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TECHNICAL NOTES

New Methodology for Laboratory Generation of Solitary Waves

Siamak Malek-Mohammadi1 and Firat Y. Testik, A.M.ASCE2

Abstract: Traditionally, solitary waves are generated in the laboratory setups using the Goring’s methodology that considers a wave of permanent form during the generation process. In this study, we propose a new methodology for solitary wave generation using pistontype wavemakers by considering the evolving nature of the wave during the generation process. This proposed methodology is tested by conducting a series of experiments in a wave tank. In the experiments, generation of solitary wave pro?les predicted by Boussinesq’s and Rayleigh’s theoretical solutions are accomplished using both the New and Goring methodologies. Waves generated using the Goring methodology served as a benchmark to assess the performance of the proposed New methodology. Comparisons of experimental observations and theoretical solutions for various wave characteristics have shown that the New methodology is capable of generating more accurate solitary waves with less attenuation as waves propagate compared to the Goring methodology. Proposed solitary wave generation methodology is expected to be useful in various laboratory investigations to study different aspects of long wave theories and to simulate tsunamis and internal waves, among others. DOI: 10.1061/?ASCE?WW.1943-5460.0000046 CE Database subject headings: Solitary waves; Tsunamis; Wave generation; Wave tanks. Author keywords: Solitary wave; Tsunami; Laboratory wave generation; Wave tanks.

Introduction

A solitary wave is a single hump of water that is entirely above still water level with an in?nite wavelength ?Goring 1979?. Solitary waves are special solution of the water wave problem in the weakly nonlinear and weakly dispersive regime, and they propagate without change in shape as these two effects balance each other ?Chow 1989?. Solitary waves were ?rst identi?ed by Russell ?1845?. Further theoretical and experimental analysis by Hammack and Segur ?1974? determined that any water surface displacement above still water level leads to the eventual emergence of a single or a series of solitary waves followed by dispersive oscillatory trailing waves. This ?nding is particularly important in the progress of tsunami science. Since tsunamis are generated by large disturbances in water surfaces mainly due to underwater geophysical dislocations such as earthquakes or landslides, solitary waves have been employed in laboratory investigations of different aspects of nearshore tsunami propagation and runup ?e.g., Goring 1979; Synolakis 1987; Li and Raichlen 2001; Jensen et al. 2003?. Recently, Madsen et al. ?2008?, studying the shoaling of long transient and periodic waves and referring to the

1 Graduate Student, Dept. of Civil Engineering, Clemson Univ., Clemson, SC 29634. 2 Assistant Professor, Dept. of Civil Engineering, Clemson Univ., Clemson, SC 29634 ?corresponding author?. E-mail: ftestik@clemson. edu Note. This manuscript was submitted on May 6, 2009; approved on December 22, 2009; published online on February 4, 2010. Discussion period open until February 1, 2011; separate discussions must be submitted for individual papers. This technical note is part of the Journal of Waterway, Port, Coastal, and Ocean Engineering, Vol. 136, No. 5, September 1, 2010. ?ASCE, ISSN 0733-950X/2010/5-286–294/$25.00.

recent Indian Ocean tsunami of 2004 ?see Synolakis and Bernard 2006?, questioned the geophysical relevance of solitary wave theory in laboratory studies of tsunamis. Despite this recent controversy among tsunami researchers, the importance of accurate laboratory generation of solitary waves is evident in various other contexts including the study of different aspects of long wave theories and laboratory simulations of internal waves in strati?ed ?uid ?Miles 1980?. The methods for laboratory generation of solitary waves include dropping weights ?e.g., Russell 1845; Wiegel 1955?, releasing prescribed amount of water behind a barrier ?e.g., Kishi and Saeki 1966?, and horizontal movement of a vertical paddle by a piston-type wavemaker ?e.g., Hall and Watts 1953; Cam?eld and Street 1969?. Among these methods, solitary wave generation using piston-type wavemakers has been the most commonly employed method, due to its various advantages over the other methods such as the shorter formation distances of the solitary waves from the wavemaker ?hence, minimum length requirement for the laboratory tank? and the accuracy of the generated waves in terms of their resemblance to the aimed wave characteristics ?e.g., height, pro?le shape, and propagation characteristics?. Hall and Watts ?1953? used a piston-type wavemaker to generate solitary waves in the laboratory for the ?rst time. Their wavemaker consisted of a vertical paddle that is moved horizontally by a mechanical system. Cam?eld and Street ?1969? in subsequent work integrated a computer to such a wavemaker system and controlled the motion of the wave paddle via computer. Goring ?1979? derived the wave-paddle trajectory to generate a solitary wave by assuming the wave to be of permanent form during the generation process, and introduced this trajectory into a computer-controlled wavemaker to generate a solitary wave of desired characteristics. Goring’s methodology, outlined below,

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has been the primary method of solitary wave generation for the last three decades. This traditional method assumes that the horizontal water particle velocity adjacent to the wave paddle, ū, is equal to the wave-paddle velocity d? =? u??, t? dt ?1?

where ? = position of the wave paddle and t = elapsed time since the start of the motion. Moreover, ? u?? , t? is assumed to be constant throughout the water depth and equal to the depth-averaged velocity derived based on conservation of mass considerations by Svendsen ?1974?. This is expressed as for long waves of permanent form ? u??, t? = c???, t? h + ???, t? ?2?

where ??? , t? = free surface elevation above still water level; h = still water depth; and c = wave celerity. By solving Eqs. ?1? and ?2? simultaneously, one can obtain the paddle trajectory. However, this procedure requires an analytical de?nition for ? to be used in ?2?. Since solitary waves are asymptotic solutions of Euler equations, the required analytical de?nition for ? can be obtained from solutions of Euler equations for solitary wave pro?les. Even though there is no exact solution of Euler equations, there exist a number of approximate solutions ?e.g., ?rst-order approximate solution ?Boussinesq 1872; Rayleigh 1876?; second-order approximate solution ?Laitone 1960?; third-order approximate solution ?Grimshaw 1971?; and other higher order solutions?. The approximate solution for the solitary wave pro?le by both Boussinesq ?1872? and Rayleigh ?1876? is as follows: ? = H sec h2?k?ct ? ??? ?3? where H = amplitude of the solitary wave and k = boundary / outskirt decay coef?cient ?k = ?3H / ?4h3? = Boussinesq-type wave and k = ?3H / ?4h2?H + h?? = Rayleigh-type wave?. Celerity for both Boussinesq- and Rayleigh-type waves are de?ned as c = ?g?h + H? ?g = gravitational acceleration?. Using Eqs. ?1?–?3?, an implicit relationship for the wave-paddle trajectory, ??t?, for solitary wave generation can be derived as follows ?see Goring 1979?: ??t? = H tanh?k?ct ? ??? kh ?4?

Boussinesq-type waves and attributed this behavior to a better description of k in Rayleigh-type waves. In the laboratory generation of solitary waves, a depression of water with superposed oscillatory trailing waves that follows the hump ?i.e., solitary wave? subsequently forms. The amplitudes of these depressions are reported to be 6–12% of the heights of the generated Boussinesq-type waves ?Goring 1979? and 3–8% of the heights of the generated Rayleigh-type waves ?Guizien and Barthelemy 2002?. Goring ?1979? suggested reducing the depression amplitudes by increasing the duration of paddle motion up to 10%. Guizien and Barthelemy ?2002? noted that increasing the duration of paddle motion, T, for a constant stroke length, S, results in smaller wave celerities, c, and heights; hence the loss of accuracy. Guizien and Barthelemy ?2002? suggested the generation of Rayleigh-type waves to reduce depression amplitudes without such adjustments. The Goring methodology has been commonly used in many laboratory studies to generate solitary waves ?e.g., Synolakis 1987; Ramsden 1993; Jensen et al. 2003?. However, solitary waves generated using this methodology attain their established form after a considerable distance from the generation zone. Indeed, in many cases, their characteristics differ considerably from the characteristics of the aimed waves as described below. The main motivation behind the present study is to develop a new methodology ?henceforth, simply referred as “the New methodology”? that can be used to generate more accurate and established solitary waves in shorter distances. In the following sections, the theoretical basis for this New methodology, the experimental setup to test the capabilities of the New methodology, the results and conclusions of this study are presented in the same order.

Wave Generation Methodology

This section encompasses the main considerations and embedded assumptions in the development of the New methodology, and the derivation of the wave-paddle trajectory, ??t?, based on the New methodology. As alluded to by Synolakis ?1990?, Goring methodology does not consider the unsteady nature of the solitary wave generation process. This methodology rather assumes that a solitary wave of permanent shape forms even during the generation stage. However, in our New methodology, we consider the evolving nature of the generated solitary wave during the generation stage. It is assumed that as the wave-paddle moves horizontally and pushes the water column out in front, a small surge forms at each instant. These small surges pile up to eventually form the smooth pro?le of the proposed solitary wave. During the wave pro?le generation, the forming wave does not propagate with the constant celerity of the developed solitary wave as it is assumed in Goring methodology. The forming wave instead propagates with a celerity that changes with time, cu?t? ?henceforth, referred as “unsteady celerity”?. Therefore, there are two unknowns to be determined when calculating the wave-paddle trajectory: ? u?t? ?horizontal water particle velocity adjacent to the wave paddle or equally wave-paddle velocity, see ?1?? and cu?t?. To determine these two unknowns at each instant of the paddle motion, mass and momentum conservation equations must be solved. The selected control volume ?CV? for our analysis is illustrated in Fig. 1. In this sketch and to maintain clarity, only the initial surge formed by the push of the wave paddle is shown. The water column pushed by the paddle enters the CV through the control surface ?CS? indicated as 1 with a depth-averaged velocity of ? u1, and the ?uid leaves the CV through CS2 that is far away

Goring ?1979? aimed to generate Boussinesq-type solitary waves in a laboratory tank for the dimensionless wave heights, ? = H / h, ranging from 0.05 to 0.65. He reported that as the ? value increases, the agreement between the characteristics of the generated and the aimed waves decreases. This ?nding was later experimentally con?rmed by Guizien and Barthelemy ?2002?, and by our experiments in this present study. Goring ?1979? reported considerable attenuation for the generated Boussinesq-type solitary waves. He also noted that the amount of observed attenuation cannot be explained by frictional wave damping proposed by Keulegan ?1948?. Guizien and Barthelemy ?2002?, by using Goring methodology to generate Boussinesq- and Rayleigh-type solitary waves, observed that characteristics of the generated Rayleigh-type waves are in closer agreement to the aimed wave characteristics. Moreover, they reported the generated Rayleightype waves to be more stable and that damping of these waves can be explained by frictional wave damping. They also observed that Rayleigh-type waves reach their established form in a shorter distance ?x = 20h, x = distance from the wave paddle? than

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cu t !

sponding CS. The ?rst ?since water is incompressible? and second terms in Eq. ?5? can be expanded as

h+η

u:

u? h

?

d dt

?

?d ? = ?

cv

d dt

?

d? = ?

cv

cu??t b = ? c u? b ?t

?6? ?7?

:

Fig. 1. De?nition sketch for the considered CV ?dashed line?. Symbols: ū1 = depth-averaged velocity at the paddle; ū2 = depth-averaged velocity far away from the wave paddle; h = still water depth; ? = surge height; and cu?t? = surge celerity.

?UA = ?? u1A1 ? ?? u2A2 = ?? u1?h + ??b ? ?? u2hb ? cs

where ?t = incremental time interval; b = wave-tank width; and subscripts 1 and 2 denote quantities for CSs 1 and 2, respectively. By substituting Eqs. ?6? and ?7? into Eq. ?5?, it is possible to retrieve the mass conservation equation in the following form: ?2 cu? ? ?h + ??? u1 = hu ?8?

from the generation zone with a depth-averaged velocity of ? u 2. This initial surge of height ??t1? propagates with a celerity of cu?t1? over the water depth of h ?subscripts for time, t, denotes the instant of time?. At the next instant, a second surge forms on top of the initial surge and these two surges form a new larger surge of height ??t2? and celerity cu?t2?. This surge formation process continues, and at each instant a part of the leading half of the solitary wave is formed as illustrated in Fig. 2. The proposed New methodology assumes the formation of surges only for the formation of the leading half of the solitary wave crest. This methodology assumes a solitary wave of permanent shape that moves with a constant celerity ?calculated by substituting ? = H in Eq. ?15? below? for the formation of the trailing one-half of the solitary wave pro?le. The integral form of the mass conservation equation for the CV presented in Fig. 1 can be written as follows ?see also Chaudhry 2008?: d dt

Similarly, the integral form of the momentum conservation equation for the CV presented in Fig. 1 can be expressed as U?UA + ? Fext = ? dt cs d

?

?Ud ?

?9?

cv

where ?Fext represents the summation of external forces acting on the system. Assuming that hydrostatic pressure distribution occurs at both CS1 and CS2, the summation of the external forces acting on the system is

? Fext = ?b

?h + ??2 h2 ? ?b 2 2

?10?

where ? = specific weight of water. The ?rst and second terms on the right side of Eq. ?9? can be expanded as

2 ? U 2A = ? u 2 ? 2bh ? ?u1b?? + h? cs

?

?d ? +

cv

? cs

?UA = 0

?5? d dt

?11?

where ? = density of water; ? = volume of the CV; U = mean ?ow velocity at the corresponding CS; and A = flow area at the corre-

?

?Ud ? = ?bcu?? u 1? h + ? ? ? ? u 2h ?

?12?

cv

By substituting Eqs. ?10?–?12? into Eq. ?9?, one would obtain the momentum conservation equation in the following form:

cuAtFB ηAtFB h cuAtDB cuAtEB ηAtDB ηAtEB

g? ?2 ?2h + ?? = hu u2 u1?h + ?? ? cu? u 2h 2 ? ?h + ??? 1 + cu? 2

?13?

h

h

A)B

A9B

A(B

By selecting a long CV such that CS2 is far away from the wave paddle ?hence, ? u2 = 0?, Eq. ?8? simpli?es to the form of mass conservation equation in Eq. ?2? given for long waves of permanent form. Therefore, depth-averaged ?ow velocity at CS1, which is equal to the wave-paddle velocity, is obtained as ? u1 = c u? ?h + ?? ?14?

H

h

u2 = 0 into Eq. ?13?, unsteady By substituting ? u1 from Eq. ?14? and ? celerity can be obtained as a function of surge height c u? t ? =

A2B

Fig. 2. Theoretical illustration of the New methodology for generation of solitary waves. Vertical dark block—wave paddle; horizontal solid line—wave-tank bottom; dashed line—surge boundary and still water level; curved solid line—generated solitary wave boundary; and horizontal arrow—direction of surge propagation. Graphs in ?a?– ?c? are three sequential instances illustrating the formation of sequential surges and graph in ?d? illustrates the formation of the leading half of the solitary wave by the sequential surges.

? ? ?? ?

g h+ ? 2 1+ ? h

?15?

Note that the surge height, ?, appearing on the right hand side of Eq. ?15?, which is the solitary wave elevation at the particular time ?see Eq. ?3? for Boussinesq- and Rayleigh-type wave pro?les?, is a function of the steady celerity of the solitary wave, c, ?c = ?g?h + H? for both Boussinesq- and Rayleigh-type waves?. From Eqs. ?14? and ?15?, the paddle velocity, ? u1, can be calculated from the following parameterization:

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Fig. 3. Schematic of the experimental setup. Numbers 1–5 represent the position of wave gauges along the wave tank that are 1 m apart.

? u1 =

d? = dt

? ? ?? ?

g ? ? h+ h 2 ? h+?

?16?

Once the wave-paddle velocity is determined at each instant, wave-paddle trajectory, ??t?, is calculated by integrating Eq. ?16?. Since wave-paddle position, ?, appears on both sides of Eq. ?16? ?? is a function of ?; i.e., ? = H sec h2 ?k?ct ? ??? for Boussinesq- and Rayleigh-type wave pro?les?, Eq. ?16? is an implicit equation and should be solved numerically. Eq. ?16? can be discretized by a simple ?nite difference method to ?nd ? at each time step. By following Goring ?1979? and Guizien and Barthelemy ?2002?, duration of paddle motion and the paddle stroke length are calculated using the analytical parameterizations ?T = 7.8 / kc + S / c and S = 2H / kh? derived by Goring ?1979?. Since the k value is unchanged in the New methodology, the paddle stroke length is the same as the one that is used in Goring methodology. To avoid complexities and to be able to make direct comparisons with the waves generated using Goring methodology, T values are calculated using the steady celerity values as in Goring methodology.

Experimental Setup

Experiments were conducted in a rectangular wave tank ?12 m ? 0.6 m ? 0.6 m? located at the Flow Physics Laboratory of Clemson University. The wave tank is composed of steel-framed rectangular modules with Plexiglas side walls and bottom for visualization purposes ?see schematic in Fig. 3?. Waves are generated by the horizontal movement of a vertical aluminum plate ?i.e., wave paddle? in the ?rst module of the tank. The wave generator functions with high accuracy and repeatability as demonstrated in Fig. 4. The wave paddle is sealed against the Plexiglas side walls of the tank by means of rubber wiper blades to minimize leakage of water during the paddle motion. The top of the wave paddle is attached to the slide of a high-precision linear actuator system that moves in the horizontal direction with a precision of 0.1 mm. The linear actuator has a maximum stroke length of 1.5 m and is capable of moving the wave paddle with velocities and accelerations up to 1.5 m/s and 10 m / s2, respectively. The actuator is powered by a servoelectric motor and commanded by a controller that is programmed using a personal computer. The principal measurement of interest is the water surface elevation pro?les. Water elevation data were collected using capacitance-type wave gauges that can sample data with up to 50-Hz acquisition frequency and with an accuracy of 0.001 m and

a measurement range of 0.005–1 m. For each experimental condition, wave elevation data were recorded at ?xed stations along the tank located from x = 2 – 6 m in equal steps of 1 m as illustrated in Fig. 3. Photographs of solitary waves were taken using a standard digital still camera and the solitary wave generation process was visualized using a high-speed video camera that is capable of recording up to 1,000 frames per second. A laser displacement sensor ?LDS? was used to monitor the wave-paddle position and velocity for each experimental run to verify the accuracy of the paddle motion. The LDS emits a laser beam that re?ects off any solid surface within its measurement range and calculates the distance with an accuracy of ?0.1 mm. Solitary wave generation experiments were conducted for the still water depth, h, of 20 cm that is uniform along the tank up to the steep sandy beach at the end of the tank in order to dampen the waves ?see Fig. 3?. In the experiments, solitary waves with dimensionless wave height values, ? = H / h, ranging from 0.3 to 0.6 were generated. Two different solitary wave generation methodologies, Goring and our New methodologies, were employed to generate waves based on two different solitary wave pro?les proposed by Boussinesq ?1872? and Rayleigh ?1876?. Photograph of a typical solitary wave generated using the New methodology is presented in Fig. 5. A total of four sets of experiments ?Rayleightype waves using Goring methodology ?GR?, Boussinesq-type waves using Goring methodology ?GB?, Rayleigh-type waves using the New methodology ?NR?, and Boussinesq-type waves using the New methodology ?NB?? were conducted. Experimental conditions are summarized in Table 1.

Results

In this section, characteristics of generated solitary waves using the proposed New methodology are presented. Comparisons with the waves generated using the traditional solitary wave generation methodology by Goring are included to demonstrate the capabilities of the New methodology. Because of space limitations, only comparisons between NR and GR are presented and comparisons with NB and GB are brie?y discussed. Main solitary wave characteristics of interest are pro?le shape, amplitude, establishment, and propagation speed. In the results presented below, the characteristics of the aimed Boussinesq- and Rayleigh-type solitary waves served as the benchmark, and the conclusions on the capabilities of the New methodology as well as the Goring methodology are drawn based on the degree of resemblance of the generated wave characteristics to the characteristics of the aimed solitary waves.

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Fig. 4. Accuracy and repeatability of the wave generator. Comparisons of the theoretical ?---? and the actual ?—? ?a? trajectory and ?b? velocity of the wave paddle. Percent errors in ?c? the actual trajectories and ?d? velocities of the wave paddle for generating the same solitary wave pro?le 20 times. The actual paddle trajectory is measured by LDS and the actual paddle velocity is calculated based on the trajectory measurements. GR waves are generated for h = 0.2 m and ? = 0.6.

Comparisons of the generated Rayleigh-type solitary wave pro?les measured at x = 10h are presented in Fig. 6. As can be seen from this ?gure, pro?les of the NR waves clearly resemble the aimed Rayleigh-type solitary wave pro?les closer than the GR waves for all ? values studied. Moreover, these ?gures along with the ?gures that are discussed below indicate that waves generated by using the New methodology can be considered as established solitary waves even at x = 10h. Wave pro?les generated by both methods resemble the aimed Rayleigh-type solitary wave pro?les closer than the aimed Boussinesq-type solitary wave pro?les. This observation is also reported by Guizien and Barthelemy ?2002? in the case of Goring methodology. The resemblance of the generated wave pro?les to the aimed Rayleigh-type solitary wave pro?les is better for larger ? values ?Fig. 6?, whereas in agreement with Goring ?1979?, the resemblance of the generated wave pro?les to the aimed Boussinesq-type wave pro?les is better for smaller ? values.

In Fig. 6, the hump of the generated wave is followed by a long water depression below the still water level that consists of a series of dispersive oscillatory trailing waves. Solitary waves generated using the New methodology are followed by smaller

Table 1. Experimental Conditions Exp. run number Methodology/ wave pro?le ?=H/h S ?m? T ?s?

Fig. 5. Photograph of a Rayleigh-type solitary wave generated by the New method. Experimental conditions: ? = 0.6; h = 20 cm; and x = 15h.

1 GB 0.3 0.25 2.16 2 GB 0.4 0.29 1.85 3 GB 0.5 0.33 1.64 4 GB 0.6 0.36 1.48 5 GR 0.3 0.29 2.47 6 GR 0.4 0.35 2.19 7 GR 0.5 0.40 2.01 8 GR 0.6 0.45 1.87 9 NB 0.3 0.25 2.16 10 NB 0.4 0.29 1.85 11 NB 0.5 0.33 1.64 12 NB 0.6 0.36 1.48 13 NR 0.3 0.29 2.47 14 NR 0.4 0.35 2.19 15 NR 0.5 0.40 2.01 16 NR 0.6 0.45 1.87 Note: Water depth, h, was 0.2 m in all experiments ?GB= Boussinesqtype waves using the Goring methodology; GR= Rayleigh-type waves using the Goring methodology; NB= Boussinesq-type waves using the New methodology; NR= Rayleigh-type waves using the New methodology?; S = paddle stroke length; T = duration of paddle motion; H = wave height; and ? = dimensionless wave height.

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? @ ;J

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Fig. 6. Comparison of the GR ?—? and the NR ?—? wave pro?les with the theoretical Rayleigh-type solitary wave pro?les ?---?. Water surface pro?les are recorded at x = 10h for h = 0.2 m and ? values ?a? 0.3; ?b? 0.4; ?c? 0.5; and ?d? 0.6. Graphs are plotted for the dimensionless water surface elevation, ? / H, versus the dimensionless time, t?g / h.

depressions, compared to those generated by the Goring methodology. Moreover, Rayleigh-type solitary waves generated by either method are associated with smaller depressions compared to Boussinesq-type solitary waves generated using the same methodology. Relative amplitudes of the depressions ?ratio of the depression amplitude and the crest height? observed at x = 10h and 30h in each experimental run are tabulated in Table 2. Amplitudes

Table 2. Relative Depression Amplitudes and Frequency Change Change of ?exp / ?theo from x = 10h – 30h ?%?

Exp. run number

at / Hexp at x = 10h

at / Hexp at x = 30h

1 0.070 0.060 8.5 2 0.078 0.070 8.5 3 0.084 0.08 8.3 4 0.060 0.057 7.6 5 0.087 0.070 10.2 6 0.083 0.065 8.4 7 0.08 0.064 7.7 8 0.09 0.08 7.2 9 0.068 0.060 6.7 10 0.076 0.071 4.4 11 0.083 0.079 3.4 12 0.055 0.052 3.4 13 0.062 0.045 7.5 14 0.044 0.038 5.5 15 0.040 0.035 5.1 16 0.045 0.041 4.7 Note: Symbols: at = depression amplitude; Hexp = experimental wave height; ?exp = experimental frequency; and ?theo = theoretical frequency.

of the depressions decrease gradually as the waves propagate along the wave tank. Furthermore, since the depression celerities are smaller than the hump celerities, depressions separate from the humps gradually as the waves propagate along the tank. This separation process can be considered as a progress toward an established solitary wave form that consists of only a hump. However, the longer the distance it takes for the separation to occur, the more dispersive effects act on the hump, changing its outskirt pro?le and reducing its celerity. The separation of depressions from the humps occurs in a shorter distance for the NR waves compared to the GR waves. Consequently, compared to the wave height values of the GR waves, wave height values of the NR waves are closer to the wave height values of the aimed waves ?see Fig. 7?. The separation process occurs over a shorter distance for the generated waves with larger ? values, since the celerities

1.0

Hexp /Htheo

0.8

0.6

0.4 0.2 0.3 0.4

ε

0.5

0.6

0.7

Fig. 7. Comparison of theoretical wave height values for the aimed Rayleigh-type waves ?Htheo? and experimental ?Hexp, measured at x = 10h? wave heights for NR ??? and GR ??? waves

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HG

Fig. 8. Attenuation of generated Rayleigh-type waves with distance and attenuation predictions by the Keulegan ?1948? formulation. Waves generated using the New methodology ??? and the corresponding Keulegan predictions ?—?; waves generated using Goring’s methodology ??? and the corresponding Keulegan predictions ?- - -? for ? = ?a? 0.3; ?b? 0.4; ?c? 0.5; and ?d? 0.6.

of the generated humps are larger. Consequently, as the ? value increases, the agreement between the characteristics ?wave height ?Fig. 7?, pro?le shape ?Fig. 9, later?, and celerity ?Fig. 10, later?? of the generated and the aimed waves improve for both of the generation methodologies. Moreover, depressions are detached from the hump in shorter distances for generated Rayleigh-type waves as compared to the generated Boussinesq-type waves for the same experimental conditions. An important consideration for large-scale laboratory studies where waves propagate tens of meters before reaching to the measurement station is the change in the pro?le characteristics of the generated waves. To examine these characteristics of the generated waves, the attenuation trends for the heights and frequencies of the generated waves as they propagate along the tank are investigated. For a solitary wave propagating in the wave tank, in addition to the wave damping inherent to the generation of waves, frictional wave damping occurs due to sidewalls and bottom wall. In order to assess the wave attenuation characteristics of the generated waves solely due to the generation mechanism in the absence of frictional damping, a comparison between the observed attenuation of the generated waves and the predicted attenuation of the solitary waves only due to frictional damping calculated using the parameterization by Keulegan ?1948? is given in Fig. 8. In this ?gure, reference wave height is selected to be the experimental wave height value recorded by the ?rst wave gauge at

x = 10h ?denoted as ?Hexp?x=10h?, and the relative experimental wave heights, Hexp / ?Hexp?x=10h, at different distances along the wave tank are presented for both NR and GR waves. In the graphs given in this ?gure, the deviation of the relative experimental wave height values ?symbols? from the wave attenuation predictions due to frictional damping ?dashed lines? may be interpreted to infer the establishment characteristics of the generated waves, i.e., the less deviation from the predictions the more established generated waves. As can be clearly seen from these graphs, the waves generated using the New methodology follow Keulegan’s predictions for the frictional damping better and undergo less attenuation compared to those generated by the Goring methodology; hence, indicating that less dispersive waves are generated using the New methodology. This observation may be explained by the fact that the wave pro?les generated using the New methodology have smaller depression amplitudes as well as demonstrate faster separation of the depressions and the humps compared to the waves generated using the Goring methodology. It is also observed that heights of Rayleigh-type waves attenuate less than heights of Boussinesq-type waves. This observation supports the observation of Guizien and Barthelemy ?2002? that attenuation of Rayleigh-type waves follows Keulegan’s predictions better than Boussinesq-type waves which undergo severe damping that cannot be explained only by friction. Guizien and Barthelemy ?2002? attributed this difference in the observed wave attenuation characteristics to different de?nitions of outskirt decay coef?cient, k, in Boussinesq- and Rayleigh-type waves. Another reason for the observed difference is the difference in the characteristics of the depressions formed in the laboratory generation of Boussinesq- and Rayleigh-type waves. As mentioned earlier, depressions have larger amplitudes and are attached to the humps for a longer distance for Boussinesq-type waves compared to Rayleigh-type waves. Therefore, depressions are expected to have more pronounced dispersive effects on Boussinesq-type waves. The second process of interest in assessing the dispersive characteristics of the generated waves is the change in wave pro?le shape as the waves propagate along the tank. The parameter that characterizes the solitary wave pro?le shape is the solitary wave frequency, ? = kc. In order to calculate the frequencies of the generated solitary wave pro?les, experimental wave elevation data are ?tted by Eq. ?17? below, which has the same functional form as elevation pro?les of the Boussinesq- and Rayleigh-type waves given in Eq. ?3?. Since wave gauge measurements are discrete and celerities of the generated waves ?1.47–1.77 m/s? are large, maximum wave elevation values measured do not necessarily correspond to the generated wave height values and the recorded wave elevation pro?les are not necessarily symmetrical with respect to the recorded highest wave elevations. Therefore, there are three unknowns in Eq. ?17? ??reg; Hreg; and time ?tting parameter, t0?, which are determined using the least-squares method ?exp = Hreg sec h2??reg?t ? t0?? ?17?

?reg, determined from the ?tting analysis, is considered as the actual experimental solitary wave frequency, and hereafter is referred to as ?exp. Eq. ?17? is ?tted to the top 95% ?i.e., wave elevation values larger than 5% of the wave height? of the recorded wave pro?le. The reason for choosing the top 95% is that the depression with the trailing waves following the hump contaminates the smooth experimental solitary wave pro?le approximately within the bottom 5% of the wave pro?le. In Fig. 9, ratio of the experimental solitary wave frequencies, ?exp, and their

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I !A

I !A

$ !"%

" #/

A !G

" #/

$ !"%

A !H

A !H

A !G

A !D

3a 8

IA IF JA JF BA

A !D

3b 8

IA IF JA JF BA

/!

I !A I !A

/!

$ !"%

" #/

A !G

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A !H

A !H

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A !D

3c8

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A !D

3d 8

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Fig. 9. Solitary wave frequency ratios ?experimental to theoretical? along the wave tank for NR ??? and GR ??? waves for ? = ?a? 0.3; ?b? 0.4; ?c? 0.5; and ?d? 0.6

theoretical values, ?theo ??theo = ?3Hg / ?4h2? for Rayleigh-type waves? at different positions along the tank are given. Note that the values of these ratios closer to unity indicate the closer resemblance of the generated wave pro?les to the aimed theoretical solitary wave pro?les and the smaller changes in these ratio values along the tank indicate less dispersion of the generated waves. Therefore, from Fig. 9, it can be concluded that pro?les of the waves generated using the New methodology resemble the aimed wave pro?les closer and disperse less as they propagate along the tank compared to the ones generated by the Goring methodology for all the experimental conditions studied. The change in the ratios of the experimental and theoretical solitary wave frequencies from x = 10h to x = 30h for each experimental run are tabulated in Table 2. Finally, the average experimental celerities of the generated waves, cexp, are compared with the theoretical celerities of the aimed solitary waves, ctheo, as shown in Fig. 10. Average experi-

mental celerities are determined from the wave gauge recordings at x = 10h and x = 30h by dividing the distance traveled by the hump by the time elapsed during this travel. As can be seen from this ?gure, like the salient characteristics discussed above, the celerities of the waves generated by the New methodology are in better agreement with the aimed wave celerities as compared to the celerities of the waves generated by the Goring methodology. Moreover, celerities of the generated Rayleigh-type waves demonstrated a better agreement with the celerities of the aimed waves in comparison to Boussinesq-type waves.

Conclusions

As pointed out by Synolakis ?1990?, the solitary wave generation methodology by Goring ?1979? considers a solitary wave of permanent form during the generation process, which leads to a compromise in the accuracy of the generated waves. In this study, we developed a new methodology for solitary wave generation using computer-controlled piston-type wavemakers by considering the evolving nature of the wave during the generation process. Taking into account this evolving nature, one has to determine the timedependent celerity of the wave in order to calculate the wavepaddle trajectory. By using the conservation of momentum principle along with the conservation of mass principle, we were able to propose a time-dependent solitary wave celerity during the generation process and the wave-paddle motion trajectory. Developed methodology is tested by conducting a series of laboratory experiments in a wave tank with h = 20 cm for ? values ranging from 0.3 to 0.6. In the experiments, Boussinesq- and Rayleigh-type waves are generated by the New and the Goring ?used as a benchmark? methodologies, and experimental observations for various wave characteristics are compared with the char-

@3<

Cexp /Ctheo

<3?

<3>

<3= <3A <3B <3=

ε

<3C

<3>

<3D

Fig. 10. Experimental to theoretical solitary wave celerity ratio versus dimensionless wave height for NR ??? and GR ??? waves

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acteristics of the aimed solitary waves. For the range of experimental conditions studied, these comparisons indicated that the resemblance of the waves generated using the New methodology to the aimed solitary waves is superior than the waves generated by the Goring methodology. Hence, it is concluded that the New methodology is capable of generating more accurate solitary waves in terms of wave height, pro?le shape, and celerity. Moreover, observed wave attenuation and pro?le change characteristics of the generated waves along the wave tank indicate that the nonlinear effects balance the dispersive effects better for the waves generated by the New methodology compared to the waves generated by the Goring methodology. Rayleigh-type waves generated using either of the generation methodologies are observed to resemble the aimed waves better than the Boussinesq-type waves. Developed solitary wave generation methodology is expected to be useful in various laboratory studies involving solitary waves. This methodology will enable solitary wave studies in both short ?note that the experimental component of this study is conducted in a wave tank of 12 m? and long laboratory setups as the generated waves approach to their permanent form faster and they are less dispersive during their propagation compared to the waves generated by the traditional solitary wave generation methodology developed by Goring ?1979?.

Acknowledgments

This research was supported by the funds provided by College of Engineering and Science at Clemson University to the second writer.

References

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