liu and Losada

2002).

It is shown in Liu and Losada (2002) that the mild slope equation is valid for both

shallow and deep water where it reduces to the linear shallow-water equation and the

Helmholtz equations respectively. Thus, the mild slope equation can be used for

propagating waves from deep to shallow water. There has been similar mild-slope

equations developed.

The parabolic approximation to the above mild slope equation has been applied

because the location of breaking waves cannot be determined. Through the parabolic

approximation, we can allow energy to travel across the wave ‘ray’ and hence,

diffraction is included (Liu et al 2002). As with the mild slope equation, we can allow

the free surface to propagate in the x direction (along the wave crest). Hence:

_ = _(x, y)eik0

x (26)

Page 32

where k0 is a reference constant wave number (Liu et al 2002). This approximation

also assumes that the amplitude varies quicker in the y direction than in the x

direction:

_2_/_y2 + (2ik0 + 1/(CCg)*(_CCg/_x))*__/_x + 1/(CCg)*(_CCg/_y)*(__/_y) +

(_2/g – k20 + (ik0)/(CCg)*(_CCg)/_x)_ = 0 (27)

Using an iterative procedure this approximation can be used weakly for backward

propagation (Liu et al 2002). Because the mild slope equation is linear as well as

parabolic approximation, the principle of superposition can be applied. The parabolic

models need a spectral input at the offshore boundary of the model (Liu et al 2002).

Then, the significant height can be computed at every grid point (Hs).

Finite amplitude waves require linearization assumptions in that

kA<<1 everywhere and A/h<<1 in shallow water (kh<<1) (28)

and will become invalid (Liu et al 2002). The finiteness of the wave has a direct effect

on the frequency dispersion and thus the phase speed. An example of the non-linear

dispersion relationship is the second order stokes wave:

_2 = gk tanh kh + _A2 + … (29)

where

_ = (k4C2)/(8sinh4 kh)*(8 +cosh 4kh – 2 tanh2 kh). (30)

Caution should be used using the extension to the Stokes wave theory into shallow

water (Liu et al 2002). Thus, as kh<<1 is approached the dispersion relationship can

be approximated to:

_2 = ghk2 (1 + 9/8*(A/h)/(k2h2)*A/h + …) (31)

and to ensure that this series converges for A/h<<1, the coefficient of the second term

must be one or smaller which gives rise to the Ursell parameter:

Page 33

Uf = O(A/h)/((kh)2) <= O(1). (32)

The requirement prescribing the Ursell parameter to be less than or equal to 1 is very

difficult to fulfil in practice. This is because with the linear theory we get A growing

proportionally to h-1/4 and kh decreasing according to √h. Therefore, the Ursell

parameter grows according to h-9/4 as water becomes shallower and should get larger

than one at some point (Liu et al 2002).

The standard Boussinesq equations can also be used in models. They are used for

variable depth:

_t + grad[_ + h)u] = 0 (33)

ut + _ grad|u|2 + g grad _ +

{h2/6*grad(grad.ut) – h/2 grad(grad.(hut))} = 0 (34)

where u is the depth averaged velocity, _ is the free surface displacement, h is the still

water depth, grad = _/_x, _/_y, the horizontal gradient operator, g gravitational

acceleration and the subscript t is the partial derivative with respect to time.

Numerical results from the Boussinesq equations have been shown to compare well

with field data and laboratory data (Liu et al 2002). However, when depth becomes

very shallow, the Boussinesq equations are not applicable as nonlinearity becomes

more important than the dispersion frequencies. Because in many engineering

applications the input consists of many different components of frequency, a lesser

depth restriction is often better (Liu et al 2002). Thus, modified forms of the above

equations are applied to shallow water and an example is given in Liu et al (2002):

_t = grad.[(_+h)u_] +

grad.{(z_2/2 – h2/6)*h*grad(grad.u_) +

(z_ + h/2)*h*grad(grad.h*u_)} = 0 (35)

u_t + _ grad |u_|2 + g*grad _ +

z_ { _ z_ grad (grad.u_t) + grad(grad.(h*u_t))} = 0 (36)

Page 34

where _ is the free surface displacement and u_ is the horizontal velocity vector at the

water depth (Liu et al 2002). The Boussinesq equation is able to model the

propagation of the wave from intermediate water to shallow water.

Another problem with depth integrated equations is the challenge of moving

shorelines and nearshore propagation. Liu et al (2002) notes that Lynett et al (2002)

have developed algorithms to counter this dilemma (Liu et al 2002).

Liu et al (2002) also explains that in most coastal problems, the energy transfer and

dissipation between water and the bottom are significant. The breaking of waves can

be included within the parabolic approximation to the mild-slope equation. There

have been many studies to understand how to incorporate breaking of waves into a

parabolic approximation model (Liu et al 2002). With incoming period, significant

wave height, mean wave direction, directional spreading and the width of the

frequency spectrum, parabolic approximations could successfully model the breaking

of a wave. It is concluded for the breaking of waves, that more specific models (such

as Reynolds Averaged Navier Stokes (RANS) Equations Model) are needed on

breaking waves. However, the RANS equations are very hard even for today’s

computing power to incorporate into a model as they are non-linear in three

dimensions. Thus little has been reported for such simulation of breaking waves (Liu

et al 2002). RANS model equations are explained in Liu et al (2002) and the

mathematical model described has been verified to be very close to experimental data.

It was also used to calculate overtopping of a caisson breakwater protected by armour

units (Hsu et al 2002).

Liu et al. (2002) also show the model capabilities over a rubble mound breakwater

structure with wave activity around the breakwater. The results correspond to the

simulation of a 1:18.4 scale model of Principe de Asturias breakwater in Gijon

(Spain). The breakwater is built of a core that is made of 14.2 kg blocks and an

armoured layer of 19.3 kg blocks. The model can incorporate these specifications (Liu

et al 2002). However, the development of structure-wave interaction models is in their

early stages. Future challenges include the integration of 2D depth integrated model

into the 3D RANS models. This incorporation would either be parameterized or direct

integration. Further, the two models could be coupled and solved simultaneously (Liu

et al 2002).

Page 35

Page 36

The literature review has shown various methods of breakwater designs, for specific

cases. It extends from simple rubble mound sizing equations to complex numerical

and analytical equations for complex multi-layered breakwaters. Because the design is

for Wilson Inlet, a contentious issue, I have adopted a simple approach to the design

process, a preliminary design to show the effect of a breakwater on the Wilson Inlet.

It is based upon the methods outlined in the Shore Protection Manual.

Page 37

Methods

The methods undertaken for the breakwater, were first to define the wave

approach to the Wilson Inlet by a ray tracing method outlined in the Shore

Protection Manual. Next, based on the direction of the waves, an alignment for

the breakwater is recommended and diffraction patterns around this were

drawn. Other methods that may be utilised in further design for the breakwater

and sediment movement are also outlined.

The quadratic friction law can be useful for modelling the effects of currents on

sediment for a long term study on the effect of a breakwater:

_o = _ CD u2 (37)

where CD is a drag coefficient or a friction factor. For investigations of sediment

transport, the velocity is generally measured at 1m above the bed. Thus:

_o = _ C100 u1002 (38)

where C100 = 3 x 10-3.

In the case of waves we get:

_o (t) = _ _ fw um2 (t) (39)

and thus

_max = _ _ fw um2 (40)

where um is the maximum current in a wave cycle and _max is the maximum shear

stress in a wave cycle. The factor fw can be determined for either the laminar or

turbulent case. However, sediment transport is less important than the wave analysis

past the breakwater and its diffraction pattern is more important.

Wave Analysis

Page 38

It must be understood that an ‘averaged wave’ that is input to a model may consist,

for example, of 14 different waves in a spectrum of waves. The period of a wave is

defined as the zero up crossing period (TZ) wherein the zero (still water level) level of

a wave crosses twice, analysed in the time domain. The height of the wave in this

period is the maximum displacement minus the minimum displacement and is called

H. As generally the waves will be analysed over a fairly long time, many

measurements of H and TZ are obtained.

The root mean square of the distribution of wave heights can be determined from the

data as:

Hrms = √((H12 + H2

2 + H32 + … + Hn

2)/n) (41)

Alternatively,

Hmax = max(H1…Hn) (42)

or arrange H1 to Hn in ascending order to get H1` to Hn` where Hn` is the maximum

and H1` is the minium. Then, the mean of the highest 1/3 of the waves is = Hs or

significant wave height. This is often what wave measurement output is desired for

design purposes. One can use equations to approximate Hs from Hrms or H1/10 (top 1/10

of wave heights):

Hs =√2 * Hrms (43)

H1/10 = 1.27 * Hs = 1.8 * Hrms (44)

This method is used (for any scalar parameter) to obtain a design wave for once in 1,

10, 50,100, 500 or 1000 years. It is recommended earlier in the literature that a design

period of 50 years should be used.

The wave generation process is worthwhile understanding as in each context this will

be slightly different. The offshore wave height is dependent on three factors:

H0 = f (u, D, F) (45)

Page 39

where u is the wind velocity, D is the duration of the wind blowing across the water,

and F is the fetch length or the length of water over which the wind is blowing. It has

been shown that the steepest wave possible is where h/L = 1/7. Where h/L goes larger

than 1/7 the wave will generally break. However, this limit also depends on the depth

of water at the locality of importance.

The wind stress factor UA can be used for calculations predicting wave heights:

UA = 0.71 * U101.23 (46)

where U10 is the wind speed at 10 meters above the sea surface. Also:

U10 = UZ * (10/z)1/7 (47)

where UZ is the velocity of the wind at z meters above sea level.

The wave motion does not displace the water in the large scale, but the individual

particles within a wave are caught in particle orbits. This orbit can be broken up into

parameters: the semi major axis and the semi minor axis:

A = H/2 * cosh [k(z+h)]/sinh [kh] (48)

B = H/2 * sinh [k(z+h)]/sinh [kh] (49)

where B is the vertical parameter and A is the horizontal parameter. It will be noted

that as we go closer to the seabed, the B value becomes zero and the particles move

left and right vacillating around a central datum with no net movement. The velocity

potential in the linear airy wave equations is given as:

_ = f(z) sin (kx – _t) (50)

where f(z) is given as:

f(z) = HC/2 * cosh[k(z+h)]/sinh[kh] (51)

Page 40

where H is the amplitude, C is the wave celerity, k is the wave number = 2pi/T, h is

the water depth, z is the depth at which f is being determined.

Further, we can find the offshore wave length and speed from the period only

according the linear airy wave theory:

Lo = gT2/(2pi) (52)

Co = gT/(2pi) (53)

where g is the gravitational acceleration due to gravity and T is the period of the

wave. The free surface elevation due to swell waves is given by:

_ = H/2 * cos(kx – _t) (54)

where k and _ are defined as

k = 2pi/L (55)

_ = 2pi/T (56)

and the dispersion relationship for a wave is thus:

_2 = gk tanh kh (57)

The general form for the wave length is given by:

L = gT2/(2pi) * tanh kh (58)

and C is equal to L/T.

To determine the wave height incident on the breakwater, offshore wave length is

determined from equation 52 as the period is already known from previous studies in

Ratcliffe Bay. Once the offshore wave length is determined, the wave length at a

depth can be determined from tables in the Shore Protection Manual (1984) of d/L

versus d/L0. Then, the equation to determine the new height is:

Page 41

H/H0` = √(1/2 . 1/n . 1/(C/C0)) = ks (59)

Where Ks is the shoaling coefficient and C/C0 is the same as L/L0. Where refraction

also occurs, the refraction coefficient KR is also necessary where KR is given by:

kR = √(b/b0) (60)

Where b0 is the offshore distance between wave orthogonals and b is the inshore

distance between orthogonals. Thus, as can also be found in the Shore Protection

Manual (1984), the new wave height is given by:

H = ks.kRH0 (61)

Wave characteristics are important in positioning of the breakwater, and therefore it is

also important to determine whether the waves will break before or after passing the

breakwater.

Storm Surges

Storm surges can cause the water level to increase that extra bit which will overtop

the breakwater in Wilson Inlet. Thus it may also be necessary to incorporate a storm

surge to add to the design wave height. We have:

u(t) = up(t) + ur(t) (62)

where u is the total response, up is the periodic response and ur is the residual storm

surge and is non-periodic. Further, we get:

∆_ = - ∆P/(_g) = - 0.993 ∆P (63)

which is the change in sea level in centimetres due to storms or air pressure change.

Thus we get that a 1 hecto-Pascal change in air pressure is equivalent to -1 cm change

in the sea level. This is the static response to a storm. The dynamic response is:

Dynamic response = static response / (1 – CA2/√(gh)) (64)

Page 42

where CA is the speed of the pressure system. The storm surge is not the dominant

forcing of the sea level in Ratcliffe Bay however, but during storm events, coupled

with high wave energy, it may become important.

Tides

From (30) we have the periodic tides which also add to the overall change in water

level. We have the equation:

up(t) = ∑ Hn cos(_nt – gn) (from n=1 to N) (65)

where Hn is the amplitude of the periodic tide. Tidal measurements may be taken from

Albany, as the distance from the Amphidromic point is similar as that of Denmark.

Tides in Wilson Inlet, although small, when incorporated with a peak wave height,

period and storm surges can cause serious damage and thus should be considered for

the breakwater design.

Longshore transport

Longshore transport may cause sediment to be transported into the opening channel to

the Inlet and should also be a part of the methods to design a breakwater in the Inlet.

Longshore transport will occur to some extent after a breakwater is installed and a

new sediment regime is established. The maximum longshore transport velocity will

generally occur at the breakers and depends on the angle of approach of the breakers,

the wave height at breaking and the slope of the beach:

VL (max) = 20.7 m (g Hb)1/2 sin (2_b) (66)

where m is the slope, Hb is the breaking wave height and _b is the angle of the

breakers orthogonal to the shoreline. Thus the maximum longshore transport occurs

when the angle of approach is pi/4 or 45 degrees.

Breaking Waves

Breaking waves are always incident on the Ocean Beach adjacent to Wilson Inlet and

thus should be understood as a mechanism for turbulence to transport large amount of

sediments and to cause damage. There are a number of maximum limits which, if the

wave characteristics reach, the wave will break. These are:

Page 43

(H/Lo)max = 0.142 ~= 1/7 (67)

(Ho/L)max = 0.142 tanh [kh] (68)

(H/h)max = 0.78

i.e.

_b = (Hb/hb)max = 0.78 (69)

However, it is also found that breaking of waves also depends upon the slope of the

beach at the point of breaking. This is readily found through measuring distance

across a number of contours to find the slope.

Wave Propagation

The propagation of the waves into Ratcliffe Bay is essential to the breakwater design

wave. This is based on the method in the Shore Protection Manual (SPM) from the

US Coastal Engineering Research Centre. The process is defined as refraction and is

analogous to Snell’s Law in other media. This is because the speed of the wave

decreases as the wave shoals.

The method as outlined in the SPM is based on Snell’s Law:

sin _2 = (C2/C1) sin _1 (70)

where _1,2 is the angle of incidence and reflection and C1,2 is the celerity in the first

region before crossing a contour line, and C2 is the celerity of the next depth. Use of a

template somewhat quickened the process and also improved accuracy (See figure 9).

Page 44

Figure 9 – Refraction template used to trace orthogonals both shoreward and

seawards (SPM)

The orthogonals are traced seawards according to the following steps:

1. Sketch a contour midway between the first two contours to be crossed, extend

the contour the this midcontour and then construct a tangent to this

midcontour at the point of crossing

2 . Lay the refraction template with the line labelled orthogonal along the

incoming orthogonal with the point marked 1.0 at the intersection of the

tangent and the contour.

3. Rotate the template about the point marked turning point until the C1/C2 value

corresponding to the contour interval being crossed intersects the tangent to

the midcontour.

4. Place a triangle along the base of the template and then construct a line

parallel to the template orthogonal line so that it intersects the incoming

orthogonal at a point B. The point B should be equidistant along the turned

orthogonal and the incoming orthogonal, which is not necessarily on the

midcontour line (SPM 1984).

5. These steps are repeated for successive contours.

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If the orthogonal is being refracted seawards, (deepening contours) then the C2/C1

value is used. Figure 9 illustrates the use of the refraction template:

Figure 10 – Usage of the refraction template to construct orthogonals (SPM)

Interaction with structures

The phenomenon of interaction of waves with structures is generally called

diffraction. The waves will diffract past and around behind the breakwater whereby

energy travels along the crest after being deformed by the structure. The equation

associated with this is:

Page 46

H = KD Hi (71)

where H is the new wave height, KD is the diffraction coefficient and Hi is the

incident wave height at the breakwater. This assumes a constant depth of water behind

the breakwater. Reflection also occurs depending on the coefficient of reflection _.

The SPM (1984) specifies diagrams derived by Wiegel to use to construct diffraction

diagrams showing wave patterns behind an impermeable breakwater for different

approach angles. The diagram used had a straight approach of incident waves (90o)

and an incident wave height of Hi. Thus, the diffracted wave height was given by

equation 70 using KD values from the diffraction diagram. The diagram is shown in

figure 10

Figure 11 – Diffraction diagram used for diffraction past breakwater (SPM)

The diagram allows one to measure in polar coordinates the wave height compared

with the incident wave height. Polar coordinates are in (wavelength, angle), i.e. where

the radius is one wavelength. Thus as the wave shoals, the radius or wavelength may

also decrease, and this should be accounted for in the diffraction diagram (SPM

1984).

Breakwater Properties

Page 47

Q = pi/6 * _s k3 is the equation that is used to determine the sizing of the rocks used in

the main breakwater. Equations 7-10 may be used if the type of breakers are known

incident upon the breakwater. Thus, when the weight of each unit is known, and a

specific density is known, the diameter of the units can be determined. The surf

similarity parameter may also be used if necessary for determining the stability of the

breakwater:

_z = tan_/√(2�Hs/gTz2)), (72)

where _ is the angle of the slope, Hs is the significant wave height, g is acceleration

due to gravity and Tz is the average wave period or the peak wave period (for

maximum design stability). Thus the nominal size of the breakwater units can be

determined as previously outlined by Van Der Meer assuming permeability, a damage

coefficient and other parameters.

The value of S/L will determine whether the waves will pass forward pass the

breakwater and break, or break into the head of the breakwater structure.

Page 48

Results

Bathymetry

Bathymetry was determined both from an A1 couple of contour maps for Ratcliffe

Bay and x, y, z data gathered from the DPI. Further 100, 50, 30, 20 and 10 contours

were also received from the DPI. This data was input into an m file called

‘bathymetry.m’ whereby a contour map in MATLAB could be produced showing

depths. The resulting contour map was output of a grid which interpolated the input

data to find the depth at each point on the grid. It was first hoped that a computer

refraction code could be developed to map orthogonals into Ratcliffe Bay, but this

process was not complete. The codes and output are presented in Appendix C.

Wave Analysis

The wave incident on Ocean Beach has been measured by Ranasinghe et al (1999).

The 50% exceedance offshore wave height was given as 1.0m. However, within the

study, the maximum Hmo is found to be about 2.5m. For design stability Hmo is taken

as 2.5m. The offshore peak period is given as 13s. This gives an offshore wavelength

of Lo = 263.68m. From the NOAA data, the average period is approximately 13s and

direction is SSW.

Page 49

Figure 12 – peak wave period and direction from the National Oceanic &

Atmospheric Administration (NOAA).

Further, waves at Denmark are visually seen to take on the shape of the bar, breaking

parallel to the shore. From personal observation, waves can be up to 2.5m on a given

day, as is also seen in figure 13.

Page 50

Figure 13 – Wave climate at Ocean Beach in Denmark, attracts many surfers

around the world. (Photo: Albany Sightworks, Sylvia Gartland)

Refraction diagrams

The refraction diagram from 30 metres to shore is shown in Appendix A and was

drawn by hand according to the method in the Shore Protection Manual. The red lines

give contours five or ten metres apart and the blue lines give contours one metre apart.

The contour data was a survey done on Ratcliffe Bay by the Department of Planning

and Infrastructure (DPI) in 1995 according to the horizontal datum AGD 1984 and a

vertical datum of 5.191m below BM A427. The contours were given every 1 metre

from 3 up until 20m in depth after which contours were every 10 metres up until the

30m contour.

However, it was realised that a depth of 30m was not offshore conditions, and it was

best to begin wave orthogonals parallel at a distance that defined offshore conditions,

i.e. when d/L was greater than 0.5. Thus, further depth contours were obtained from

the DPI in ‘.dxf’ format for contours of 100, 50, 30, 20 and 10 metres in depth.

The study by Ranasinghe et al (1991) in Ratcliffe Bay obtained data indicating that at

a depth of 17m the offshore wave height could be determined as 1.0m for 50%

exceedance. The peak spectral period was 13s. The period is all that is necessary to

determine the refraction diagram, as given an inshore depth and the d/Lo ratio we can

Page 51

determine the wavelength at that depth using equation 52 and Appendix C-1 in the

Shore Protection Manual.

The refraction diagram of 100m to shore is included to show how the incoming wave

from the data given by Ranasinghe et al (1999) (10o) to the west from south was

propagated seawards to give an offshore approach angle.

This angle was then used to propagate several wave rays into Ratcliffe Bay and to

observe the refraction diagram. Values used to determine the refraction diagram are

also found in Appendix A. The diagram indicates how the orthogonals spread to take

on the shape of the bay, and is also evident from aerial photography of the bay.

Refraction/Diffraction diagrams

The refraction/diffraction diagram outlines how the waves pass the breakwater and

are incident on the beach near the Wilson Inlet channel berm. The diagram is given in

Appendix B. On it is shown the K’ diffraction coefficients for waves refracting

around the breakwater as shown. This diagram shows the location of the proposed

breakwater and the extra breakwater to minimise sediment deposition at the entrance

to the Wilson Inlet channel.

The main breakwater is labelled ‘B1’ and the secondary breakwater is labelled ‘B2’.

The position of B2 is so that the longshore transport that does eventuate from the

primary breakwater does not deposit at the entrance of the channel as it is transported

along the shore (although this value is expected to be small compared with deposition

due to incident waves without a breakwater). Ranasinghe et al (1999) noted that the

numerical model with only longshore transport did not close the channel, and this

condition could be mimicked by a breakwater which effectually dissipates incoming

energy from waves.

The diffraction diagram in Appendix B indicates a K’ value of about 0.1 opposite the

mean position of the western opening of the bar for Wilson Inlet. Further, the level

lines for K’ bend towards the Nullaki point as they approach the shore, indicating that

increasing proximity to the shoreline will also decrease the wave height (neglecting

wave shoaling for the present). The value of S/L is given by the slope length divided

by the wavelength at the toe of the breakwater. Since the depth is approximately 13

metres at the toe, and the slope is 2, we get S=14.5m and a wavelength of 138m.

Page 52

Thus, S/L = 0.105 which is considered small. Hence, waves will propagate forward

past the breakwater into the near shore area.

Page 53

Synthesis

Sizing of Armour Units and Breakwater Specifications

The sizing armour units are essential to the stability of the breakwater. Due to the

prototype nature of this project, a single layer breakwater with armour units is

suggested. From the diffraction diagram, the chosen breakwater alignment is parallel

to the monochromatic wave climate indicated previous which will maximise energy

dissipation from incident wave energy. The incoming wave height at the toe of the

breakwater is given by

H = ks.kRH0

And n is given by:

n = _ [1 + 2kh/sinh kh] = 0.52

ks is thus equal to 1.35 and kR is equal to √(bo/b) = √(30/40) = 0.866 determined from

the difference of measurement of the distance between orthogonals offshore and at the

breakwater toe.

Thus, assuming an offshore wave height of 2.5m, we get the height at the breakwater

as:

H = (1.35).(0.866).(2.5) = 2.92m

Hence, we get that the wave height opposite the Wilson Inlet is approximately 30cm.

Because this dominant wave motion with a breakwater is seen to be alongshore, with

its energy being broken by breakwater B2, this result is appreciably small.

The weight of armour units is determined from the wave height, the surf similarity

parameters, breaking wave height and the slope of the breakwater, as well as other

parameters which may or may not be included.

Page 54

The slope of the bathymetry in the near shore locality behind the breakwater can be

determined by measuring distance across six contours (1m between each). This

distance is measured as 250m. Thus, the slope is rise over run:

Slope = 6/350 = 0.024

It is noted that the slope is a lot smaller in the region before the breakwater than after.

To utilise figure 7-3 in the SPM the parameter Ho’/(gT2) must be determined. This is

thus given as 2.5/(9.81*132) = 0.001508 which corresponds to a value on the diagram

for Hb/Ho’ of 1.45. Thus, Hb is given as 1.45 * 2.5 = 3.625m. Then, Hb/(gT2) is equal

to 0.00219 and using figure 7-2 in the SPM we find that db/Hb is equal to 1.125.

Hence, db is equal to 4.08 metres depth. Hence, the waves break past the breakwater

and the weight of the units according to Hedar (1985) is determined by the down rush.

First, the weight of each block is determined by the Hudson equation assuming

variables for _s = 1760 kg/m3; H = 2.92m; kD = 2.4; _w = 1030 kg/m3; and cot _ = 2.

Then, we get a weight for the trunk units of 25 x 103 kg. And thus a diameter of 3.03

metres is necessary for the trunk blocks. Blocks in the lower stability 60o sector of the

head of the breakwater should have a weight 3 times the weight of the trunk values,

i.e. approximately 75 x 103kg. The value of 1760 is the density of medium to low

density limestone, because the limestone quarry nearby may provide materials for the

breakwater if cost efficiency is the first priority.

Secondly, we can determine the weight of each block according to Hedar by the down

rush equation for a pervious under layer. Application of equation nine give the

outcome for the diameter of the blocks as 4.48m assuming a slope of cot _ = 2 and the

angle of repose as 45o. For stability, units at the head of the breakwater should have a

diameter corresponding to about 3 times the weight of a unit with a diameter of 4.5m.

These units are placed in the 60o sector as previously described elsewhere in this

paper.

Page 55

Conclusions

The breakwater design for Wilson Inlet as presented above is a preliminary design.

Further investigation into the detail of the breakwater is necessary as the design must

be done right because of the expected lifespan of a breakwater.

The diffraction and refraction diagrams show that a breakwater will decrease onshore

transport of sediment and decrease the probability of a bar forming in front of the

channel. Further, because the rays traced into the bay are relatively perpendicular to

the bottom contours, time to reach equilibrium with the installation of a breakwater

will be small. More importantly, longshore transport will be minimised and the

hydraulic integrity of the channel (in that it transports sea water to the inlet and) is

maintained because of this.

Wave heights at the breakwater are sufficiently small so that breaking occurs past the

breakwater. This is important for the surf life saving club and those interested in

surfing at Ocean Beach because wave height does not diminish to a large degree to

the eastern leeward side of the breakwater although a reef may impinge upon the

surfing area.

A breakwater is a major change to the hydraulic regime of Ratcliffe Bay as large

wave energy will not be incident upon all of the shoreline. However, longshore

transport may prove important as equilibrium is reached and help to maintain some of

the freshwater regime of Wilson Inlet.

Breakwater unit sizes are sufficiently small to enable use of local material for most of

the breakwater, and because the breakwater is connected at an accessible point,

construction costs are also minimised, to increase the benefits for the environment (as

the Wilson Head environment will not be impinged upon to a great extent) and for

people involved in procuration of materials and finance.

Page 56

Recommendations

As the design of a breakwater for Wilson Inlet is not yet seen to be sustainable at this

stage (WRC 2002) as nutrient release from sediments in the inlet is not fully

understood, further design of the breakwater parameters such as core components

(e.g. an impervious caisson or horizontal plate) and other design features such as a

groin near the channel to the inlet are not discussed in any detail. However, in future

studies, features such as internal plates, which can reduce the size of the breakwater,

and maintain its integrity should be looked at.

Further, the diffraction of the full spectral directions might also be of benefit as the

spectral variation is important in the resulting wave patterns past a breakwater.

Further analysis of waves breaking past the breakwater, for use by surfers and tourists

should be analysed, so that a safe swimming beach may still be present.

Finally, analysis of the quarry stone near Denmark and its logistical sustainability for

use in a breakwater should be done. Tests done should include drop tests and test of

density variability.

Page 57

Appendices

Page 58

Appendix A

Figure A1 Ray Tracing from 100m contour shorewards

Page 59

Figure A2 – Ray Tracing from near shore area

Page 60

Table C2 - Shoreward Ray Tracing

d d/Lo tanh khc1/c

2 c2/c1 d4 0.01516 0.3022 4

1.12 0.892

5 0.01895 0.3386 5

1.096 0.913

6 0.02274 0.371 6

1.067 0.937

7 0.02653 0.396 7

1.062 0.942

8 0.03032 0.4205 8

1.06 0.943 �

9 0.034109 0.4457 9

� 1.053 0.95 �

10 0.037899 0.4691 10

� 1.047 0.955 �

11 0.04169 0.4911 11

1.036 0.965 �

12 0.04548 0.5088 12

� 1.034 0.967 �

13 0.04927 0.5263 13

� 1.035 0.966 �

14 0.05306 0.5449 14

� 1.032 0.969 �

15 0.05685 0.5626 15

� 1.026 0.975 �

16 0.06064 0.577 16

� 1.028 0.973 �

17 0.06443 0.593 17

� 1.023 0.977 �

18 0.068218 0.6069 18

� 1.024 0.976 �

19 0.072008 0.6217 19

� 1.023 0.978 �

20 0.0758 0.6359 20

� 1.172 0.854 �

Page 61

30 0.1137 0.745 30

Table C1 - Seaward Ray Tracing

d d/Lo tanh kh c1/c2c2/c

1 d10 0.037899 0.4691 10

1.3556 0.7377

20 0.07579771 0.6359 20

1.1716 0.8536

30 0.11369666 0.745 30

1.1758 0.8505

50 0.18949443 0.876 50

1.1236 0.89

100 0.37898886 0.9843 100

Page 62

Appendix B

Page 63

Figure B1 – Diffraction Diagram for Wilson Inlet around the Breakwater

Page 64

Appendix C

Figure C1 – Output contour from running bathymetry.m for 100 by 75 uniformgrid

Page 65

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Page 70

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