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Wave Shoaling on Steep Slopes and Breaking Criteria Ching-Piao Tsai, Hong-Bin Chen and Ming-Jen Huang Department of Civil Engineering, National Chung-Hsing University Taichung 402, Taiwan, China ABSTRACT Estimation of the transformation of wave shoaling and breaking isessential for the nearshore hydrodynamics and the design of coastalstructures. Many empirical formulas have been well recognized to thewave transformation on gently sloping beaches. This paper reports theapplicability of previous formulas of wave shoaling and breakingindices for the situation of steep slopes based on comparing with the present experimental results. Two steep bottom slopes of 1/3 and 1/5,and one gently slope of 1/10 were conducted in the present experiments.It was found that the shoaling distance on steep slope becomes short,and the surface waves may be partially reflected from the steep bottom.The coefficient of wave reflection is well related to the surf similarity parameter. The estimations of wave shoaling based on the well-knownformula by Shuto (1974) did not conform completely to theexperimental results for steep slopes. The empirical formula for the breaking criteria proposed in Goda (1975) was slightly modified in this paper for applying to the steep beaches. A time-dependent mild-slopeequation model finally was adopted to calculate the wave heighttransformation in the surf zone by introducing the modified breakingindex. KEY WORDS: Wave shoaling; breaking index; wave reflection;time-dependent mild-slope equation. INTRODUCTION Wave breaking is one of the most interesting phenomena of the wavetransformation in the nearshore region. It is not only produces a largeforce on coastal structures, but also induces nearshore currentcirculation that affects bottom topography. Therefore, the prediction of wave breaking is an important theme for nearshore hydrodynamics, aswell as for the design of coastal structures. Many theories and empiricalformulas, including the calculation of wave shoaling and the predictionof breaking criteria, have been proposed in the literature.Priori to wave breaking, the energy flux method employing linear or nonlinear wave theory was presented to predict the wave heighttransformation under the shoaling process (Le Méhauté and Webb,1964; Svendsen and Brink-Kjaer, 1972; Stiassnie and Peregrine, 1980).Using a perturbation method, Shuto (1974) presented an approximateformula for wave height change of shoaling, on the basis of the K-dVequation. Shuto’s results have been found to agree reasonably withexperimental data for a wide range of wave steepness. Thus they areoften used for practical application (Horikawa, 1988). Recently,numerical models based on the mild slope equation have also been proposed to compute the nearshore wave transformation (Watanabe andDibajnia, 1988; Tsai et al., 2001).The shoaling waves will break when their height reaches a certain limit.The criteria of waves at breaking in terms of breaking height and breaking depth have been analyzed by many investigators, such as inMcCown (1894), Miche (1951), Weggel (1972), Goda (1975),Svendsen and Hansen (1976), Van Dorn (1978), Peregrine (1983) andSvendsen (1987) etc.. There were many works expressing the breakingwave height in terms of a given deepwater wave condition, as inLeMéhauté and Koh (1967), Komar and Gaughan (1973) andSunamura (1983) etc..Most of works, however, were investigated for the wave transformationon gently sloping beaches. For the situation of east coast in Taiwan,where the bottom slope is very slope to 1/5, even to 1/3. Because of thedistance of wave shoaling on a steep bottom becomes shorter than on agently slope, that the shoaling waves may result in breaking early.Besides, the surface wave may be partially reflected from the steepslope. Thus those previous empirical formulas for wave shoaling andthe wave breaking criteria should be examined for the situation of steep beaches. In this paper, the previous formulas for the estimation of coefficient of wave reflection, wave transformation of shoaling, and the breaking index are compared with the present experimental results.Then, a mild-slope equation model proposed by Tsai et al. (2001) isadopted to calculate the wave height transformation by introducing amodified breaking index . EXPERIMENTS The experiment was conducted in a two-dimensional wave channel thatwas 100 m long, 2 m wide and 2 m high. The sidewalls at theobservation sections of the channel were consisted of strengthenedglass plates. A piston-type wave generator system, controlled by a D/A 61 7 Proceedings of The Twelfth (2002) International Offshore and Polar Engineering ConferenceKitakyushu, Japan, May 26 – 31, 2002Copyright © 2002 by The International Society of Offshore and Polar Engineers ISBN 1-880653-58-3 (Set); ISSN 1098-6189 (Set) converter and a personal computer, was mounted at the end of thechannel. Only regular waves were considered in the experiments. Thesketch of the experimental setup is shown in Fig. 1, in which the modelof a slope beach was installed at the other end of the channel. The bottom slope followed by a horizontal section was made in theexperiments, which is similar to Horikawa and Kuo (1966) . The experimental wave conditions were varied from 1.0 sec to 2.5 sec periods corresponding with many different wave heights ranged from0.057 m to 0.33 m, from which the steepness of incident waves in deepwater ( H o / L o ) ranged from 0.005 to 0.08. For forcing the incident waves just breaking at the end of the sloping part and in front of the horizontalregion, the water depth ( h ) is adjusted by ranging from 0.86 m to 1.10m. A series of capacitance-type wave gauges were used to measure thewave profiles. An A-D converter at a sampling frequency of 20Hzdigitized all records.Fig. 1 Sketch of Experimental Setup RESULTS AND COMPARISONS Wave Reflection Coefficient For the wave reflection from a sloping beach, Battjes (1974) obtainedan empirical formula for estimating the reflection coefficient as 2 1.0 ξ = r K (1)in which ξ is referred as the surf similarity parameter and defined as oo L H //tan β ξ = (2) 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.30.60.91.2 ξ K r Battjes (1974)Slope = 1/10Slope = 1/3Slope = 1/5 Fig. 2 Comparisons of the wave reflection coefficientswhere tan β is the bottom slope, and H o and L o are the wave height andlength in deep water, respectively.As shown in Fig. 2, the experimental results of the wave reflectioncoefficients are in good agreement with the relationship indicated in Eq.1. It is shown that the value of reflection coefficient for the bottomslope of 1/3 in the present experiments could reach to 0.5 about. It isnoted that the analysis of the reflection coefficients of the experimentaldata was based on the method proposed in Goda and Suzuki (1976). Comparisons of Wave Shoaling Shuto (1974) has proposed a set of approximate formulas for waveshoaling on a sloping beach based on the K-dV equation, shown as ≤=−≤≤=≤= r r r r o U const U Hh U HhU khn H H 50for .)32( 5030for const. 30for tanh121 2/57/2 (3) which has often been used for practical applications (Horikawa, 1988).In Eq. 3, U r = gHT 2 /h 2 is the local Ursell parameter defined in Shuto(1974), H is the wave height, T is the wave period, k is the local wavenumber, h the local water depth and n is defined as += khkhn 2sinh2121(4) Noted that Eq. 3 implies energy flux is not a single-value function of the wave height and the shoaling coefficient calculated from theserelationships is identical to the small amplitude wave theory when30 ≤ r U .The approximately sound relationships are adopted for comparing withthe present experimental data. For the slope of 1/10, it is shown in Fig.3 that the calculated results are well in agreement with the experiments.However, for the steep slopes of 1/5 and 1/3 shown in Figs. 4 and 5, thecalculated results did not conform completely to the experimentalresults. The discrepancy especially occurs in the region of the smaller values of h/L o or larger values of U r . It may be resulted from the reasonthat the shoaling distance on steep slope becomes shorter than that of gently slope. Noted that ( H/L ) max = 0.142 tanh kh was used as the breaking criteria shown in these figures. The experimental resultsdepicted that the breaking wave heights were much smaller than theestimations, for the steep beaches of 1/5 and 1/3. Comparisons of Breaking Wave Indices There were many criteria presented in the literature to predict the wave breaking. Goda (1970, 1975) expressed the breaking criteria graphicallyand then presented an approximate expression for the curves givenusing)]}tan1(5.1exp[1{ 3/4 β π B Lh A L H oob +−−= (5)in which H b is the height of the breaking wave, A and B are empiricalconstants taken to be 0.17 and 15, respectively. 61 8 0.1 0.2 0.30.60.81.01.21.41.61.8 h/L o H/H o breaking criteriaShuto (1974)H o /L o =0.020H o /L o =0.048H o /L o =0.060H o /L o =0.074H o /L o =0.020H o /L o =0.048H o /L o =0.060H o /L o =0.074 tan β = 1/10 Fig. 3 Comparisons of the wave shoaling coefficients, tan β = 1/1 0 10 -2 10 -1 0.51.01.52.02.53.0 h/L o H/H o H o /L o =0.010H o /L o =0.005H o /L o =0.020H o /L o =0.040H o /L o =0.080breaking criteriaShuto (1974)H o /L o =0.005H o /L o =0.010H o /L o =0.020H o /L o =0.040H o /L o =0.080 tan β = 1/5 Fig. 4 Comparisons of the wave shoaling coefficients, tan β = 1/5 10 -2 10 -1 0.51.01.52.02.53.0 h/L o H/H o breaking criteriaShuto (1974)H o /L o =0.005H o /L o =0.010H o /L o =0.020H o /L o =0.040H o /L o =0.080H o /L o =0.005H o /L o =0.010H o /L o =0.020H o /L o =0.040H o /L o =0.080 tan β = 1/3 Fig.5 Comparisons of the wave shoaling coefficients, tan β = 1/3Good agreement is found in Fig. 6 upon comparison with theestimations and the experimental results for the slopes of 1/10 and 1/5.However, the estimations are much higher than the experimental datafor the very steep slope of 1/3 shown in Fig. 7. The discrepancy might be due to the short shoaling distance, which forcing wave breakingearly. It could be suggested for this case that the empirical constants A and B are taken as 0.16 and 7 in Eq. 5, respectively, fitting in theexperimental data.The other breaking criterion was expressed by the ratio of breakingwave height to the water depth. The earliest formula was proposed inMcCowan (1894) given using ( H/h ) b = 0.78, as the bottom is horizontal,in which the subscript b denotes the value at breaking. Svendsen (1987) presented the breaking criterion versus the beach slope and localrelative water depth, given using 20.0 05.1 S b = γ (6)where γ b = ( H/h ) b and S = tan β /(h/L) b . Svendsen (1987) interpreted thisapproximation is reasonable to the data for 0.25 < S < 1. As introducingthe laboratory results of the steep slopes shown in Fig. 8, it shows thatEq. 5 could be extended for larger values of S , though the experimentaldata have slightly scattering from the regression line. 0.00 0.02 0.04 0.06 0.08 0.10 0.120.000.040.080.120.16 h b / L o H b / L o Goda (1975), tan β = 1/5MeasuredGoda (1975), tan β = 1/10Measured Fig. 6 Comparisons of the wave breaking criterion, tan β = 1/10, 1/5 0.00 0.02 0.04 0.06 0.08 0.10 0.120.000.040.080.120.16 h b / L o H b / L o Goda (1975), tan β = 1/3measuredH b /L o = 0.16 {1-exp[-1.5 π h b /L o (1+7 tan 4/3 β )]}modified formula, tan β = 1/3 Fig. 7 Comparisons of the wave breaking criterion, tan β = 1/3 61 9 10 -1 10 0 10 1 0.511.522.53 S γ b Van Dorn, Slope = 1/45Van Dorn, Slope = 1/25Svendsen (1987)Van Dorn, Slope = 1/12ISVA, Slope = 1/35Measured, Slope = 1/5Measured, Slope = 1/10 γ b = 0.78Measured, Slope = 1/3 Fig. 8 Comparisons of the wave breaking criterion Calculations of Wave Height Transformation Wave height transformation including the wave decay can be calculatedfrom the numerical model presented in Tsai et al. (2001). In the model,the set of time-dependent mild-slope equation was incorporated anapproximate nonlinear shoaling corrector and an energy dissipationfactor. The accurate use of the breaking criterion is required in thecalculations, in which Eq. 5 was well adopted in Tsai et al. (2001).Based on the calculations, the numerical results are found to agreereasonable well with experimental data, as shown in Figs. 9-11. Notedthat the modified empirical constants A = 0.16 and B = 7 in Eq. 5 wereused for the solutions of the slope of 1/3 shown in Fig. 11. It was foundthat the fluctuation of wave height transformation before wave breakingappears in the numerical and experimental results, which is due to thewave reflection from the bottom. Though the mild-slope equationmodel is based on the assumption of “mild’ slope bottom, the resultsshows that the model is applicable for the calculated examples. CONCLUSIONS The wave transformation were presented in the literature, but most of them were paid attention to waves propagating on the gently sloping beaches. This paper reported the laboratory investigation for the two-dimensional wave transformation on steep beaches involving the waveshoaling and the breaking indices. The wave reflection from the bottom,the approximate formula and the breaking indices were compared withthe experimental results. The wave reflection from the steep bottom has been compared well with the relationship presented in Battjes (1974),from which good agreement was found between the experiment dataand the empirical formula. The wave height transformation wascompared with the approximate formula proposed by Shuto (1974).Owing to the shoaling distance of the steep slope is very short and thewave breaking occur earlier, the estimations were found to deviate fromthe experiments, in the region with larger Ursell number. For the slopesof 1/10 and 1/5, the experimental results of the breaking criterion werein good agreement with the empirical formula of Goda (1975). But theestimated breaking height was much higher than the experimental datafor the steep slope of 1/3, from which the empirical constants of Goda’sformula was modified in the paper for this steep slope. Based on thenumerical model of Tsai et al. (2001), the calculated results of waveheight transformation across the surf zone was found to agreereasonably with the experimental results. -5 0 5 10 15 20 250.000.100.200.300.40 x(m)H(m)numerical solutionmeasuredT = 2.6 sec, H o = 0.213 mh = 0.980 m, tan β = 1/10 Fig. 9 Comparisons of the wave height transformation, tan β = 1/10 -5 0 5 10 15 200.000.100.200.30 x(m)H(m)numerical solutionmeasuredT = 2.6 sec, H o = 0.106 mh = 0.885 m, tan β = 1/5 Fig. 10 Comparisons of the wave height transformation, tan β = 1/5 -5 0 5 10 15 200.000.100.200.300.40 x(m)H(m)numerical solutionmeasuredT = 1.8 sec, H o = 0.202 mh = 0.970 m, tan β = 1/3 Fig. 11 Comparisons of the wave height transformation, tan β = 1/3 6 20
