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Wavepressurereductionon

verticalseawalls/caissonsdueto

anoffshorebreakwater

ArticleinIndianJournalofGeo-MarineSciences·December2004

DOI:10.1115/OMAE2003-37074

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Ocean Engineering xx (xxxx) 1–18

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www.elsevier.com/locate/oceaneng

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M.G. Muni Reddya, S. Neelamanib,*

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a

Department of Ocean Engineering, Indian Institute of Technology Madras, Chennai-600 036, India

Coastal Engineering and Air Pollution Department, Environmental and Urban Development Division,

Kuwait Institute for Scientific Research, P.O. Box 24885, 13109 Safat, Kuwait

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Received 21 January 2004; accepted 9 July 2004

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This paper presents results obtained from a series of experiments conducted in wave flume to

assess the influence of the offshore low-crested breakwater as a defence structure in reducing the

wave forces on vertical seawall. The main aim of the tests was to know the effect of crest elevation of

the offshore low-crested breakwater as a rehabilitation structure for the existing damaged shore

protection structures. In this study five relative breakwater heights are used and associated flow

evolution was analyzed. With the sections proposed in this study, it is possible to achieve

considerable reduction of wave force on the seawall. Modification factor is proposed to estimate the

shoreward force on the seawall defenced by low-crested breakwater.

q 2005 Published by Elsevier Ltd.

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Abstract

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Keywords: Low-crested breakwater; Shoreward force; Overtopping; Submerged breakwaters; Seawall;

Modification factor

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Coastal erosion is one of the challenging coastal engineering problems faced by human

being around the world. This calls for the proper remedial measures to protect valuable

properties situated along the coast. Many seawalls and vertical caisson breakwaters

(CIRIA, 1986b; Oumeraci, 1994) around the world are being damaged. Such failures are

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1. Introduction

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Hydrodynamic studies on vertical seawall

defenced by low-crested breakwater

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* Corresponding author. Tel.: C965 483 6100x5351; fax: C965 481 5192.

E-mail addresses: reddy_muni@hotmail.com (M.G. Muni Reddy), nsubram@kisr.edu.kw (S. Neelamani).

0029-8018/$ - see front matter q 2005 Published by Elsevier Ltd.

doi:10.1016/j.oceaneng.2004.07.008

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mainly caused by extreme wave actions, through displacement of the entire structure, or

progressive failure starting from locally weak point, or through overall foundation failures,

or through overtopping and toe erosion. It may be economical to allow the less frequent

storm wave to spill over the crest of the seawall rather than to its full height to reflect fully

all the waves. The disadvantage, however, is that overtopping waves plunge over the crest

and inundates lee side leading to high economical loss.

The need for force reduction on these structures to increase the life span has resulted in

different force reduction techniques like, introduction of porosity at the front face of the

caisson, slotted seawalls, construction of horizontally composite caissons and construction

of low-crested caissons etc. Introduction of porosity into the structure leads to reduction of

the strength of the structure. Construction of horizontally composite structure in dynamic

environment is risky. Low-crested breakwater attracts lesser forces but the overtopping of

waves create significant disturbance on the lee side. These drawbacks can be overcome by

constructing a low-crested breakwater in front of these structures to reduce the incident

wave energy levels. The offshore breakwater can be constructed after installation of

caisson without much risk for floating vessels and caisson. For existing weak or damaged

structures construction of a protection structure such as submerged offshore breakwater is

relatively an easy task.

Submerged breakwaters with deeper submergence would give larger wave energy

transmission, which might eventually lead to failure in sheltering function of the

breakwaters. Therefore how to reduce the incident wave energy levels becomes a great

challenge for coastal engineers. In the present study an offshore low-crested rubble mound

breakwater is considered as a defence structure to reduce the incident wave energy levels

that reach the vertical impervious structure viz., seawall/caisson. This type of protection

can also be used in situations wherein it is required to reduce the wave forces to enhance

the functional life of protection structures that are damaged by extreme wave forces, as a

rehabilitation structure. A theoretical analysis of the present problem is cumbersome. Due

to the complexity of the physical processes at the submerged breakwaters, physical

modeling is necessary to define the site-specific interactions between the structure and the

local wave climate. The defence structure may become submerged or emerged during the

tidal variation

Low-crested rock structures can be classified (van der Meer and Daemen, 1994) as

dynamically stable reef breakwaters, statically stable low-crested breakwaters and

statically stable submerged breakwaters. A reef breakwater is low-crested homogenous

pile of stones without a filter layer or core and is allowed to be reshaped by wave attack

(Ahrens, 1987).

Statically stable low-crested breakwaters are close to non-overtopping structures, but

are more stable due to the fact that large part of the wave energy can pass over the

breakwater (Powell and Allsop, 1985). All waves overtop statically stable submerged

breakwaters and the stability increases remarkably if the crest height decreases.

Submerged breakwaters have been widely used as wave energy dissipaters. Efficiency

of the submerged breakwaters depends on the crest free board, crest width and permeable

material characteristics. Many investigators like Newman (1965); Dick and Brebner

(1968); Dattatri et al. (1978); Losada et al. (1997), have studied the wave transmission

and reflection characteristics. The stability and wave transmission characteristics of

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the low-crested rubble mound breakwaters were investigated by Allsop (1983); Ahrens

(1989); van der Meer and Pilarczyk (1990); van der Meer and d’Angremond (1991);

Seabrook and Hall (1998), and Yamashiro et al. (2000) and, the design formulae were

developed by van der Meer and Daemen (1994) by analyzing the data sets of various

investigations. Behavior of the deeply submerged breakwaters with multi vertical sliced

permeable structure was investigated by Twu et al. (2000).

Based on the monitoring results of a submerged breakwater and resulted model

studies, Dean et al. (1997) have reported that detached breakwater modifies both the

wave and current fields depending substantially on the crest elevation relative to the

still water level. However, not much study on the present topic except the work by

Gonzleg Madrigal and Olivares Prud’homme (1990) on the reduction of forces on

vertical breakwater defenced by seaward submerged breakwater. For partial barrier of

any configuration, irrespective of the porosity and flexibility, full reflection always

occurs when the distance between the end-wall and the barrier is an integer multiple of

half-wave length and hence overturning and moment will vanish (Yip et al., 2002).

Many investigators have studied analytically and numerically the wave transmission

and reflection characteristics of the submerged breakwaters. Yet these mathematical

models cannot reproduce some of the features observed such as strong mean water

level gradients on the submerged breakwater, pumping effect of the submerged

breakwater and vertical circulation induced by breaking waves on the submerged

breakwater.

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2. Experimental procedure and investigation

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Experiments have been carried out in a 30 m length 2 m wide and 1.7 m deep

wave flume at Indian Institute of Technology Madras, Chennai, India. Seawall was

fixed (Fig. 1) over a six-component force balance (GmbH R67). Top level of the

force balance is flushed with the flume bed. The sensitivity of the transducers (strain

gauge type) of six-component force balance at rated loading is about G2 mV/V.

Force balance consists of a stainless steel platform 850!850 mm size, below which

force transducers were fixed to a rigid frame of 900!900 mm2. This frame was

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Fig. 1. Experimental set-up for the present study.

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tightly fixed to the flume sidewalls to arrest the movement of force balance. Seawall

model was mounted on top of the steel platform so that the force on the seawall will

be transferred to the transducers. The height of the seawall model was fixed based on

the theoretically estimated maximum run-up over the seawall, to ensure no

overtopping of waves. Crest width of the offshore low-crested breakwater was chosen

as 0.40 m. The stable weight of the armour unit of the breakwater was estimated by

using the van der Meer (1987) formulae for statically stable low-crested and

submerged breakwater. Here our aim was not the damage of the low-crested

breakwater, so a stable armour weight was used. Breakwater was constructed with two

layers, an armour layer and core. Weight of the armour stone was 14.70–19.62 kN

[This was arrived at for the inputs significant wave height ‘Hs’Z0.29 m, zero crossing

period ‘Tz’Z3.0 s, damage level ‘S’Z2, number of waves ‘N’Z3000 and gsZ

26.5 kN/m3 for plunging breaking]. The weight of the core stone was 1.96–2.45 kN

(gsZ29.5 kN/m3). Five crest level configurations (two emerged, two submerged and

one at still water level) were used in this study. A stable slope of 2H:1V was adopted

as the effects of breakwater slope on the wave transformation were found to be

relatively unimportant (Seabrook and Hall, 1998). Ratio of the breakwater height to

water depth h/d is varied from 0.66 to 1.33, keeping the water depth ‘d’ constant at

0.30 m and varying the height of the breakwater, ‘h’ from 0.20 to 0.40 m with 0.05 m

increment. This simulates the investigation on site where the tidal fluctuations are

insignificant. Two pool lengths, Lp (Lp is the distance between the toe of the lowcrested breakwater and seawall, Fig. 1) 0.50 and 1.0 m were used.

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2.1. Data collection and analysis procedure

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The wave synthesizer (WS4) involving an application software package, along with

analogue-digital and I/O modules installed in personal computer was employed in the

measurement and analysis. The software is capable of controlling the wave paddle and at

the same time acquires data from sensors used in the tests. The force balance transducers

are connected to the data acquisition system through carrier frequency amplifiers. Each set

of data for regular wave was sampled at frequency of 40 Hz. The filtered signals are

analyzed using the wave synthesizer. It contains the options for synthesis of regular and

random 2D waves. Regular waves of different predetermined wave period and wave

amplitude combinations are generated for the testes. The horizontal force (force in the

direction of wave propagation), vertical force on the seawall, run-up on the wall and wave

elevations in front of the model were acquired.

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2.1.1. Range of inputs

Relative wave height, Hi/d

Relative depth, d/L

Wave steepness, Hi/L

Relative breakwater height, h/d

Non-dimensional pool length, Lp/L

Relative breakwater width, B/d

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M.G. Muni Reddy, S. Neelamani / Ocean Engineering xx (xxxx) 1–18

0.15–0.51

0.025–0.192

0.003–0.058

0.66–1.33

0.035–0.641

1.33

Here L is the deep-water wavelength and Hi is the incident wave height.

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2.1.2. Data analysis

The data collected were converted to physical variables by using the corresponding

calibration constants/coefficients. The raw data (in the form of time series) were analyzed

in time domain to get the clear understanding of the phenomenon under investigation. The

measured wave height, wave periods and forces were obtained by analyzing the measured

time histories of wave surface elevations and force amplitudes using the thresholdcrossing analysis. The threshold-crossing option is a generalization of classical zerocrossing analysis. For a pre-defined reference level, the input time series is divided into

events. For each event, the peak–peak value, the minimum and maximum values, and the

duration are determined.

The time series of the different parameters stated earlier were viewed to pickup the part

of time series with regular trend by omitting the transient part. This also ensures that no rereflected waves were present in the selected window of the time series. The regular time

series of force was then subjected to threshold-crossing analysis to get the mean amplitude

of the time history. The mean of the all amplitudes above the reference level in a time

series is taken as a positive or shoreward force. Similarly mean of all the amplitudes below

the reference level on a time series is taken as negative or seaward force. The mean

amplitudes of measured hydrodynamic force were obtained using the above procedure for

each test run.

½F x shore is the ratio of shoreward force in the direction of wave propagation in the

absence of the low-crested breakwater to shoreward force in the direction of wave

propagation in the presence of low-crested breakwater. ½F x sea is the ratio of seaward force

in the direction opposite to wave propagation in the absence of the breakwater to the

seaward force in the direction opposite to wave propagation in the presence of the

breakwater. These forces are obtained using procedure for the respective case of with and

without low-crested breakwater.

Incident wave elevations are measured using DHI capacitance wave gauges in the

absence of model in the flume, for pre-determined sets of different wave period and wave

height combinations. This procedure is repeated thrice and the average value is taken for

the wave height for that particular combination. It is done with a view to check the

repeatability of wave heights at the same point later when tests are conducted with the

model in position.

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3. Results and discussion

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3.1. General

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The non-breaking wave forces on seawalls are pulsating. A substantial portion of the

horizontal momentum of the wave is imparted to the wall. Methods to calculate the wave

forces for simple vertical structures and pulsating wave conditions are relatively well

established and are described by Goda (1985).

According to Goda (1985)

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Fh Z 0:5ðp1 C p2 Þh C 0:5ðp1 C p4 ÞhÃc

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p1 Z 0:5ð1 C cos bÞða1 C a2 cos2 bÞrw gHmax

(2)

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p2 Z p1 =½coshð2ph=LÞ

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p 3 Z a 3 p1

(4)

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p1mod Z 0:5ð1 C cos bÞa1 rw gH

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where Fh, total horizontal force per meter length of the wall/caisson; hb, water depth at a

location at a distance of 5H1/3 seaward of the breakwater; a2, pressure coefficient (varies

from 0 to 1.0); b, angle between the direction of wave approach and a line normal to the

breakwater; d, depth above the armour layer of the rubble foundation; hc, crest elevation of

the breakwater above the bottom of the upright section; h*, elevation at which the wave

pressure exerted; hÃc , min{h*, hc}; Hmax, maximum or design wave height.

p1, p2, p3 and p4 are the representative wave pressure intensities. Pressure coefficient a2

represents the tendency of the pressure to increase with the height of the rubble mound

foundation. The coefficient a2 (Eq. (6)) becomes zero, as hb and water depth d are the same

in the present study. Hence the Eqs. (1) and (2) can be written as

(7)

(8)

p1mod is less than p1 in Eq. (2) because of additive term a2 cos2 b vanishes. From Eqs. (3)

and (4) it can be observed that the magnitude of p2 and p3 reduces hence horizontal force

Fh in Eq. (8).

The measured shoreward forces (without breakwater) are compared (Fig. 2) with

Eq. (8) for validation of the present shoreward force measurements. The measured

forces are more than the estimated forces. Increase of wave pressure/force due to the

presence of a rubble foundation may regarded as the result of the change in the

behavior of wave from non-breaking to breaking although actual waves never exhibit

such marked changes.

Most design methods for caisson and the other vertical wall concentrate on forces that

act landward, usually termed as positive forces. It has however, been shown that some

breakwaters/walls failed by sliding or rotation seaward indicating that net seaward forces

may indeed be greater than positive forces.

The time series of incident wave height and wave force on the wall for different

relative breakwater height ‘h/d’ ratios are shown in Fig. 3. Quantitative reduction in

force on the seawall with increased h/d is very clear. The time series of wave forces on

the seawall defenced by an low-crested breakwater show that the wave breaking on the

breakwater generates high frequency waves on the lee side of breakwater, which results

in irregular force time series consisting of superposition of fundamental wave

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(6)

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a2 Z min ½ðhb K dÞ=3hb ðHmax =dÞ2 ; 2d=Hmax

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p4 Z

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Fig. 2. Comparison of non-dimensional shoreward force on vertical seawall with Goda’s (1974) formulae [dZ

0.30 m, Hi/dZ0.29K0.48].

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frequencies and the higher wave frequencies. It would be worth mentioning at this

point, the effect of wave set-up or pumping effect (Drei and Lamberti, 2000) or the

piling-up (Diskin, 1970) of water behind the protected area creates a difference in mean

water level inside the protected area and that of open sea. This component is inherent

in the time series shown in the Fig. 3. It is difficult to quantify this component in the

force measurement, because the force balance measures total effect. For laboratory

measurements this effect is unavoidable due to the fact that the water will confine

between the sidewalls of the flume and between two structures and there will be very

little scope for water to escape. In the field situations, in open sea this effect will not be

of much significant as there will be sufficient space for water to escape laterally

between the two structures. It should be noted that experiments were conducted in the

two-dimensional flume, and thus the values of mean water levels may be overestimated

in comparison with the values of mean water levels in three-dimensional wave field.

About 14% deviation observed from the forces estimated by Eq. (8) and the forces

measured from the experiments.

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Fig. 4 provides the effect of h/d on fore ratio ½F x shore for different incident wave

steepness. Force ratio 1.0 means that the breakwater has no effect on the reduction of

forces on the caisson and zero means 100% protection of the caisson by low-crested

breakwater. The value of force ratio lies in-between zero and 1.0. Oscillatory nature of

force ratio ½F x shore is observed when the h/d is varied from 0.66 to 1.33. The amplitude

of the oscillation decreases with increase of h/d. The high value of force ratio for

h/dZ0.83 is due to wave jetting on the seawall after overtopping over the low-crested

breakwater. This increased force is unwarranted for the general presumption that as the

barrier height increases force will have to decrease correspondingly. Designers and

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3.2. Effect of relative height of the breakwater, h/d on the normalized wave

forces on the seawall

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Fig. 3. Typical force time series for different relative breakwater height h/d [HiZ0.152 m, dZ0.3, d/LZ0.059,

B/dZ1.33, Lp/LZ0.071–0.64].

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coastal engineers should take care of this while decision making in choosing the range

h/d values. For h/dO1.0, the wave energy is effectively dissipated which result in

significant wave force reduction on the seawall. When h/dZ1.0, the reduction in

average shoreward or positive force is 66% (standard deviation is 0.097) as Hi/L is

varied from 0.003 to 0.058 for the range of Lp/LZ0.035–0.321. Percentage decrease in

the magnitude of peaks of force ratio is ½F x shore found to increase with h/d. The

following wave-structure interaction processes were identified during the experimental

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Fig. 4. Variation of shoreward force ratio with relative reakwater height h/d for three different wave steepness

[Lp/LZ0.198, B/dZ1.33, d/LZ0.059].

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investigations, which are explained below for the type of normalized wave force trend

observed:

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(a) For offshore breakwater with more submergence (say h/dZ0.66), the wave transmit

freely, reflects from the seawall. These reflected waves contribute significantly for

the amplification of waves and the corresponding wave forces on the seawall/

caisson.

(b) For offshore breakwater with smaller submergence (say h/dZ0.83), the propagating

wave on the breakwater attains the characteristics of wave breaking and the

overtopping jet of mass acts on the seawall/caisson resting behind the breakwater

and imparts higher order of forces.

(c) For the case of offshore breakwater with crest level flushing with still water level

(h/dZ1.0), most of the interacting energy is expected to be dissipated on the crest

of the breakwater and hence the wave force reduction is significant.

(d) For the offshore breakwater with less emergence i.e. crest located just above the

still water level (here h/dZ1.16), the dominant mode of wave transmission is by

run-up and overtopping and the efficiency of transmission process increase as wave

height increases. The energy available with this overtopping water mass imparts

forces on the seawall. The wave energy dissipation due to the interaction with the

breakwater reduces to the significant overtopping processes.

(e) For the offshore breakwater with significant emergence of the crest (h/dZ1.33),

overtopping will be prevented for most of the waves and the waves may be allowed

to transmit through the pores of the breakwater. The energy available with this

transmitted wave imparts forces on the rear side structures.

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A through analysis of Fig. 4 with this understanding gives a clear answer why a force

ratio variation is oscillatory with increased h/d. It was observed that the force ratio at any

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Table 1

Measured mean force ratios for two pool lengths

h/d

Lp/LZ0.

035–0.

32, Hi/

LZ0.

003–0.09

½F x shore

Standard

deviation

½F x sea

Standard

deviation

Lp/LZ0.

071–0.

641, Hi/

LZ0.

003–0.09

½F x shore

Standard

deviation

½F x sea

Standard

deviation

1.33

1.16

1.00

0.83

0.66

0.17

0.31

0.33

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h/d is the minimum when the waves acting on the system are steeper. This point must be

given due attention, since the design is carried out for steeper waves. The results plotted in

Fig. 4 are typical for a d/L value of 0.058. The results for the other values were also

observed to follow the same trend.

In order to bring out the cumulative effect of range of heights and periods used in the

study, a table containing the force ratio information is prepared. Table 1 is provided to

visualize the force ratio for two pool lengths and for a wide range of wave steepness. Both

shoreward and seaward force ratios are provided in the table. The mean and the standard

deviation of the wave force ratios are tabulated. Mean of the shoreward force amplitude

and seaward force amplitude and the corresponding standard deviations are provided in

this table. For example, the shoreward force ratio value for h/dZ1.33 is given as 0.17,

which is the average value for a number of wave heights and periods. The standard

deviation for this set is 0.066 (i.e., the ratio of standard deviation and the average is about

38.8%). This table is mainly provided for an overall understanding of the effect of pool

length and relative height of the breakwater on wave force reduction on the seawall. From

this table it is clear that the shoreward force ratio for the case of smaller pool length (Lp/

LZ0.035–0.32), is from 0.17 to 0.55 when h/d is varied from 1.33 to 0.66. This means that

the mean wave force reduction of the order of 83–45% is possible for this case. For the

large pool length ratio (Lp/LZ0.07–0.64), the shoreward force ratio ranges from 0.2 to

0.48 for the same range of h/d. The seaward force ratio is ranging from 0.29 to 0.60, which

is significantly higher than the shoreward force ratio. The average value of force ratio

along with the standard deviation can be used for the selection for appropriate vale of h/d.

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3.3. Effect of wave period and wave height on wave force ratio

Fig. 5 shows the effect of variation of wave period on shoreward force ratio. The

variation of wave period is presented in terms of relative water depth, d/L. This plot is

given for the case of larger pool length (Lp/LZ0.07–0.64), h/dZ1.0 and for three different

range of wave heights in terms of relative wave heights, Hi/d.

The force ratio has oscillating character when the d/L changes from 0.021–0.192.

Theoretical results of Yip et al. (2002) on the interaction of wave on vertical walls

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protected by a thin porous barrier also shows similar trend on wave reflection. Since the

wave reflection and wave forces are related, the present trend can say to be acceptable.

Also, it is to be noted that the response of the seawall depends entirely on the response of

the pool, which is bounded by two bodies (wall and breakwater). Hence the pool is

expected to resonate when situation arises.

In general, it is found that the force ratio is smaller for high waves compared to the

wave of smaller heights. This is due to the predominant wave breaking and the consequent

dissipation of waves. This sort of trend is good for the design of seawall/caisson, since the

design is governed by high wave actions. It is also reported that the wave damping effects

of breakwaters increases with increasing wave steepness (Johnson et al., 1951). This

phenomenon suggests the submerged breakwater behaves as a filter, attenuating steeper

waves with higher energies.

The peak value of force ratio is about 0.4, which occurs at d/LZ0.083. For high waves,

i.e., Hi/dZ0.45 this clearly proves that the force can be reduced to the order of 50% when

h/dZ1.0.

A cost estimate of the seawall without defence structure and that with defence structure

and its comparison is required for finalizing the selection of the option with defence

structure. Further investigation and analysis is required in this direction.

Fig. 6 shows the variation of seaward or negative force ½F x sea ratio for the same input

condition. The trend of variation of force ratio is similar to that of Fig. 5. The difference is

the maximum value of the force ratio, which is of the order of 0.8 for smaller Hi/d and is

about 0.4 for higher Hi/d.

Fig. 7 is similar plot as shown in Fig. 5, but for h/dZ0.66. Again the oscillating nature

of wave force with increased d/L persists. However, the major difference between the

Figs. 5 and 7 is the value of force ratio, which is about 0.5 for Hi/dZ0.45, whereas for

the same condition with h/dZ1.0, the force ratio is only about 0.4. That means the force

ratio has increased by an about 10% due to the submergence of the breakwater from h/dZ

1.0 to h/dZ0.66. Fig. 8 is a similar to Fig. 6, but for h/dZ0.66. Here again it is found that

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Fig. 5. Variation of shoreward force ratio with relative water depth d/L [h/dZ1.0, B/dZ1.33, Lp/LZ0.071–

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Fig. 6. Variation of seaward force ratio with relative water depth d/L [h/dZ1.0, Lp/LZ.071–0.641, B/dZ1.33].

the maximum force ratio for Hi/dZ0.45 is about 0.7, compared to 0.4 for the same Hi/d for

h/dZ1.0.

Once the value of h/d is selected, one can select the value of force ratio for a

given wave period and height. The actual force acting on the seawall can now be

estimated by multiplying the force ratio with the force on the seawall without defence

structure.

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3.4. Influence of pool length (Lp) on wave forces on the seawall defenced

by low-crested breakwater

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Some of the hydrodynamic phenomena found in the area between the breakwater

and seawall are wave height and period evolution, wave reshaping and possible

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Fig. 7. Variation of shoreward force ratio with relative water depth d/L [h/dZ0.66, Lp/LZ0.071–0.641, B/dZ

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Fig. 8. Variation of seaward force ratio with relative water depth d/L [h/dZ0.66, Lp/LZ0.071–0.641, B/dZ1.33].

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wave breaking, interference with waves reflected from seawall. Wave reflection and

re-reflection between the caisson and breakwater depends on distance between the

barrier and end wall is called pool length and this is required for effective location of

offshore breakwater.

A typical plot Fig. 9 shows the effect of relative pool length, Lp/L on wave forces.

In general, we have the wave force reduces to the extent of 10–30% when then

relative pool length is reduced from 0.16 to 0.08. It is to be recalled that in the real

field situation, it is always better to place the breakwater closer to the seawall, since

the water depth is expected to be small and hence the quantity of stones required for

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Fig. 9. Effect of pool length (Lp) on shoreward force with relative Breakwater height, h/d [B/dZ1.33, Hi/LZ0.015

and d/LZ0.048].

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the construction will be less, which will be economical. Fig. 9 is only typical plot for

one wave height and period. In order to zero down to the cumulative effect of pool

length, Fig. 10 is plotted by keeping the force ratio on the x-axis and cumulative

probability of force ratio on y-axis for two pool lengths. In this plot all the ranges of

wave heights, wave periods and relative breakwater heights of the present study are

considered. From this plot, it is found that 2% exceedence value of force ratio is 0.75

for Lp/LZ0.035–0.32 and 0.78 for Lp/LZ0.07–0.64.

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Fig. 10. Cumulative probability of shoreward force ratio ½F x shore for different relative breakwater heights (h/dZ

0.66–1.33, including all wave heights and wave periods which are used for the investigation, thick line for Lp/LZ

0.071–0.641 and dotted line for Lp/LZ0.035–0.32.

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3.5. Probability analysis on force ratio

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The probability of non-exceedence of force ratio ½F x shore for all wave heights

(Hi/dZ0.15–0.51) and wave periods T(d/LZ0.021–0.192) is given in Fig. 11. The

value corresponding to 98% non-exceedence (2% exceedence) can be taken for

the purpose of design of the seawall. It is seen that when the seawall is defenced by

the breakwater, 2% exceedence value of ½F x shore is 0.71, 0.81, 0.42, 0.38 and 0.29

when h/dZ0.66, 0.83, 1.0, 1.16 and 1.33, respectively, for Lp/LZ0.035–0.32. This

clearly brings out the relative benefit of increasing the height of breakwater for the

purpose of reduction of wave loads on the seawall. It is also clear that one has to

avoid h/d around 0.83, which induce plunging breaking over the breakwater and

causes more force than the case for h/dZ0.66.

Fig. 12 shows the shoreward force ratio for 2% exceedence against h/d for the two pool

lengths studied. This is simple and consolidated plot but can be used reliably by the coastal

community for the design of seawalls. A cost benefit analysis is required to select a

suitable h/d value. It is to be remembered that if h/d is increasing the force ratio on the

seawall will reduce, which will result in economic design of seawall but the cost of

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Fig. 11. Cumulative probability of shoreward force ratio ½F x shore for different relative heights (h/dZ0.66–1.33)

of breakwater [Lp/LZ0.035–0.32].

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defence structure will increase with increase in h/d. Further optimization study for a

typical site will be helpful for the user.

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3.6. Modification factor (Sn) for shoreward force

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Modification factor is proposed to estimate the shoreward force on the seawall defenced

by low-crested breakwater. After analyzing the influence of the non-dimensional

parameters on shoreward force a modification factor (Sn) is derived from a non-linear

optimization algorithm. Modification factor is the ratio of shoreward force on the wall

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Fig. 12. 2% non-exceedence of shoreward force ratio with relative breakwater height h/d for two pool lengths

[Lp/LZ0.035–0.32 and Lp/L 0.071–0.64].

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Fig. 13. Comparison of observed and predicted modification factor for different wave periods and wave heights.

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½Fx shore Z Sn ½Fx Goda

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½Sn shore Z 0:1ðh=dÞK0:41 ðHi =dÞK0:60 ðLp =LÞK0:40

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Measurements of wave elevations and forces on the structures reveal that the flow

behavior changes depending on the relative height of the breakwater for a given water

depth, resulting in five characteristic phases: freely transmitting wave, overtopping, crest

dissipation, predominant wave breaking and transmission over the breakwater as the

breakwater crest level reduces form emergent to submerged. Relative height of the

breakwater, h/d, associated with the formation of standing wave and resonant conditions

between the structures, is found to be important parameters for the oscillatory behavior of

the force ratios, which also depends on wave period.

Average shoreward force ratio ½F x shore is more for h/dZ0.83 when compared to

h/dZ0.66 for the range of pool length chosen in the experimental investigation. This was

not expected because as the relative breakwater height increases force ratio decreases, but

the reason for such peculiar behavior is found out based on the experimental observations.

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(10)

Fig. 13 shows the comparison of observed and measured modification factor. This

factor is more sensitive to relative breakwater height, h/d. In the above equation Lp/L is

influence of the wave period since Lp has taken as constant and, also decides the offshore

location of the low-crested breakwater. As shown in Fig. 4 force ratio increases at

h/dZ0.83, but in the Fig. 13 the modification factor is not depicting same trend because of

combined influence of the other non-dimensional parameters. Scatter in the points

includes the same inherent error in measurement of force as explained in the Fig. 2.

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(with low-crested breakwater) to force estimated from Goda formulae (Eq. (8)).

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Amplitudes of the force ratio decrease with increase in relative breakwater height and

relative water depth. Force ratios are small for steeper waves as the damping effect of

breakwater increases for steeper waves due to depth limited breaking over submerged

breakwater.

Influence of the pool length on reduction of force ratios is observed to be small (to the

order of 5–10%) for two different ranges studied. This needs investigation for more pool

lengths to substantiate further. Finally a modification factor is presented to estimate the

shoreward force on the vertical structure defenced by an offshore low-crested rubble

mound breakwater. The results of this study can be used for rehabilitating the partially

damaged seawalls and caissons or for the design of new seawall and caissons with offshore

breakwater as a defence structure. A cost benefit analysis by using the present results is

required to select the optimum h/d values.

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Ahrens, J.P., 1987. Characteristics of reef breakwaters. Technical Report CERC-87-17, Vicksburg.

Ahrens, J.P., 1989. Stabilty of reef breakwaters. J. WW, Port, Coastal Ocean Eng. 115 (2), 221–234.

Allsop, N.W.H., 1983. Low-crested breakwaters, studies in random waves, Proceedings of Coastal Structure

1983, Arlington, Virginia 1983 pp. 94–107.

CIRIA, 1986b. Sea walls: survey of performance and design practice. CIRIA (Construction Industry Research

and Information Association), London, Technical Note 125.

Dattatri, J., Raman, H., Shankar, J.N., 1978. Performance characteristics of submerged breakwaters. Proc. 16th

ICCE, ASCE 1978;, 2153–2171.

Dean, R.G., Renjie Chen, Browder, A.E., 1997. Full scale monitoring study of a submerged breakwater, Palm

Beach, Florida, USA. Coastal Eng. 129, 291–315.

Dick, T.M., Brebner, A., 1968. Solid and permeable submerged breakwaters. Proc. 11th Conf. Coastal Eng.,

ASCE 1968;, 1141–1158.

Diskin, M.H., 1970. Piling-up behind low and submerged permeable breakwaters. J. WW Harbour Division,

ASCE, WW2 96, 359–372.

Drei, E., Lamberti, A., 2000. Wave pumping effect of a submerged barrie, Coastal Structures 99, vol. 1 2000 pp.

667–673.

Goda, Y., 1985. Random Seas and Design of Maritime Structures. University of Tokyo Press, Tokyo, Japan.

Gonzleg Madrigal, B., Olivares Prud’homme, J., 1990. in: Edge, Billy L. (Ed.), Reduction of wave forces and

overtopping by submerged structures in front of a vertical breakwater Coastal Engineering Proceedings, vol. I

and II, pp. 1349–1361.

Johnson, J.W., Fuchs, R.A., Morison, J.R., 1951. The damping action of submerged breakwaters. Trans. Am.

Geoph. Union 32 (5), 704–717.

Losada, I.J., Patterson, M.D., Losada, M.A., 1997. Harmonic generation past a submerged porous step. Coastal

Eng. 31, 281–304.

Newman, J.N., 1965. Propagation of water waves past long two-dimensionalobstacles. J. Fluid Mech.,

Cambridge, UK 23, 23–29.

Oumeraci, H., 1994. Review and analysis of vertical breakwater failures—lessons learned. Coastal Eng. 22, 3–29.

Powell, K.A., Allsop, N.W.H., 1985. Low-crested breakwaters, hydraulic performance and stability. Report SR

57, HR Wallingford, England.

Seabrook, S.R., Hall, K.R., 1998. Effect of crest width and geometry on submerged breakwater performance. 26th

Int. Conf. Coastal Eng., Copenhagen, Denmark 1998;, 144–145.

Twu, S.W., Liu, C.C., Hsu, W.-H., 2000. Wave damping characteristics of deeply submerged breakwaters.

J. WW, Port, Coastal Ocean Eng. 127 (2), 97–105.

van der Meer, J.W., 1987. Stability of breakwater armour layers—design formulae. Coastal Eng. 11, 219–239.

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J. WW, Port, Coastal Ocean Eng. 120 (1), 1–19.

van der Meer, J.W., d’Angremond, K., 1991. Wave transmission at low-crested structures, Coastal structures and

breakwaters, ICE, Coastal structures and breakwaters, ICE 1991 pp. 25–41.

van der Meer, J.W., Pilarczyk, K.W., 1990. Stability of low-crested and reef breakwaters. Proc. 22th ICCE, ASCE

1990;, 1375–1388.

Yamashiro, M., Yoshida, A., Irie, I., 2000. Experimental study on wave field behind a submerged breakwater.,

Coastal Structures 1999. Balkema, Rotterdam pp. 675–682.

Yip, T.L., Sahoo, T., Chwang, A.T., 2002. Trapping of surface waves by porous and flexible structures. J. Wave

Motion 35, 41–54.

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https://www.researchgate.net/publication/237227831

Wavepressurereductionon

verticalseawalls/caissonsdueto

anoffshorebreakwater

ArticleinIndianJournalofGeo-MarineSciences·December2004

DOI:10.1115/OMAE2003-37074

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Ocean Engineering xx (xxxx) 1–18

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www.elsevier.com/locate/oceaneng

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M.G. Muni Reddya, S. Neelamanib,*

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a

Department of Ocean Engineering, Indian Institute of Technology Madras, Chennai-600 036, India

Coastal Engineering and Air Pollution Department, Environmental and Urban Development Division,

Kuwait Institute for Scientific Research, P.O. Box 24885, 13109 Safat, Kuwait

b

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Received 21 January 2004; accepted 9 July 2004

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This paper presents results obtained from a series of experiments conducted in wave flume to

assess the influence of the offshore low-crested breakwater as a defence structure in reducing the

wave forces on vertical seawall. The main aim of the tests was to know the effect of crest elevation of

the offshore low-crested breakwater as a rehabilitation structure for the existing damaged shore

protection structures. In this study five relative breakwater heights are used and associated flow

evolution was analyzed. With the sections proposed in this study, it is possible to achieve

considerable reduction of wave force on the seawall. Modification factor is proposed to estimate the

shoreward force on the seawall defenced by low-crested breakwater.

q 2005 Published by Elsevier Ltd.

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Abstract

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Keywords: Low-crested breakwater; Shoreward force; Overtopping; Submerged breakwaters; Seawall;

Modification factor

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Coastal erosion is one of the challenging coastal engineering problems faced by human

being around the world. This calls for the proper remedial measures to protect valuable

properties situated along the coast. Many seawalls and vertical caisson breakwaters

(CIRIA, 1986b; Oumeraci, 1994) around the world are being damaged. Such failures are

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1. Introduction

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Hydrodynamic studies on vertical seawall

defenced by low-crested breakwater

8

* Corresponding author. Tel.: C965 483 6100x5351; fax: C965 481 5192.

E-mail addresses: reddy_muni@hotmail.com (M.G. Muni Reddy), nsubram@kisr.edu.kw (S. Neelamani).

0029-8018/$ - see front matter q 2005 Published by Elsevier Ltd.

doi:10.1016/j.oceaneng.2004.07.008

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mainly caused by extreme wave actions, through displacement of the entire structure, or

progressive failure starting from locally weak point, or through overall foundation failures,

or through overtopping and toe erosion. It may be economical to allow the less frequent

storm wave to spill over the crest of the seawall rather than to its full height to reflect fully

all the waves. The disadvantage, however, is that overtopping waves plunge over the crest

and inundates lee side leading to high economical loss.

The need for force reduction on these structures to increase the life span has resulted in

different force reduction techniques like, introduction of porosity at the front face of the

caisson, slotted seawalls, construction of horizontally composite caissons and construction

of low-crested caissons etc. Introduction of porosity into the structure leads to reduction of

the strength of the structure. Construction of horizontally composite structure in dynamic

environment is risky. Low-crested breakwater attracts lesser forces but the overtopping of

waves create significant disturbance on the lee side. These drawbacks can be overcome by

constructing a low-crested breakwater in front of these structures to reduce the incident

wave energy levels. The offshore breakwater can be constructed after installation of

caisson without much risk for floating vessels and caisson. For existing weak or damaged

structures construction of a protection structure such as submerged offshore breakwater is

relatively an easy task.

Submerged breakwaters with deeper submergence would give larger wave energy

transmission, which might eventually lead to failure in sheltering function of the

breakwaters. Therefore how to reduce the incident wave energy levels becomes a great

challenge for coastal engineers. In the present study an offshore low-crested rubble mound

breakwater is considered as a defence structure to reduce the incident wave energy levels

that reach the vertical impervious structure viz., seawall/caisson. This type of protection

can also be used in situations wherein it is required to reduce the wave forces to enhance

the functional life of protection structures that are damaged by extreme wave forces, as a

rehabilitation structure. A theoretical analysis of the present problem is cumbersome. Due

to the complexity of the physical processes at the submerged breakwaters, physical

modeling is necessary to define the site-specific interactions between the structure and the

local wave climate. The defence structure may become submerged or emerged during the

tidal variation

Low-crested rock structures can be classified (van der Meer and Daemen, 1994) as

dynamically stable reef breakwaters, statically stable low-crested breakwaters and

statically stable submerged breakwaters. A reef breakwater is low-crested homogenous

pile of stones without a filter layer or core and is allowed to be reshaped by wave attack

(Ahrens, 1987).

Statically stable low-crested breakwaters are close to non-overtopping structures, but

are more stable due to the fact that large part of the wave energy can pass over the

breakwater (Powell and Allsop, 1985). All waves overtop statically stable submerged

breakwaters and the stability increases remarkably if the crest height decreases.

Submerged breakwaters have been widely used as wave energy dissipaters. Efficiency

of the submerged breakwaters depends on the crest free board, crest width and permeable

material characteristics. Many investigators like Newman (1965); Dick and Brebner

(1968); Dattatri et al. (1978); Losada et al. (1997), have studied the wave transmission

and reflection characteristics. The stability and wave transmission characteristics of

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the low-crested rubble mound breakwaters were investigated by Allsop (1983); Ahrens

(1989); van der Meer and Pilarczyk (1990); van der Meer and d’Angremond (1991);

Seabrook and Hall (1998), and Yamashiro et al. (2000) and, the design formulae were

developed by van der Meer and Daemen (1994) by analyzing the data sets of various

investigations. Behavior of the deeply submerged breakwaters with multi vertical sliced

permeable structure was investigated by Twu et al. (2000).

Based on the monitoring results of a submerged breakwater and resulted model

studies, Dean et al. (1997) have reported that detached breakwater modifies both the

wave and current fields depending substantially on the crest elevation relative to the

still water level. However, not much study on the present topic except the work by

Gonzleg Madrigal and Olivares Prud’homme (1990) on the reduction of forces on

vertical breakwater defenced by seaward submerged breakwater. For partial barrier of

any configuration, irrespective of the porosity and flexibility, full reflection always

occurs when the distance between the end-wall and the barrier is an integer multiple of

half-wave length and hence overturning and moment will vanish (Yip et al., 2002).

Many investigators have studied analytically and numerically the wave transmission

and reflection characteristics of the submerged breakwaters. Yet these mathematical

models cannot reproduce some of the features observed such as strong mean water

level gradients on the submerged breakwater, pumping effect of the submerged

breakwater and vertical circulation induced by breaking waves on the submerged

breakwater.

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2. Experimental procedure and investigation

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Experiments have been carried out in a 30 m length 2 m wide and 1.7 m deep

wave flume at Indian Institute of Technology Madras, Chennai, India. Seawall was

fixed (Fig. 1) over a six-component force balance (GmbH R67). Top level of the

force balance is flushed with the flume bed. The sensitivity of the transducers (strain

gauge type) of six-component force balance at rated loading is about G2 mV/V.

Force balance consists of a stainless steel platform 850!850 mm size, below which

force transducers were fixed to a rigid frame of 900!900 mm2. This frame was

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Fig. 1. Experimental set-up for the present study.

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tightly fixed to the flume sidewalls to arrest the movement of force balance. Seawall

model was mounted on top of the steel platform so that the force on the seawall will

be transferred to the transducers. The height of the seawall model was fixed based on

the theoretically estimated maximum run-up over the seawall, to ensure no

overtopping of waves. Crest width of the offshore low-crested breakwater was chosen

as 0.40 m. The stable weight of the armour unit of the breakwater was estimated by

using the van der Meer (1987) formulae for statically stable low-crested and

submerged breakwater. Here our aim was not the damage of the low-crested

breakwater, so a stable armour weight was used. Breakwater was constructed with two

layers, an armour layer and core. Weight of the armour stone was 14.70–19.62 kN

[This was arrived at for the inputs significant wave height ‘Hs’Z0.29 m, zero crossing

period ‘Tz’Z3.0 s, damage level ‘S’Z2, number of waves ‘N’Z3000 and gsZ

26.5 kN/m3 for plunging breaking]. The weight of the core stone was 1.96–2.45 kN

(gsZ29.5 kN/m3). Five crest level configurations (two emerged, two submerged and

one at still water level) were used in this study. A stable slope of 2H:1V was adopted

as the effects of breakwater slope on the wave transformation were found to be

relatively unimportant (Seabrook and Hall, 1998). Ratio of the breakwater height to

water depth h/d is varied from 0.66 to 1.33, keeping the water depth ‘d’ constant at

0.30 m and varying the height of the breakwater, ‘h’ from 0.20 to 0.40 m with 0.05 m

increment. This simulates the investigation on site where the tidal fluctuations are

insignificant. Two pool lengths, Lp (Lp is the distance between the toe of the lowcrested breakwater and seawall, Fig. 1) 0.50 and 1.0 m were used.

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2.1. Data collection and analysis procedure

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The wave synthesizer (WS4) involving an application software package, along with

analogue-digital and I/O modules installed in personal computer was employed in the

measurement and analysis. The software is capable of controlling the wave paddle and at

the same time acquires data from sensors used in the tests. The force balance transducers

are connected to the data acquisition system through carrier frequency amplifiers. Each set

of data for regular wave was sampled at frequency of 40 Hz. The filtered signals are

analyzed using the wave synthesizer. It contains the options for synthesis of regular and

random 2D waves. Regular waves of different predetermined wave period and wave

amplitude combinations are generated for the testes. The horizontal force (force in the

direction of wave propagation), vertical force on the seawall, run-up on the wall and wave

elevations in front of the model were acquired.

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2.1.1. Range of inputs

Relative wave height, Hi/d

Relative depth, d/L

Wave steepness, Hi/L

Relative breakwater height, h/d

Non-dimensional pool length, Lp/L

Relative breakwater width, B/d

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0.15–0.51

0.025–0.192

0.003–0.058

0.66–1.33

0.035–0.641

1.33

Here L is the deep-water wavelength and Hi is the incident wave height.

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2.1.2. Data analysis

The data collected were converted to physical variables by using the corresponding

calibration constants/coefficients. The raw data (in the form of time series) were analyzed

in time domain to get the clear understanding of the phenomenon under investigation. The

measured wave height, wave periods and forces were obtained by analyzing the measured

time histories of wave surface elevations and force amplitudes using the thresholdcrossing analysis. The threshold-crossing option is a generalization of classical zerocrossing analysis. For a pre-defined reference level, the input time series is divided into

events. For each event, the peak–peak value, the minimum and maximum values, and the

duration are determined.

The time series of the different parameters stated earlier were viewed to pickup the part

of time series with regular trend by omitting the transient part. This also ensures that no rereflected waves were present in the selected window of the time series. The regular time

series of force was then subjected to threshold-crossing analysis to get the mean amplitude

of the time history. The mean of the all amplitudes above the reference level in a time

series is taken as a positive or shoreward force. Similarly mean of all the amplitudes below

the reference level on a time series is taken as negative or seaward force. The mean

amplitudes of measured hydrodynamic force were obtained using the above procedure for

each test run.

½F x shore is the ratio of shoreward force in the direction of wave propagation in the

absence of the low-crested breakwater to shoreward force in the direction of wave

propagation in the presence of low-crested breakwater. ½F x sea is the ratio of seaward force

in the direction opposite to wave propagation in the absence of the breakwater to the

seaward force in the direction opposite to wave propagation in the presence of the

breakwater. These forces are obtained using procedure for the respective case of with and

without low-crested breakwater.

Incident wave elevations are measured using DHI capacitance wave gauges in the

absence of model in the flume, for pre-determined sets of different wave period and wave

height combinations. This procedure is repeated thrice and the average value is taken for

the wave height for that particular combination. It is done with a view to check the

repeatability of wave heights at the same point later when tests are conducted with the

model in position.

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3. Results and discussion

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3.1. General

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The non-breaking wave forces on seawalls are pulsating. A substantial portion of the

horizontal momentum of the wave is imparted to the wall. Methods to calculate the wave

forces for simple vertical structures and pulsating wave conditions are relatively well

established and are described by Goda (1985).

According to Goda (1985)

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Fh Z 0:5ðp1 C p2 Þh C 0:5ðp1 C p4 ÞhÃc

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p1 Z 0:5ð1 C cos bÞða1 C a2 cos2 bÞrw gHmax

(2)

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p2 Z p1 =½coshð2ph=LÞ

(3)

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p 3 Z a 3 p1

(4)

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p1mod Z 0:5ð1 C cos bÞa1 rw gH

Fhmod Z 0:5ðp1mod C p2mod Þh 0 C 0:5ðp1mod C p4mod ÞhÃc

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where Fh, total horizontal force per meter length of the wall/caisson; hb, water depth at a

location at a distance of 5H1/3 seaward of the breakwater; a2, pressure coefficient (varies

from 0 to 1.0); b, angle between the direction of wave approach and a line normal to the

breakwater; d, depth above the armour layer of the rubble foundation; hc, crest elevation of

the breakwater above the bottom of the upright section; h*, elevation at which the wave

pressure exerted; hÃc , min{h*, hc}; Hmax, maximum or design wave height.

p1, p2, p3 and p4 are the representative wave pressure intensities. Pressure coefficient a2

represents the tendency of the pressure to increase with the height of the rubble mound

foundation. The coefficient a2 (Eq. (6)) becomes zero, as hb and water depth d are the same

in the present study. Hence the Eqs. (1) and (2) can be written as

(7)

(8)

p1mod is less than p1 in Eq. (2) because of additive term a2 cos2 b vanishes. From Eqs. (3)

and (4) it can be observed that the magnitude of p2 and p3 reduces hence horizontal force

Fh in Eq. (8).

The measured shoreward forces (without breakwater) are compared (Fig. 2) with

Eq. (8) for validation of the present shoreward force measurements. The measured

forces are more than the estimated forces. Increase of wave pressure/force due to the

presence of a rubble foundation may regarded as the result of the change in the

behavior of wave from non-breaking to breaking although actual waves never exhibit

such marked changes.

Most design methods for caisson and the other vertical wall concentrate on forces that

act landward, usually termed as positive forces. It has however, been shown that some

breakwaters/walls failed by sliding or rotation seaward indicating that net seaward forces

may indeed be greater than positive forces.

The time series of incident wave height and wave force on the wall for different

relative breakwater height ‘h/d’ ratios are shown in Fig. 3. Quantitative reduction in

force on the seawall with increased h/d is very clear. The time series of wave forces on

the seawall defenced by an low-crested breakwater show that the wave breaking on the

breakwater generates high frequency waves on the lee side of breakwater, which results

in irregular force time series consisting of superposition of fundamental wave

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(6)

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a2 Z min ½ðhb K dÞ=3hb ðHmax =dÞ2 ; 2d=Hmax

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: h Ã % hc

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p1 ð1 K hc =hÃ Þ : hÃ O hc

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p4 Z

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Fig. 2. Comparison of non-dimensional shoreward force on vertical seawall with Goda’s (1974) formulae [dZ

0.30 m, Hi/dZ0.29K0.48].

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frequencies and the higher wave frequencies. It would be worth mentioning at this

point, the effect of wave set-up or pumping effect (Drei and Lamberti, 2000) or the

piling-up (Diskin, 1970) of water behind the protected area creates a difference in mean

water level inside the protected area and that of open sea. This component is inherent

in the time series shown in the Fig. 3. It is difficult to quantify this component in the

force measurement, because the force balance measures total effect. For laboratory

measurements this effect is unavoidable due to the fact that the water will confine

between the sidewalls of the flume and between two structures and there will be very

little scope for water to escape. In the field situations, in open sea this effect will not be

of much significant as there will be sufficient space for water to escape laterally

between the two structures. It should be noted that experiments were conducted in the

two-dimensional flume, and thus the values of mean water levels may be overestimated

in comparison with the values of mean water levels in three-dimensional wave field.

About 14% deviation observed from the forces estimated by Eq. (8) and the forces

measured from the experiments.

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Fig. 4 provides the effect of h/d on fore ratio ½F x shore for different incident wave

steepness. Force ratio 1.0 means that the breakwater has no effect on the reduction of

forces on the caisson and zero means 100% protection of the caisson by low-crested

breakwater. The value of force ratio lies in-between zero and 1.0. Oscillatory nature of

force ratio ½F x shore is observed when the h/d is varied from 0.66 to 1.33. The amplitude

of the oscillation decreases with increase of h/d. The high value of force ratio for

h/dZ0.83 is due to wave jetting on the seawall after overtopping over the low-crested

breakwater. This increased force is unwarranted for the general presumption that as the

barrier height increases force will have to decrease correspondingly. Designers and

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3.2. Effect of relative height of the breakwater, h/d on the normalized wave

forces on the seawall

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Fig. 3. Typical force time series for different relative breakwater height h/d [HiZ0.152 m, dZ0.3, d/LZ0.059,

B/dZ1.33, Lp/LZ0.071–0.64].

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coastal engineers should take care of this while decision making in choosing the range

h/d values. For h/dO1.0, the wave energy is effectively dissipated which result in

significant wave force reduction on the seawall. When h/dZ1.0, the reduction in

average shoreward or positive force is 66% (standard deviation is 0.097) as Hi/L is

varied from 0.003 to 0.058 for the range of Lp/LZ0.035–0.321. Percentage decrease in

the magnitude of peaks of force ratio is ½F x shore found to increase with h/d. The

following wave-structure interaction processes were identified during the experimental

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Fig. 4. Variation of shoreward force ratio with relative reakwater height h/d for three different wave steepness

[Lp/LZ0.198, B/dZ1.33, d/LZ0.059].

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investigations, which are explained below for the type of normalized wave force trend

observed:

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(a) For offshore breakwater with more submergence (say h/dZ0.66), the wave transmit

freely, reflects from the seawall. These reflected waves contribute significantly for

the amplification of waves and the corresponding wave forces on the seawall/

caisson.

(b) For offshore breakwater with smaller submergence (say h/dZ0.83), the propagating

wave on the breakwater attains the characteristics of wave breaking and the

overtopping jet of mass acts on the seawall/caisson resting behind the breakwater

and imparts higher order of forces.

(c) For the case of offshore breakwater with crest level flushing with still water level

(h/dZ1.0), most of the interacting energy is expected to be dissipated on the crest

of the breakwater and hence the wave force reduction is significant.

(d) For the offshore breakwater with less emergence i.e. crest located just above the

still water level (here h/dZ1.16), the dominant mode of wave transmission is by

run-up and overtopping and the efficiency of transmission process increase as wave

height increases. The energy available with this overtopping water mass imparts

forces on the seawall. The wave energy dissipation due to the interaction with the

breakwater reduces to the significant overtopping processes.

(e) For the offshore breakwater with significant emergence of the crest (h/dZ1.33),

overtopping will be prevented for most of the waves and the waves may be allowed

to transmit through the pores of the breakwater. The energy available with this

transmitted wave imparts forces on the rear side structures.

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A through analysis of Fig. 4 with this understanding gives a clear answer why a force

ratio variation is oscillatory with increased h/d. It was observed that the force ratio at any

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Table 1

Measured mean force ratios for two pool lengths

h/d

Lp/LZ0.

035–0.

32, Hi/

LZ0.

003–0.09

½F x shore

Standard

deviation

½F x sea

Standard

deviation

Lp/LZ0.

071–0.

641, Hi/

LZ0.

003–0.09

½F x shore

Standard

deviation

½F x sea

Standard

deviation

1.33

1.16

1.00

0.83

0.66

0.17

0.31

0.33

0.63

0.55

0.066

0.069

0.097

0.113

0.108

0.25

0.35

0.39

0.62

0.56

0.106

0.106

0.110

0.180

0.110

0.20

0.34

0.31

0.52

0.48

0.08

0.12

0.09

0.16

0.12

0.29

0.46

0.43

0.64

0.60

0.15

0.18

0.15

0.13

0.13

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h/d is the minimum when the waves acting on the system are steeper. This point must be

given due attention, since the design is carried out for steeper waves. The results plotted in

Fig. 4 are typical for a d/L value of 0.058. The results for the other values were also

observed to follow the same trend.

In order to bring out the cumulative effect of range of heights and periods used in the

study, a table containing the force ratio information is prepared. Table 1 is provided to

visualize the force ratio for two pool lengths and for a wide range of wave steepness. Both

shoreward and seaward force ratios are provided in the table. The mean and the standard

deviation of the wave force ratios are tabulated. Mean of the shoreward force amplitude

and seaward force amplitude and the corresponding standard deviations are provided in

this table. For example, the shoreward force ratio value for h/dZ1.33 is given as 0.17,

which is the average value for a number of wave heights and periods. The standard

deviation for this set is 0.066 (i.e., the ratio of standard deviation and the average is about

38.8%). This table is mainly provided for an overall understanding of the effect of pool

length and relative height of the breakwater on wave force reduction on the seawall. From

this table it is clear that the shoreward force ratio for the case of smaller pool length (Lp/

LZ0.035–0.32), is from 0.17 to 0.55 when h/d is varied from 1.33 to 0.66. This means that

the mean wave force reduction of the order of 83–45% is possible for this case. For the

large pool length ratio (Lp/LZ0.07–0.64), the shoreward force ratio ranges from 0.2 to

0.48 for the same range of h/d. The seaward force ratio is ranging from 0.29 to 0.60, which

is significantly higher than the shoreward force ratio. The average value of force ratio

along with the standard deviation can be used for the selection for appropriate vale of h/d.

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3.3. Effect of wave period and wave height on wave force ratio

Fig. 5 shows the effect of variation of wave period on shoreward force ratio. The

variation of wave period is presented in terms of relative water depth, d/L. This plot is

given for the case of larger pool length (Lp/LZ0.07–0.64), h/dZ1.0 and for three different

range of wave heights in terms of relative wave heights, Hi/d.

The force ratio has oscillating character when the d/L changes from 0.021–0.192.

Theoretical results of Yip et al. (2002) on the interaction of wave on vertical walls

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protected by a thin porous barrier also shows similar trend on wave reflection. Since the

wave reflection and wave forces are related, the present trend can say to be acceptable.

Also, it is to be noted that the response of the seawall depends entirely on the response of

the pool, which is bounded by two bodies (wall and breakwater). Hence the pool is

expected to resonate when situation arises.

In general, it is found that the force ratio is smaller for high waves compared to the

wave of smaller heights. This is due to the predominant wave breaking and the consequent

dissipation of waves. This sort of trend is good for the design of seawall/caisson, since the

design is governed by high wave actions. It is also reported that the wave damping effects

of breakwaters increases with increasing wave steepness (Johnson et al., 1951). This

phenomenon suggests the submerged breakwater behaves as a filter, attenuating steeper

waves with higher energies.

The peak value of force ratio is about 0.4, which occurs at d/LZ0.083. For high waves,

i.e., Hi/dZ0.45 this clearly proves that the force can be reduced to the order of 50% when

h/dZ1.0.

A cost estimate of the seawall without defence structure and that with defence structure

and its comparison is required for finalizing the selection of the option with defence

structure. Further investigation and analysis is required in this direction.

Fig. 6 shows the variation of seaward or negative force ½F x sea ratio for the same input

condition. The trend of variation of force ratio is similar to that of Fig. 5. The difference is

the maximum value of the force ratio, which is of the order of 0.8 for smaller Hi/d and is

about 0.4 for higher Hi/d.

Fig. 7 is similar plot as shown in Fig. 5, but for h/dZ0.66. Again the oscillating nature

of wave force with increased d/L persists. However, the major difference between the

Figs. 5 and 7 is the value of force ratio, which is about 0.5 for Hi/dZ0.45, whereas for

the same condition with h/dZ1.0, the force ratio is only about 0.4. That means the force

ratio has increased by an about 10% due to the submergence of the breakwater from h/dZ

1.0 to h/dZ0.66. Fig. 8 is a similar to Fig. 6, but for h/dZ0.66. Here again it is found that

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Fig. 5. Variation of shoreward force ratio with relative water depth d/L [h/dZ1.0, B/dZ1.33, Lp/LZ0.071–

0.641].

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Fig. 6. Variation of seaward force ratio with relative water depth d/L [h/dZ1.0, Lp/LZ.071–0.641, B/dZ1.33].

the maximum force ratio for Hi/dZ0.45 is about 0.7, compared to 0.4 for the same Hi/d for

h/dZ1.0.

Once the value of h/d is selected, one can select the value of force ratio for a

given wave period and height. The actual force acting on the seawall can now be

estimated by multiplying the force ratio with the force on the seawall without defence

structure.

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3.4. Influence of pool length (Lp) on wave forces on the seawall defenced

by low-crested breakwater

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Some of the hydrodynamic phenomena found in the area between the breakwater

and seawall are wave height and period evolution, wave reshaping and possible

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Fig. 7. Variation of shoreward force ratio with relative water depth d/L [h/dZ0.66, Lp/LZ0.071–0.641, B/dZ

1.33].

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Fig. 8. Variation of seaward force ratio with relative water depth d/L [h/dZ0.66, Lp/LZ0.071–0.641, B/dZ1.33].

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wave breaking, interference with waves reflected from seawall. Wave reflection and

re-reflection between the caisson and breakwater depends on distance between the

barrier and end wall is called pool length and this is required for effective location of

offshore breakwater.

A typical plot Fig. 9 shows the effect of relative pool length, Lp/L on wave forces.

In general, we have the wave force reduces to the extent of 10–30% when then

relative pool length is reduced from 0.16 to 0.08. It is to be recalled that in the real

field situation, it is always better to place the breakwater closer to the seawall, since

the water depth is expected to be small and hence the quantity of stones required for

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Fig. 9. Effect of pool length (Lp) on shoreward force with relative Breakwater height, h/d [B/dZ1.33, Hi/LZ0.015

and d/LZ0.048].

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the construction will be less, which will be economical. Fig. 9 is only typical plot for

one wave height and period. In order to zero down to the cumulative effect of pool

length, Fig. 10 is plotted by keeping the force ratio on the x-axis and cumulative

probability of force ratio on y-axis for two pool lengths. In this plot all the ranges of

wave heights, wave periods and relative breakwater heights of the present study are

considered. From this plot, it is found that 2% exceedence value of force ratio is 0.75

for Lp/LZ0.035–0.32 and 0.78 for Lp/LZ0.07–0.64.

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Fig. 10. Cumulative probability of shoreward force ratio ½F x shore for different relative breakwater heights (h/dZ

0.66–1.33, including all wave heights and wave periods which are used for the investigation, thick line for Lp/LZ

0.071–0.641 and dotted line for Lp/LZ0.035–0.32.

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3.5. Probability analysis on force ratio

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The probability of non-exceedence of force ratio ½F x shore for all wave heights

(Hi/dZ0.15–0.51) and wave periods T(d/LZ0.021–0.192) is given in Fig. 11. The

value corresponding to 98% non-exceedence (2% exceedence) can be taken for

the purpose of design of the seawall. It is seen that when the seawall is defenced by

the breakwater, 2% exceedence value of ½F x shore is 0.71, 0.81, 0.42, 0.38 and 0.29

when h/dZ0.66, 0.83, 1.0, 1.16 and 1.33, respectively, for Lp/LZ0.035–0.32. This

clearly brings out the relative benefit of increasing the height of breakwater for the

purpose of reduction of wave loads on the seawall. It is also clear that one has to

avoid h/d around 0.83, which induce plunging breaking over the breakwater and

causes more force than the case for h/dZ0.66.

Fig. 12 shows the shoreward force ratio for 2% exceedence against h/d for the two pool

lengths studied. This is simple and consolidated plot but can be used reliably by the coastal

community for the design of seawalls. A cost benefit analysis is required to select a

suitable h/d value. It is to be remembered that if h/d is increasing the force ratio on the

seawall will reduce, which will result in economic design of seawall but the cost of

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Fig. 11. Cumulative probability of shoreward force ratio ½F x shore for different relative heights (h/dZ0.66–1.33)

of breakwater [Lp/LZ0.035–0.32].

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defence structure will increase with increase in h/d. Further optimization study for a

typical site will be helpful for the user.

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3.6. Modification factor (Sn) for shoreward force

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Modification factor is proposed to estimate the shoreward force on the seawall defenced

by low-crested breakwater. After analyzing the influence of the non-dimensional

parameters on shoreward force a modification factor (Sn) is derived from a non-linear

optimization algorithm. Modification factor is the ratio of shoreward force on the wall

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Fig. 12. 2% non-exceedence of shoreward force ratio with relative breakwater height h/d for two pool lengths

[Lp/LZ0.035–0.32 and Lp/L 0.071–0.64].

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Fig. 13. Comparison of observed and predicted modification factor for different wave periods and wave heights.

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½Fx shore Z Sn ½Fx Goda

(9)

½Sn shore Z 0:1ðh=dÞK0:41 ðHi =dÞK0:60 ðLp =LÞK0:40

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Measurements of wave elevations and forces on the structures reveal that the flow

behavior changes depending on the relative height of the breakwater for a given water

depth, resulting in five characteristic phases: freely transmitting wave, overtopping, crest

dissipation, predominant wave breaking and transmission over the breakwater as the

breakwater crest level reduces form emergent to submerged. Relative height of the

breakwater, h/d, associated with the formation of standing wave and resonant conditions

between the structures, is found to be important parameters for the oscillatory behavior of

the force ratios, which also depends on wave period.

Average shoreward force ratio ½F x shore is more for h/dZ0.83 when compared to

h/dZ0.66 for the range of pool length chosen in the experimental investigation. This was

not expected because as the relative breakwater height increases force ratio decreases, but

the reason for such peculiar behavior is found out based on the experimental observations.

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(10)

Fig. 13 shows the comparison of observed and measured modification factor. This

factor is more sensitive to relative breakwater height, h/d. In the above equation Lp/L is

influence of the wave period since Lp has taken as constant and, also decides the offshore

location of the low-crested breakwater. As shown in Fig. 4 force ratio increases at

h/dZ0.83, but in the Fig. 13 the modification factor is not depicting same trend because of

combined influence of the other non-dimensional parameters. Scatter in the points

includes the same inherent error in measurement of force as explained in the Fig. 2.

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(with low-crested breakwater) to force estimated from Goda formulae (Eq. (8)).

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Amplitudes of the force ratio decrease with increase in relative breakwater height and

relative water depth. Force ratios are small for steeper waves as the damping effect of

breakwater increases for steeper waves due to depth limited breaking over submerged

breakwater.

Influence of the pool length on reduction of force ratios is observed to be small (to the

order of 5–10%) for two different ranges studied. This needs investigation for more pool

lengths to substantiate further. Finally a modification factor is presented to estimate the

shoreward force on the vertical structure defenced by an offshore low-crested rubble

mound breakwater. The results of this study can be used for rehabilitating the partially

damaged seawalls and caissons or for the design of new seawall and caissons with offshore

breakwater as a defence structure. A cost benefit analysis by using the present results is

required to select the optimum h/d values.

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Ahrens, J.P., 1987. Characteristics of reef breakwaters. Technical Report CERC-87-17, Vicksburg.

Ahrens, J.P., 1989. Stabilty of reef breakwaters. J. WW, Port, Coastal Ocean Eng. 115 (2), 221–234.

Allsop, N.W.H., 1983. Low-crested breakwaters, studies in random waves, Proceedings of Coastal Structure

1983, Arlington, Virginia 1983 pp. 94–107.

CIRIA, 1986b. Sea walls: survey of performance and design practice. CIRIA (Construction Industry Research

and Information Association), London, Technical Note 125.

Dattatri, J., Raman, H., Shankar, J.N., 1978. Performance characteristics of submerged breakwaters. Proc. 16th

ICCE, ASCE 1978;, 2153–2171.

Dean, R.G., Renjie Chen, Browder, A.E., 1997. Full scale monitoring study of a submerged breakwater, Palm

Beach, Florida, USA. Coastal Eng. 129, 291–315.

Dick, T.M., Brebner, A., 1968. Solid and permeable submerged breakwaters. Proc. 11th Conf. Coastal Eng.,

ASCE 1968;, 1141–1158.

Diskin, M.H., 1970. Piling-up behind low and submerged permeable breakwaters. J. WW Harbour Division,

ASCE, WW2 96, 359–372.

Drei, E., Lamberti, A., 2000. Wave pumping effect of a submerged barrie, Coastal Structures 99, vol. 1 2000 pp.

667–673.

Goda, Y., 1985. Random Seas and Design of Maritime Structures. University of Tokyo Press, Tokyo, Japan.

Gonzleg Madrigal, B., Olivares Prud’homme, J., 1990. in: Edge, Billy L. (Ed.), Reduction of wave forces and

overtopping by submerged structures in front of a vertical breakwater Coastal Engineering Proceedings, vol. I

and II, pp. 1349–1361.

Johnson, J.W., Fuchs, R.A., Morison, J.R., 1951. The damping action of submerged breakwaters. Trans. Am.

Geoph. Union 32 (5), 704–717.

Losada, I.J., Patterson, M.D., Losada, M.A., 1997. Harmonic generation past a submerged porous step. Coastal

Eng. 31, 281–304.

Newman, J.N., 1965. Propagation of water waves past long two-dimensionalobstacles. J. Fluid Mech.,

Cambridge, UK 23, 23–29.

Oumeraci, H., 1994. Review and analysis of vertical breakwater failures—lessons learned. Coastal Eng. 22, 3–29.

Powell, K.A., Allsop, N.W.H., 1985. Low-crested breakwaters, hydraulic performance and stability. Report SR

57, HR Wallingford, England.

Seabrook, S.R., Hall, K.R., 1998. Effect of crest width and geometry on submerged breakwater performance. 26th

Int. Conf. Coastal Eng., Copenhagen, Denmark 1998;, 144–145.

Twu, S.W., Liu, C.C., Hsu, W.-H., 2000. Wave damping characteristics of deeply submerged breakwaters.

J. WW, Port, Coastal Ocean Eng. 127 (2), 97–105.

van der Meer, J.W., 1987. Stability of breakwater armour layers—design formulae. Coastal Eng. 11, 219–239.

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van der Meer, J.W., Daemen, I.F.R., 1994. Stability and transmission at low-crested rubble mound structures.

J. WW, Port, Coastal Ocean Eng. 120 (1), 1–19.

van der Meer, J.W., d’Angremond, K., 1991. Wave transmission at low-crested structures, Coastal structures and

breakwaters, ICE, Coastal structures and breakwaters, ICE 1991 pp. 25–41.

van der Meer, J.W., Pilarczyk, K.W., 1990. Stability of low-crested and reef breakwaters. Proc. 22th ICCE, ASCE

1990;, 1375–1388.

Yamashiro, M., Yoshida, A., Irie, I., 2000. Experimental study on wave field behind a submerged breakwater.,

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Motion 35, 41–54.

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