Hydrodynamic studies on vertical seawall defenced by lowcrested breakwater

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

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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: (M.G. Muni Reddy), (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 Šshore 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 Šsea 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=Lފ

(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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È
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a2 Z min ½ðhb K dÞ=3hb ŠðHmax =dÞ2 ; 2d=Hmax

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: h à % hc

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p1 ð1 K hc =hÃ Þ : hà 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 Šshore 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 Šshore 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 Šshore 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 Šshore

Standard
deviation

½F x Šsea

Standard
deviation

Lp/LZ0.
071–0.
641, Hi/
LZ0.
003–0.09
½F x Šshore

Standard
deviation

½F x Šsea

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

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0.56

0.106
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0.110
0.180
0.110

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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 Šsea 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 Šshore 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 Šshore 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 Šshore 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 Šshore 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 Šshore Z Sn ½Fx ŠGoda

(9)

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

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4. Concluding remarks

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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 Šshore 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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References

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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
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Motion 35, 41–54.

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