Reflection and Transmission of Waves over Submerged Breakwaters

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REFLECTION AND TRANSMISSION OF WAVES OVER

SUBMERGED BREAKWATERS

By Dimitrios G. Stamos1 and Muhammad R. Hajj,2 Member, ASCE

ABSTRACT: This work has a twofold objective. First, a new method to separate incident and reflected wavecomponents, using measurements from one wave probe, is presented. This technique is based on local variationsin amplitude and phase of the measured wave and uses wavelet analysis. Second, the method is applied toperform a parametric study to compare the reflection and transmission characteristics of flexible and rigidbreakwaters. Results are discussed for different depths of submergence of the models, different internal pressuresin the case of the flexible breakwater, and a wide range of wave steepnesses. The results show that, in general,the rigid model has a higher reflection coefficient than the flexible model. On the other hand, the flexible modelhas a much higher energy-loss coefficient. Optimal breakwater widths for reflection, transmission, and energy-loss coefficients for waves with different wavelengths are also presented.

INTRODUCTION

Detached, segmented, reef, floating, and submerged break-waters are variations of coastal structures built to protect theshoreline. The primary function of such structures is to inter-cept the incident waves and cause them to break or reflect. Inspite of their usefulness, the concept of having permanentcoastal structures for wave attenuation is a subject of contro-versy due to environmental and aesthetic reasons. Some of theproblems associated with permanent breakwaters include re-tardation in beach buildup during periods of calm waves,breakwater subsidence when the seabed is composed of softmaterial, and formation of tombolos. To overcome these andother problems, fluid-filled flexible structures have been pro-posed as breakwaters. These structures are usually submergedand can be either suspended under the water surface or an-chored on the seabed. Their main advantage over permanentbreakwaters is that they can be deployed in different config-urations for optimal wave reflection with minimal adverse ef-fects. Another advantage is that they can be used as temporarysacrificial breakwaters to reduce the size of storm waves im-pacting harbors, coastal areas, or fixed breakwaters. In addi-tion, they have a lower construction cost compared to the con-ventional massive structures. Because their orientations can bevaried, they can be used more efficiently to control beach ero-sion. Moreover, they can be deflated in calm seas, which re-sults in a less-adverse impact.

Extensive numerical, analytical, and experimental investi-gations have been conducted to examine the performances ofdifferent flexible membranes as breakwaters. These studiesdealt with several issues related to performance, includingtransmission and reflection characteristics, structures’ re-sponses to the wave forcing, and effects of parameters suchas internal pressure and membrane elasticity on different as-pects of these structures. Frederiksen (1971) conducted exper-iments on the effectiveness of floating and bottom-mountedair-filled membranes as submerged breakwaters. Ohyama et al.(1989) conducted a numerical study, based on linear potentialtheory, to investigate the transmission and reflection charac-

1Res. Asst., Dept. of Engrg. Sci. and Mech., Virginia Polytechnic Inst.and State Univ., Blacksburg, VA, 24061. E-mail: [email protected]

2Assoc. Prof., Dept. of Engrg. Sci. and Mech., Virginia PolytechnicInst. and State Univ., Blacksburg, VA. E-mail: [email protected]

Note. Associate Editor: Alexander Cheng. Discussion open until July1, 2001. To extend the closing date one month, a written request mustbe filed with the ASCE Manager of Journals. The manuscript for thispaper was submitted for review and possible publication on February 15,2000. This paper is part of the Journal of Engineering Mechanics, Vol.127, No. 2, February, 2001. qASCE, ISSN 0733-9399/01/0002-0099–0105/$8.00 1 $.50 per page. Paper No. 22232.

J. Eng. Mech. 20

teristics of a bottom-mounted flexible structure that has theshape of an arc. They complemented the study with modelexperiments. They also presented results on the effects of sev-eral parameters such as the internal added pressure in themembrane, elasticity of the membrane, and submergence ratioon the wave reflection and transmission. Phadke and Cheung(1999) developed a 2D numerical model to investigate hydro-dynamic characteristics of bottom-mounted fluid-filled mem-branes and computed transmission coefficients.

In laboratory studies on wave reflection from breakwaters,incident and reflected wave parameters are usually obtainedfrom experiments in wave tanks. The situation is, however,complicated because of end reflections and their effects on thepropagating wave. Several methods have been proposed to ob-tain the reflection coefficient of regular waves over breakwa-ters. One method was proposed by Dean and Dalrymple(1991) and involves traversing one wave probe in the directionof the wave propagation to measure the maximum Hmax andminimum Hmin wave heights of the composite wave field. Thevalues of Hmax and Hmin correspond to wave heights at a quasi-antinode and node, respectively, of the corresponding com-posite wave system. The incident wave height Hi is calculatedas the average of Hmax and Hmin, and reflection wave height Hr

is calculated as half the difference between Hmax and Hmin. Asa simplification to this approach, Dean and Dalrymple (1991)also proposed placing two wave probes at fixed distances ofL/4 and L/2 from the reflecting structure, where L is the wave-length. Both these methods have slight difficulties and disad-vantages. The first method has the disadvantage of being cum-bersome to execute when many measurements are required.The second method may be inaccurate because of uncertaintyof the locations of L/4 and L/2 from the structure, particularlyfor a perforated or sloping structure. The latter method is alsocumbersome when tests are carried out for a series of wave-lengths.

Other methods that involve the use of two and three fixedwave probes ahead of the breakwater have also been devel-oped to provide explicit resolutions of the incident wave heightand of the height and phase of the reflected wave component.One of these methods, called the ‘‘resolution technique,’’ usestwo fixed wave probes at adjacent locations in front of thestructure. It has been described by Thornton and Calhoun(1972) and modified by Goda and Suzuki (1976). In thismethod, the measured wave parameters are two wave heightsand one phase angle. The method has reasonable accuracy,except when the spacing between adjacent probes approachesan integer number of half wavelengths. Another method, de-veloped by Mansard and Funke (1980), uses three fixed wave

JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 2001 / 99

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probes in front of the breakwater. It involves a least-squaresfit to the measured wave parameters, which are three heightsand two phase angles. This method fails for the case of equalprobe spacing, when the spacing approaches an integer numberof half wavelengths. As for the case of unequal probe spacing,the method fails at shorter wavelengths. Isaacson (1991) de-scribed yet another method to obtain the incident and reflectedwave heights, using three fixed wave probes to measure threewave heights. This method fails for the case of equal probespacing, when the spacing approaches an integer number ofquarter wavelengths. Isaacson (1991) performed a comparisonof the range of application and accuracy of the above methodsand showed that the least-squares fit to the measured waveparameters from three fixed wave probes is the most accurate.

The objective of this work is twofold. First, a parametricexperimental study is performed to compare the reflection andtransmission characteristics of a water-filled flexible break-water model to those of a rigid model with the same dimen-sions. The study involves monochromatic waves propagatingin the normal direction to the models, which are anchored tothe bottom of a wave tank. Reflection, transmission, and en-ergy-loss coefficients are measured for a variety of wave con-ditions over a rigid rectangular model as well as for differentinternal pressures in a flexible rectangular model for severalspecific water depths. Second, a new method is presented todetermine the reflection wave component from the variationsin the amplitude as well as phase of the wave in a time recordfrom one fixed wave probe located upstream of the model.The novelty of this method is that it requires data over a shortperiod before end reflections start affecting the data. The rec-ord starts at the time the experiment is initiated. Thus, theprobe would initially measure the incident wave component.After a certain time lapse, it would measure the total wavedue to incidence and reflection from the submerged breakwatermodel. Wavelet analysis is used to separate the two parts ofthe wave record, which are then used to determine the reflec-tion coefficient. The transmission coefficient is measured usinga second wave probe, which is set behind the breakwatermodel.

WAVELET ANALYSIS

The separation of a wave record to identify incident andreflected wave components is performed with the use of acontinuous wavelet, namely, the Morlet wavelet. The contin-uous wavelet transform W(a, t) of a function g(t) is definedas the inner product integral between g(t) and the complexconjugate of a wavelet family C(a, t). It is given by

`

W(a, t) = g(t)C*(a, t 2 t) dt (1)E2`

The wavelet family C(a, t) is given by continuous translationsand dilations of a mother wavelet C(t) according to

t 2 tC(a, t 2 t) = p(a)C (2)S Da

where a = dilation parameter; t = translation parameter; andp(a) = scale weighting function. In this work, p(a) is chosen

2pÏp(a) = 2 (3)S Da

This choice makes the magnitude of the wavelet coefficientsindependent of the peak frequency of the mother wavelet (Teo-lis 1997). Transforming the convolution integral in (1) to thefrequency domain and applying the inverse Fourier transform,the wavelet coefficients can be calculated

100 / JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 2001

J. Eng. Mech. 20

`

1 ˆˆW(a, t) = g( f )(C*(a, f ))exp(i2p f t) df (4)E2p 2`

where = Fourier transform of C* [(t 2 t)/a]; andC*(a, f )g( f ) = Fourier transform of g(t). The wavelet coefficientW(a, t) represents the contribution to g(t) of scale a at timet. Eq. (4) is very useful for efficient numerical computationof the wavelet coefficients, which can be real or complex. Forthe inverse Fourier transform to exist, the wavelet functionC(t) must satisfy the admissibility condition (Kaiser 1994).For an integrable wavelet function, this condition implies thatthe function must have a zero mean.

One common complex wavelet is the Morlet wavelet, wherethe mother wavelet is given by a sinusoidal wave with a fre-quency of fm multiplied by a Gaussian function centered att = 0 and a standard deviation st equal to 1/(2p fc). The Morletwavelet is given by

2(2p f t)cC(t) = exp(i2p f t)exp 2 (5)m F G2

The ratio fm/fc is a dimensionless parameter, denoted here asg. Note that the Morlet wavelet is only marginally admissiblebecause it has a zero mean only when a correction term isadded. However, when g is set to 6, the correction term isunnecessary because it is of the same order as a computerround-off error. The Fourier transform of the complex conju-gate of the Morelet wavelet is real and is given by

21 ( f 2 f )mC*( f ) = exp 2 (6)F G22p 2 f cÏ

Based on (2) and (3), the Morelet wavelet family is given by

22p i2p f t (2p f t)Ï m cC(a, t) = 2 exp exp 2 (7)S D F G2a a 2a

The Morlet wavelet family in the frequency domain is thengiven by

2(af 2 f )mC*(a, f ) = 2 exp 2 , f $ 0 (8)F G22 f c

C*(a, f ) = 0, f < 0 (9)

Note that, in implementing the wavelet, we perform thesampling in the frequency domain, not the time domain (Sta-mos et al. 1999). Moreover, and because we are dealing withone wave component only, we perform the analysis for onescale only by considering a mother wavelet, a = 1, with a peakfrequency fm that matches the frequency of the fundamentalwave component of the measured signal.

IMPLEMENTATION TO NUMERICAL EXAMPLE

This section provides a numerical example to show how oneuses the Morlet wavelet to separate incident and reflected wavecomponents. This example is obtained by considering two su-perimposed cosine waves that represent propagating incidentand reflected waves in a wave tank

h(x, t) = h (x, t) 1 h (x, t) = a cos(st 2 kx 1 f )i r i i

1 a cos(st 1 kx 1 f )r r (10)

where h(x, t) = surface elevation from the mean water level;k(=2p/L) = wave number with L being the wavelength;s(=2p f ) = angular frequency of the wave with f being thewave frequency; a = wave amplitude; t = time; x = horizontaldistance of the measuring point from the piston; f = phase;and subscripts i and r denote the incident and reflected wavecomponents, respectively. In a wave tank experiment, a wave

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FIG. 1. (a) Signal Generated by Numerical Example Given byEq. (10) and Magnitude of Wavelet Coefficient at fm = 0.7 Hz; (b)Phase of Wavelet Coefficient at fm = 0.7 Hz

probe located ahead of the breakwater model will first measurethe incident wave. After a certain time lapse, the reflectedwave will reach the probe, which will then measure the totalelevation of the composite wave. The wave periods of theincident and reflected wave components will be the same.However, when the reflected wave component reaches themeasuring probe, it will introduce a phase change in the wave.In this application, this phase change will be used to separatethe incident and composite waves.

One generates the numerical signal as given in (10) by let-ting ai = 0.01 m, ar = 0.005 m, fi = p, fr = p/2, x = 10 m,and f = 0.7 Hz. The wave number is computed to be k = 3.361rad/m, using the dispersion relationship.

2s = gk tanh kh (11)

where g = 9.81 m/s2; and h = water depth set to 0.2 m.One also assumes that the incident wave is recorded undis-

turbed for 25 s. After this instant, one starts the reflected wave,which results in a composite wave. As explained above, thissimulates the history of an experiment in a wave tank. For thenumerical values chosen, the signal is presented in Fig. 1(a).Also shown in Fig. 1(a) is the magnitude of the wavelet co-efficient at fm = 0.7 Hz as defined in (4). As expected, it is theenvelope of the signal. Note that for t < 25 s, the magnitudeof the wavelet coefficient uW u is equal to the amplitude of theincident wave component, and for t > 25 s, the magnitude ofthe wavelet coefficient is equal to the amplitude of the com-posite wave, as given in (10).

The phase of the wavelet coefficient is independent of itsmagnitude and can thus provide additional information aboutthe signal. If this phase is unwrapped and the 2p ft componentis subtracted, the result fW is the phase of the signal h(x, t)and is shown in Fig. 1(b). For t < 25 s, the phase of the waveletcoefficient is equal to the total phase of the incident wave,given by 2kx 1 fi. Because the wave number k and positionx are known, one can recover the incident phase fi = p. Thephase of the wavelet coefficient for t > 25 s is equal to thephase of the composite wave h(x, t). Note that the slight de-parture from actual values near t = 0, 25, and 50 s are due toend effects, which result from the use of continuous wavelets.

The reflected wave component can be recovered by sub-tracting the incident wave component from the compositewave vectorially. A vectorial representation of the sum of theincident and reflected waves, such as the one given in (10), isshown in Fig. 2. In this figure, uW u and fW represent the am-plitude and phase, respectively, of h (x, t). For t < 25 s, the

J. Eng. Mech.

FIG. 2. Vector Representation of Incident and Reflected WaveComponents

FIG. 3. Numerical Example of Reflected Wave Component: (a)Amplitude; (b) Phase

magnitude of the wavelet coefficient uW u is equal to ai andthe phase fW is equal to 2kx 1 fi. For t > 25 s, the magnitudeof the wavelet coefficient uW u is the amplitude of the com-posite wave, which includes incident and reflected wave com-ponents, and fW is the phase of the composite wave. As shownin Fig. 2, from the two sets of uW u and fW, one can determinethe amplitude ar and total phase, kx 1 fr, of the reflectedwave component. Fig. 3(a) shows the amplitude of the re-flected wave ar. Note that, for t < 25 s, it is zero because onerecords only the incident wave component. For t > 25 s, theamplitude of the reflected wave ar is equal to 0.005, which isthe value given to the reflected wave component in the ex-ample. Fig 3(b) shows the total reflected phase, which is equalto kx 1 fr after 25 s. For t < 25 s, the phase does not haveany significant meaning because there was no reflected com-ponent. Because k and x are known, one is able to recover thereflected phase fr = p/2 as set in the example. This numericalexample shows how wavelet analysis enables one to separatea reflected wave component from a composite wave using onesignal.

EXPERIMENTAL SETUP

The experiments were carried out in the wave tank of theEngineering Science and Mechanics Department at VirginiaPolytechnic Institute. The tank is 27.93-m long, 1.83-m wide,and 1.22-m deep. The wavemaker is capable of producing reg-ular surface waves with frequencies ranging from 0.2 to 2 Hz.


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FIG. 4. Experimental Setup

A digital function generator is used to run the wavemaker. Thetank is also equipped with a permeable wave-absorbing beachat the downstream end to reduce end reflections. A schematicof the experimental setup is shown in Fig. 4. Two wave probeswith capacitance transducers were used to measure the watersurface elevation, with a resolution of 0.02 cm. Calibration ofeach wave probe was carried out before each set of experi-ments. The first probe was located at 5.27 m in front of themodel, and the second was 2.44 m behind it. The model wasplaced at a distance of 13.60 m from the piston. The upstreamprobe measured the incident wave component hi(x, t) initiallyand the composite wave h(x, t) after a time lapse. The down-stream probe measured the transmitted wave ht(x, t).

Two breakwater models were used in these experiments.The first model is a rigid prism that has dimensions of 183 330.48 3 17.14 cm. It is made of polyvinyl-chloride materialmounted on an aluminum plate. The second model consists ofa flexible membrane also mounted on an aluminum plate. Themembrane was made of high-grade neoprene rubber, whichwas attached to the plate’s sides with stainless steel clamps. Itwas reinforced at the top with a rectangular plastic sheet thatwas placed inside the model. The ends were capped with rec-tangular sheets of plexiglass. When the flexible model is filledwith water, this design assumes a rectangular shape with di-mensions close to those of the rigid prism.

The flexible membrane was also equipped with two hosefittings. The first, which is attached to one side at the bottom,is used to recover water losses during the experiment. This isachieved with a pressure head device that was placed 6.76-mbehind the flexible model. The second fitting is on the top ofthe other side of the membrane and allowed reading of theinternal pressure before each set of experiments. The rigid andflexible structures were anchored to the tank floor with fourpad eyes, two at each side on the aluminum plate, and steel-nylon wires. There was no contact between the models’ endsand wave tank walls during the experiment. Wave recordswere sampled at a rate of 10 Hz over 102.4 s.

RESULTS AND DISCUSSION

Representation in Nondimensional Variables

The wave reflection and transmission in the presence of arigid or flexible fluid-filled breakwater structure would occurwith the loss of some portion of the incident wave energy bythe following processes: losses due to wave breaking over thestructure, loses due to turbulence induced by flow separationover and near the structure, and internal losses due to turbu-lence of the fluid contained inside the structure as well asnonelastic deformations of the structure itself.

The relationship between the characteristics of wave reflec-tion, wave transmission, and energy dissipation for the rigid


J. Eng. Mech.

and flexible breakwater models can best be presented in termsof dimensionless parameters. The dimensional parameterscharacterizing the problem and presented in Fig. 4 are the in-cident Hi, reflected Hr, and transmitted Ht wave heights; lossin wave height HL due to the processes of energy loss; depthof submergence z; internal pressure head in the flexible modely; incident wavelength L; water depth h; and model width b.These variables can be expressed in the dimensionless param-eters Hr/Hi, Ht/Hi, HL/Hi, Hi/L, kh, z/h, y/h, and b/L where k= 2p/L is the wave number. Note that HiL is the wave steep-ness, Hr/Hi = Cr is the reflection coefficient, Ht/Hi = Ct is thetransmission coefficient, and HL/Hi = CL is the energy-loss co-efficient. The expression represents the energy loss, and CL

2CL

is used as a coefficient to represent this loss. The coefficientsof reflection Cr, transmission Ct, and energy loss CL are thusfunctions of five variables

H z y biC = f , kh, , , (12)r r S DL h h L

H z y biC = f , kh, , , (13)t t S DL h h L

H z y biC = f , kh, , , (14)L L S DL h h L

Based on energy conservation, the energy-loss coefficientCL can be calculated from the following relation (Thorntonand Calhoun 1972):

2 2 2C 1 C 1 C = 1 (15)r t L

Tests for different wave frequencies and amplitudes, result-ing in different wave steepnesses Hi/L, were conducted at wa-ter depths h of 22.5 cm (z/h = 0.24) and 27.5 cm (z/h = 0.38)for both models. Moreover, for the flexible model, one con-siders the ratios of pressure head to water depth y/h of 0.007,0.141, and 0.282 at each water depth.

Typical Results for One Set of Measurements

Typical profiles of regular waves recorded ahead and behindthe rigid breakwater model are shown in Fig. 5. The wave-maker frequency was set at 0.6 Hz. The first record, shown inFig. 5(a), consists initially of he incident wave componenthi(x, t). After an initial period, it contains the sum of the in-cident and reflected wave components (composite wave)h(x, t). It should be noted here that the initial portion is not a‘‘clean’’ harmonic. This is most likely due to the responsefunction of the wavemaker to the input from the function gen-erator. However, it should be stressed that, with these varia-tions, the largest wave component in the signal remains at theexcitation frequency. The second record, shown in Fig. 5(b),

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FIG. 5. Typical Regular Wave Profiles at Two Locations: (a)Ahead of Rigid Breakwater Model; (b) behind Rigid BreakwaterModel

FIG. 6. Wavelet Coefficient at Frequency of FundamentalWave Component in Record Shown in Fig. 5(a): (a) Magnitude;(b) Phase

FIG. 7. Real Part of Estimated Incident Wave Component(Solid Line) and Real Part (Dashed Line) of Wavelet Coefficientat Frequency of Fundamental Wave Component

J. Eng. Mech.

FIG. 8. Magnitude of Wavelet Coefficient at Frequency of Fun-damental Wave Component in Record Shown in Fig. 5(b)

contains the transmitted wave ht(x, t). Analysis similar to theone presented in the ‘‘Implementation’’ section was conductedfor the first record. The magnitude and unwrapped phase ofthe wavelet coefficient at the wave component with fm = 0.6Hz are presented in Figs. 6(a and b), respectively. Estimatesfor the amplitude ai and total phase (2kx 1 fi) of the incidentwave component are obtained by computing averages for themagnitude and phase of this wavelet coefficient between t =5.5 s and 9.3 s. The first time, t = 5.5 s, corresponds to a well-defined incident wave peak approaching the upstream probe.The second time, t = 9.3 s, corresponds to that peak arrivingat the breakwater. Based on these averages, one estimates ai

to be 1.21 cm and 2kx 1 fi to be 4.905 rad. Fig. 7 comparesthe estimated incident wave component (dashed line) obtainedwith these values for ai and 2kx 1 fi and the real part (solidline) of the wavelet coefficient at fm = 0.6 Hz of the recordshown in Fig. 5(a). Note the good match over the time periodbetween 5 and 10 s. This shows that the estimated incidentwave component compares very well with the wave compo-nent of the measured signal over the part of the record thatconsists of the incident component.

The reflected wave component is determined by subtractingvectorially the incident wave component, as determined above,from the composite wave. Estimates for the amplitude ar andtotal phase (kx 1 fr) of the reflected wave component areobtained by computing averages for the amplitude and phaseof the reflected component obtained from the vectorial sub-traction between t = 13 and 24.8 s. The first time, t = 13 s,corresponds to a composite wave peak approaching the up-stream probe, and the second time, t = 24.8 s, correspondsroughly to a re-reflected wave peak from the piston approach-ing it again. Based on these averages, one calculates ar to be0.54 cm and 2kx 1 fr to be 2.06 rad.

Fig. 8 shows the magnitude of the wavelet coefficient at thewave component at fm = 0.6 Hz in the transmitted signal [Fig.5(b)]. An estimate for the amplitude at of the transmitted waveis obtained by computing an average value for the magnitudeof the wavelet coefficient between t = 11.2 and 18.1 s. Thefirst time, t = 11.2 s, corresponds to a well-defined wave peakapproaching the probe behind the breakwater model, and thesecond time, t = 18.1 s, corresponds to that peak arriving atthe wave tank end. Based on this average, one calculates at tobe 0.80 cm. Based on the above estimates for ai, ar, and at,the reflection coefficient Cr is calculated to be 0.44 and thetransmission coefficient Ct is 0.66. Based on (15), the energyloss coefficient CL is found to be about 0.61.


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FIG. 9. Variations of Wave Coefficients with kh (Water Depth,22.5 cm) (● Rigid, C y/h = 0.007, ▫ y/h = 0.141, L y/h = 0.282)

Reflection and Transmission Coefficients

The above analysis was performed on several records forrigid and flexible breakwater models, under different internalpressures in the flexible breakwater, wave frequencies, and wa-ter depths. This section compares the effectiveness of the rigidand flexible breakwaters and presents the results in terms ofthe parameters already presented.

Variations of reflection, transmission, and energy-loss co-efficients with kh are given in Fig. 9. The results show thatthe reflection is most effective with a coefficient value of 0.55near kh values of 0.60 and 1.86. Between these values, Cr issmaller, with a minimum of 0.05 near kh = 0.98 for the flexiblemodel with y/h ratios of 0.141 and 0.282. As for the differencein the reflection characteristics of the rigid and flexible break-waters, it is found that the average reflection for the rigidmodel is about 38% for the range of kh considered here. Forthe flexible model, Cr is near 32, 33, and 36% for y/h ratiosof 0.007, 0.141, and 0.282, respectively. These results indicatethat the reflection coefficient increases with the increase in thestiffness of the model. For the transmission coefficient Ct, itis noted that it varies around a value of 0.5, with a slight dipnear kh = 0.98. The rigid model has an average transmissionof about 64%, whereas the flexible model has an average trans-mission of 35, 53, and 52% for y/h ratios of 0.007, 0.141, and0.282, respectively, for the range of kh studied. These resultsshow that Ct decreases with decreasing stiffness of the break-water.

For the energy loss coefficient CL, the results show that itincreases for kh # 0.84 and decreases for kh $ 1.28. Betweenthese values, CL is almost independent of the variations in kh.It should also be noted that the energy loss tends to increasewith a decrease in stiffness. The corresponding average valuesfor the rigid and flexible breakwaters with y/h ratios of 0.007,0.141, and 0.282 are 87, 74, and 74%, respectively. This showsthat the energy dissipation is highest for the flexible modelwith y/h = 0.007. It is important to also note that, when kh =0.98, energy loss is maximum, and at about 0.90, it coincideswith the kh at which the transmission is minimum for bothmodels. Moreover, at the same kh value, the reflection is min-imal, indicating that most of the wave energy is dissipatedrather than reflected. This result emphasizes the significanceof the energy losses in reducing the transmitted waves.

Variations of reflection, transmission, and energy-loss co-efficients with kh for z/h = 0.38 are presented in Fig. 10. Theresults show that Cr varies near 0.5 for kh # 0.95 and kh $1.95. Between these values, the values of Cr are lower, with a


J. Eng. Mech.

FIG. 10. Variations of Wave Coefficients with kh (Water Depth,27.5 cm) (● Rigid, C y/h = 0.007, ▫ y/h = 0.141, L y/h = 0.282)

minimum near 0.2 at kh = 1.48. Moreover, the reflection co-efficient values are very close for all models for kh # 1.11.For kh $ 1.29, the reflection by the rigid model is found tobe significantly higher when compared to reflection by theflexible model. For the transmission coefficient Ct, the resultsshow values >0.45 for all models. The results also show thatCt tends to increase with an increase in stiffness in that therigid model transmits higher wave energy than the flexiblemodel. The energy dissipation is maximum for the flexiblemodel with y/h = 0.007 and minimum for the rigid modelthroughout the range of kh.

For the rigid model, the average value of Cr is found to beabout 40% and that of Ct and CL are about 69 and 57%, re-spectively, for the range of kh studied. For the flexible modelwith y/h = 0.007, the average value of Cr is found to be about29% and that of Ct and CL are about 51 and 79%, respectively.Although the rigid model reflects more wave energy, the flex-ible model is a more effective breakwater because the trans-mitted waves are smaller. This is most likely because the flex-ible model has the ability of oscillating in different modes andpossibly breaking the incoming waves. For the flexible modelwith y/h = 0.141 and 0.282, the average values are 29 and33% for the reflection coefficient, 60 and 61% for the trans-mission coefficient, and 72 and 70% for the energy-loss co-efficient. By comparing the average values of the wave coef-ficients for the higher wave reflections of the flexible models,lower wave transmission and higher energy loss are achievedfor the lower submergence depth ratio of 0.24 than the highersubmergence depth ratio of 0.38.

To determine the effect of model width, variations of allthree coefficients are discussed at the constant submergencedepth ratio z/h, substituting the kh parameter with the ratio ofmodel width to wavelength b/L. The model width for bothbreakwaters was kept constant at 0.30 m, and the incidentwavelength L is varied from 2.3 to 0.75 m for the case ofz/h = 0.24 and from 2.5 to 0.77 m for the case of z/h = 0.38.For the case of z/h = 0.24, values of b/L were varied between0.130 and 0.401. The optimum b/L ratio for both models isbetween 0.210 and 0.242, where the wave transmission is min-imum and the energy loss is maximum, and is most noticeablefor the flexible model with y/h = 0.007. For the case of z/h =0.38, the value of b/L was varied between 0.119 and 0.392.The optimum b/L ratio for this case is between 0.344 and0.392, where the transmission varies around 0.5 and the wavereflection (especially for the rigid model) and energy loss (es-pecially for the flexible model with y/h = 0.007) are high.

2001.127:99-105.

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CONCLUSIONS

In this work, the reflection and transmission characteristicsof regular waves and the energy dissipation by rigid and flex-ible rectangular breakwater models, which are anchored to thebottom of a wave tank, are investigated experimentally. Thisstudy is performed with a new technique for separation ofincident and reflected wave components. This technique isbased on local variations in amplitude and phase of the mea-sured wave and uses wavelet analysis. Transmission and en-ergy-loss coefficients are also obtained. Studies are conductedfor different depths of submergence of the models, differentinternal pressures in the case of the flexible breakwater, andfor a wide range of wave steepnesses.

The results show that, in general, the rigid model has ahigher reflection coefficient than the flexible model. On theother hand, the flexible model has a much higher energy losscoefficient. This is most likely because the flexible model hasthe ability of oscillating in different modes and possibly break-ing the incoming waves. In the range of internal pressuresconsidered here, the energy-loss coefficient is higher at thelower pressures. The results also provide optimal breakwaterwidths for reflection, transmission, and energy-loss coefficientsfor waves with different wavelengths.

J. Eng. Mech. 20

APPENDIX. REFERENCESDean, R. G., and Dalrymple, R. A. (1991). Water wave mechanics for

engineers and scientists, World Scientific, River Edge, N.J.Frederiksen, H. D. (1971). ‘‘Wave attenuation by fluid-filled bags.’’ J.

Wtrwy., Harb. and Coast. Engrg. Div., ASCE, 97(1), 73–90.Goda, Y., and Suzuki, Y. (1976). ‘‘Estimation of incident and reflected

waves in random wave experiment.’’ Proc., Conf. Coast. Engrg.,ASCE, Vol. 1, 828–845.

Isaacson, M. (1991). ‘‘Measurement of regular wave reflection.’’ J.Wtrwy., Port, Coast., and Oc. Engrg., ASCE, 117(6), 553–569.

Kaiser, G. (1994). A friendly guide to wavelets, Birkhauser, Boston.Mansard, E. P. D., and Funke, E. R. (1980). ‘‘The measurement of inci-

dent and reflected spectra using a least squares method.’’ Proc., 17thCoast. Engrg. Conf., ASCE, Vol. 1, 154–172.

Ohyama, T., Tanaka, M., Kiyokawa, T., Uda, T., and Murai, Y. (1989).‘‘Transmission and reflection characteristics of waves over a submergedflexible mound.’’ Coast. Engrg. in Japan, Tokyo, 32, 53–68.

Phadke, A. C., and Cheung, K. F. (1999). ‘‘Response of bottom-mountedfluid-filled membrane in gravity waves.’’ J. Wtrwy., Port, Coast., andOc. Engrg., ASCE, 125(6), 294–303.

Stamos, D. G., Lusk, C. P., Hajj, M. R., and Telionis, D. P. (1999).‘‘Wavelet analysis of reflection and transmission characteristics of sub-merged flexible breakwaters.’’ Proc., ASME Summer Meeting,FEDSM99-7163.

Teolis, A. (1997). ‘‘Identification of noisy FM signals using non-orthog-onal wavelet transforms.’’ SPIE, 3078, 590–601.

Thornton, E. B., and Calhoun, R. J. (1972). ‘‘Spectral resolution of break-water reflected waves.’’ J. Wtrwy., Harb. and Coast. Engrg. Div.,ASCE, 98(4), 443–460.


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Reflection and Transmission of Waves over Submerged Breakwaters