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:
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(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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