Accuracy of solitary wave generation by a piston wave maker.

A new experimental procedure to generate solitary waves in a flume using a piston type wave maker is derived from Rayleigh's (1876, [18]) solitary wave solution. Resulting solitary waves fordimensionless amplitudes £ ranging from 0.05 to 0.5 are as pure as the ones generated using Goring's (1978, [7]) procedure which is based on Boussinesq (1871a, [1]) solitary wave, with trailing waves of amplitude lower than 3 % of the main pulse amplitude. In contrast with Goring's procedure, the new procedure results in very little loss of amplitude in the initial stage of the propagation of the solitary waves. We show that solitary waves generated using this new procedure are more rapidly established. This is attributed to the better description of the outskirts decay coefficient in a solitary wave given by Rayleigh's solution rather than by a Boussinesq expression. Two other generation procedures based on first-order (KdV) and second order shallow water theories are also tested. Solitary waves generated by the latter are of much lower quality than those generated with Rayleigh or Boussinesq-based procedures.


Introduction
The aim of this study is to assess solitary wave generation proce dures. Hammack and Segur (1974, [10]) showed experimentally and theoretically that from any net positive volume of water above the still water level, at least one solitary wave will emerge followed by a train of (dispersive) waves. Thus, different proce dures have been used to generate solitary waves. Scott Russel (1845, [20]) generated solitary waves by allowing a solid weight to fall from near the surface to the bottom of a tank. Similarly, Daily and Stephan (1952, [6]) displaced a given mass of water by the vertical motion of a piston rising from the bottom of a tank. They also tested a method consisting in releasing a mass of water behind a moveable banier at one end of a flume. Goring (1978, [7]) studied in detail the generation of solitary waves using a pis ton-type wave maker.
Our concern in this study is to generate solitary wave as 'pure' as possible. This means we have focussed our efforts in generating waves with minimised trailing waves but also of stable amplitude during propagation. These concerns have scientific practical im portance. For instance, the study of solitary wave reflection (Renouard et al., 1985 [19]) or of interaction of either solitary waves with solitary waves or monochromatic waves with solitary waves (Guizien and Barthélemy, 2000 [9]) require waves as pure Revision received June 7, 2001. Open for discussion till October 31, 2002. as possible, especially if phase shifts are to be measured. Since long waves are associated to quasi vertically uniform hori zontal velocity, the piston-type wave maker seems a natural gen eration device. However it is technically more difficult to set up than the other generation means cited above. Without paying much attention to the way in which the generator is displaced (mass falling or rising, barrier, wave maker), the success of all these methods is based on the fact that the largest solitary wave outruns any transient dispersive disturbance and also any other solitary wave of lower amplitude for sufficient propagation dis tances. Indeed, as the dimensionless amplitude of a solitary wave increases up to e = 0.796 (e = A/h () where A is the solitary wave amplitude and h 0 is the mean water depth), its phase speed in creases. However, above this value, the phase speed tends to de crease (Longuet-Higgins and Fenton, 1974 [15], Byatt-Smith and Longuet-Higgins, 1976 [4]). Regarding generation, this feature raises a major problem, as highlighted by Longuet-Higgins (1981, [14]) because, together with the phase speed, the total displaced mass decreases over the upper range of solitary wave amplitude. Hence, there are actually two waves of different amplitude with the same displaced mass. Thus, using the aforementioned proce dures, two distinct solitary waves may emerge. However, in most applications, generating the largest solitary wave is not so impor tant.
The usual procedure for long-wave and more specifically solitarywave generation consists in matching the paddle velocity at each position in time with the vertically averaged horizontal velocity of the wave. Mathematically this is expressed in the following way: where \=X is the paddle position along the x-axis and u(x,t) is the long wave depth-averaged horizontal velocity in the labora tory frame of reference. The X-axis is the centreline of the flume, taking the origin x = 0 at the back position of the piston stroke as defined on figure 1. This contrasts with the approach taken by Synolakis (1990, [23]) who solved an inverse evolution problem of the Korteweg-De Vries (KdV) equation in order to generate arbitrary long waves at any location in a horizontal flume. In the present paper, we discuss Goring's procedure which is based on Boussinesq (1871a, [1 ]) solitary wave solution in com parison with four generation laws derived from other existing solitary wave solutions. Two generation laws are obtained using Rayleigh's (1876, [18]) solitary wave solution. Indeed, by assum ing a small displacement, an analytical law of motion can be de rived from the integration of equation (1) in which the Rayleigh solitary wave solution is used. This analytical law of motion is compared with a law of motion obtained after fitting the numeri cal integration of equation (1), also using Rayleigh's solution, by a hyperbolic tangent. The other two laws of motion tested in this study are derived from linearization of the Lagrangian formula tion in the first-order (or Korteweg and De Vries) and secondorder shallow water theories (Temperville, 1985 [24]). In our ex periments, we use a very similar device to the one used by Gor ing, that is described in section 3. In section 4, performances of the tested laws of motion are compared. In order to understand the good performance of both Goring procedure and the law of motion based on Rayleigh's solitary wave solution, paddle laws of motion are discussed in section 5 in comparison with the one deduced from Byatt-Smith's (1970, [3]) numerical solution.

Wave maker laws of motion
In this section, we present the different laws of motion to be pre scribed to the paddle in order to generate a solitary wave. A soli tary wave is a steady solution in the wave co-moving frame trav elling at the wave phase speed c. Hence, equation (1), that gives the paddle position X in the laboratory frame of reference, can be written after a change of variables from (x,t) to (6 = ct -X,t) in the general form: dX_ dQ -umxy) (2) In equation (2), the solitary wave depth-averaged horizontal ve locity can be given by various theories. Amongst them, the Boussinesq (1871 a, [ 1 ]) and Rayleigh (1876, [18]) solitary waves have the following same functional form: where A is the solitary wave amplitude, h Q the mean water depth, c the phase speed, p the outskirts decay coefficient, T)(8) the free surface elevation from rest and w(6) the depth-averaged horizon tal velocity. The two solitary wave expressions differ in terms of the values of c and p. Then, integrating (2) with (3) and (4) yields: 2/1 X(t) = -tanh[p(rt-X(0)/2] From (5), the total stroke of the paddle S can be deduced: The duration! of the paddle motion can be determined after trun cation of the infinite theoretical law of motion: second set resistive probes positions

Goring/ Boussinesq
In his study, Goring (1978, [7]) first tackled the question of minimising trailing waves in solitary waves generation. He chose a procedure based on the Boussinesq (1871a, [1]) solitary wave expression. Following Daily and Stephan's (1952, [6]) conclu sions regarding the best agreement of this expression with experi ments, he chose the following values for c and p in (5): 2 V4hl (8) In his study, Goring used both a paddle law of motion derived from the full equation (5) and a simplified paddle law of motion taking u(X,t) to be equal to «(0,/) in equation (1). This yields the following expression: where 5=4 3 is the total stroke of the paddle.
These two laws of motion were prescribed for the paddle by means of a hydraulic servo-system. The solitary waves generated experimentally using the simplified procedure were followed by a dispersive tail of 10 % of the amplitude of the main pulse. Us ing the paddle law of motion derived from the full equation, trail ing wave amplitude is drastically reduced. Yet, it was noted that by taking a duration of motion 10 % longer than that given by (7) with (8) and (9), the amplitude of the trailing dispersive waves was even more reduced. The two experimental records plotted by Goring ( [7], pp 123 and 130) show nearly pure solitary waves generated using this longer duration. Yet, the resulting solitary waves show a rapid decrease in amplitude, larger than what can be attributed to friction.

Shallow water second order
In the same way and following calculations from Temperville (1985, [24]), a second-order shallow-water paddle law of motion can be derived. For a solitary wave of amplitude A, the paddle law of motion X SH -,(t) is: The paddle stroke is thus S SK2 = 4 In practice this law is truncated with respect to the precision of experimental device and the last term can be neglected.

Rayleigh
Shallow water approximation relies on both long waves and small amplitude assumptions. Avoiding the latter restriction of small amplitude, it is possible to derive a set of equations for non linear waves (Mei, 1992 [16]). Indeed, assuming the Ursell number U r =e/a to be of order 1 (e=o~ with a=h {) /A where A=2/p is a hori zontal length scale of the solitary wave), Whitham (1974, [25]) derived the Boussinesq equations (see eq. (13.101) in Whitham, 1974 [25]). Without making this assumption, namely allowing e to be of order 1, the following set of equations is derived after truncating at the order a 4 ( Serre, 1953 [21], Su and Gardner, 1969 [22]):

Shallow water first order/KdV
Clamond and Germain (1999, [5]) used a paddle law of motion X Kllv (t) derived from the KdV (or first-order shallow water) solu tion and written: where P s (10) stroke is thus 5 + --. The paddle 11 , the same as that derived from Boussinesq's solitary wave form in Goring's procedure. This is obtained from calculations in Lagrangian form after linearization around X = 0. (14) Serre (1953, [21]) found a solitary wave solution for this set of equations which is actually the solitary wave solution that Ray leigh (1876, [18], see also Lamb, 1932 [ 13]) found for the steady progressive solution described by Russel and has the form (3) taking: It should be noted here that Rayleigh's solitary wave solution dif fers from the conjectured Boussinesq solitary wave form by the outskirts decay coefficient P (Miles, 1980 [17]). The paddle posi tion X is thus given by equation (5) incorporating (15) and (16).

Hence, the paddle stroke is given by 5 = 4 A{A + h")
Equation (5) together with (15) and (16) can be solved either nu merically or, if small displacements are assumed, explicitly after linearization: A hyperbolic tangent function of the form X(t) = a tanh(8f) can be fitted in the least square sense on the numerical integration of equation (5), where c and p are given by (15) and (16). The coef ficients a and P corresponding to the experiments are reported in table 1.

--
For practical purposes, the paddle laws of motion are all truncated using the following same criterion, namely by imposing tanh(5/) = 0.9999. Figure 2 shows these different laws of motion for the same expected resulting solitary wave. The two paddle laws of motion derived from Rayleigh's solution lead to larger paddle displacements than paddle laws of motion derived from KdV or second-order shallow-water theory. As paddle displace ments are limited by jack length, the maximum dimensionless amplitude generated using paddle laws of motion derived from Rayleigh's solution would be smaller than those derived from KdV.

Experiments description
The  ton is linked to a hydraulic jack capable of a 550 mm stroke. The control system is monitored by a computer. Different motions of the paddle can be prescribed by the computer, enabling the gener ation of either solitary waves or sinusoidal waves. Nevertheless the piston-type wave maker is more appropriate for long wave generation since it displaces the entire water column uniformly. However, we need to prescribe the appropriate law of motion for the paddle in order to produce solitary waves that are as pure as possible. The different laws of motion detailed in the previous sections are tested. A solitary wave of expected amplitude A is generated using each law. The two Rayleigh laws require larger paddle displacement than the shallow-water laws. With regard to the finite stroke of the jack, the laws deduced from the Rayleigh solution lead to smaller solitary wave amplitudes. Probe precision is estimated to be 0.5 mm for free surface elevations lower than 5 cm and at 1 mm beyond this limit. This is due to probe calibration (Guizien, 1998 [8]). This means that the relative error in the dimensionless amplitude is about 3 % for the smallest solitary waves (e = 0.05) and is less than 2 % for the others.   All solitary waves generated using a paddle law of motion de rived from KdV or second-order shallow water theory exhibit a trough (dispersive wave) after the main pulse whereas using pad dle laws of motion derived from Rayleigh, a bump (maybe a smaller solitary wave) trails it. It is noticeable that the bumps are of smaller amplitude than the troughs, except for the largest waves with e = 0.6 for h () = 0.2 m. Indeed, this case behaves dif ferently. There is sometimes breaking or two solitary waves are clearly generated for laws of motion giving good results for lower amplitudes and the worst law of motion suddenly gives better results. This latter feature was already observed for e = 0.5 with the second-order shallow water procedure. In fact, the limit of the dynamic servo-controller is reached for such waves. The discrep ancy between the prescribed and actual paddle motion is no lon ger negligible, due to the very large acceleration required. On figure 5, the paddle laws of motion corresponding to the hyper bolic tangent function fitted to a Rayleigh numerical integration are plotted for e ranging from 0.1 to 0.6. Greatest accelerations occur somewhere between the beginning of motion (zero veloc ity) and mid-stroke (maximum velocity). Qualitatively, the larger the maximum velocity and the shorter the duration of motion, the greater the acceleration. From figure 5, it may be expected that the maximum acceleration will increase with the amplitude of the solitary wave and may reach the limit of the servo-controller. From figure 2, it may also be noted that for a prescribed ampli tude, the law of motion based on the second-order shallow water theory requires greater acceleration than the other procedures. This explains why for this law of motion the servo-controller limit is already reached when e = 0.5. Thus, we will exclude from our discussion the e = 0.6 experiment for all laws and the e = 0.5 experiment for the second-order shallow water theory based pro cedure. Indeed, except for these cases, the actual paddle law of motion is sufficiently close to the prescribed one.

Trailing waves
Confident in the assumption that the system is capable of follow- ing the desired law of motion, our interest is to minimise the trail ing waves. On fig. 6.a and .b, we present the amplitude a of these bumps or troughs with respect to the main pulse amplitude measured at x = 67h 0 (resp. x = \00h 0 ) away from the paddle end of the flume for h Q = 0.3 m (resp. h {) = 0.2 m). The trough ampli tudes (on average 5 %) are sometimes more than twice those of the bumps (on average 3 %). We also discuss the four procedures tested in our experiments in comparison to Goring's (1978, [7]). The

Main pulse stability
By reproducing the same generation sequence several times, it becomes clear that the generated wave is highly reproducible, so that for a given paddle law, the solitary wave amplitude could be determined at any location in the flume together with the relative size of the trailing waves given the probe accuracy. During the second set of experiments, we studied the changes occurring in the first moments of propagation of the solitary wave emerging from the KdV and Rayleigh solutions, between x = 17/? () and x = 83ft 0 for h 0 = 0.3 m (see figure 8). In the first and second set of experiments, the amplitude of solitary waves generated using KdV or the second-order shallow water theory clearly decreases more as it propagates than it does when using Rayleigh. This can not be attributed only to damping by viscous friction as estimated by the Keulegan formula (1948, [ 11]) although this formula gives a very good prediction of this dissipation, as tested by Renouard et al. (1985, [19]). Indeed, from Keulegan's formula, the ampli tude decrease due to viscous dissipation over Ax = 45/z () would be 3.3 % whereas the measured damping over the same distance for solitary waves generated using KdV or the second-order shallow water theory varies between 5 and 6 %. For a solitary wave gen erated using Rayleigh, this damping is no more than 3 %, and on average 1. lower that the prescribed amplitude e = 0.2), Goring gives a damping over Ax = 100A 0 (between x = 10/? () and x = 1 \0h {) ) of 17 %, whereas the estimation from Keulegan's formula is 11 %. In similar conditions (e = 0.2, h () = 0.2 m), our experiments give a damping over At = 67.5/? () for solitary waves generated using Rayleigh-based procedures of 5.5 % (4.6 % for the tanh fit and 6.2 % for the linearization), and 8 % for shallow water based pro cedures, while Keulegan's formula predicts a viscous damping of 4.8 %. Figure 9 shows the loss of amplitude for all the sets of ex periments performed. These plots contain values given for all the generation laws tested. The trend discussed above appears very clearly. Rayleigh based procedures (and especially the numerical one) result in solitary wave primary pulses whose amplitude de creases less than in the case of those produced by shallow water. Goring noticed that the Boussinesq based procedure produced severely damped solitary waves as well. Moreover the Boussinesq, KdV and second-order shallow water procedures produce trailing waves that need a longer distance to separate from the main pulse. This indicates that energy exchanges be tween these two parts of the wave train last longer, resulting in less stable primary pulses. For all amplitudes, it is shown that a solitary wave generated using Rayleigh based procedures is stable beyond x = 20/; 0 while for other procedures the distance can be estimated at x = 80/z". The good performance of laws of motion derived from Rayleigh's solution are probably due to the accuracy with which Rayleigh's solution describes certain characteristics of the exact solitary waves. These aspects are discussed below.

Discussion
The Byatt-Smith (1970, [3j) numerical solution will be used as a reference for the exact solitary wave. It is indeed known to be one of the most accurate solitary wave solution within the potential flow theory. From this solution, the depth-averaged horizontal velocity is easily computed and substituted in (1) to compute a law of motion for the paddle. It should be noted that no experi ments were performed using this law. However it is felt that a Byatt-Smith procedure is not very practical since each wave ve locity field needs to be computed numerically which is very timeconsuming. On figure 10, the dimensionless paddle laws of mo tion with time for Byatt-Smith, Rayleigh and Boussinesq (Goring procedure) solitary waves and for shallow-water theories are plot ted for e = 0.306. For the same wave we also plot on figure 11 the dimensionless paddle velocity corresponding to these laws of mo tion. In the example shown on figure 10 it appears that the total paddle stroke is better described by the second-order shallow wa ter solution than by any other solution. Figure 12 plots the depthaveraged net mass displacement L in a solitary wave computed from Byatt-Smith's numerical solution compared to the ones as sociated to KdV, shallow water second-order and Rayleigh solu tions. This net mass displacement is the total stroke of the paddle prescribed in each procedure. Hence, it is confirmed that the total to" stroke in the second-order shallow-water theory procedure is the closest to the Byatt-Smith solution mean net mass displacement, at least in the solitary wave dimensionless amplitude range con sidered in the experiments. Nevertheless, it seems that this good agreement is not sufficient to produce 'pure' solitary waves. In deed, trailing waves are smaller when using Goring or Rayleigh procedures and yet, total strokes are smaller and larger respec tively in Goring and Rayleigh procedures compared to the Byatt-Smith mean net displacement. Figure 11  This outskirts decay coefficient describes the way free surface elevation tends towards the mean level at infinity. Stokes showed that P is a solution of the following equation, also used by Byatt-Smith: tan(p) P (18) It should be underlined here that the Boussinesq solitary wave expression is neither a solution of the KdV nor of the Boussinesq equations (1872, [2]). It appears to be a mixture of the Rayleigh phase speed and of the KdV outskirts decay coefficient and con sequently, net mass transport. Concerning the outskirts decay co efficient, the same hierarchy as for the phase speed is observed on figure 14.b. This explains why the paddle velocity in Rayleigh's procedure matches the paddle velocity based on Byatt-Smith so lution both at the maximum and the outskirts. As a matter of fact none of these approximate solutions matches the Byatt-Smith reference with regard to all the criteria used in this study. Second-order shallow water theory correctly predicts the mean net displacement. The Rayleigh and Boussinesq solu- tions agree with Byatt-Smith' s numerical solution concerning the maximum velocity at mid-stroke. And finally paddle velocity de rived from Byatt-Smith's numerical solution is matched by Rayleigh-based paddle velocity at the outskirts. We can now understand why, by increasing the duration of mo tion for a given stroke, Goring reduces the waves trailing the first solitary wave. Indeed, the duration of motion as expressed by (7) is a function of both the phase speed c and the outskirts decay coefficient (3 of the solitary wave. To a given stroke corresponds a given outskirts decay coefficient. Thus, to prolong the duration of motion it would be necessary to take a smaller phase speed than the one corresponding to the outskirts decay coefficient in Boussinesq form. This is in agreement with the relation in Byatt-Smith's numerical solution between outskirts decay coefficient, phase speed and solitary wave amplitude. Thus, prolonging the duration of motion tends to match both outskirts decay coefficient and phase speed to the Byatt-Smith solution but for a smaller am plitude than the design one. As a consequence, the relation be tween the net displacement (paddle stroke) and solitary wave am plitude will tend to be fulfilled since in Boussinesq's solitary wave the net displacement was underestimated. But how much should we prolong motion duration to get the purest solitary waves? Goring suggested 10 % but this value actually depends on the solitary wave amplitude and hence is not constant. Therefore, we suggest using Rayleigh's procedure rather than Goring's be cause it is then unnecessary to modify the duration of paddle mo tion arbitrarily and the resulting solitary waves are as pure and more rapidly established.

Conclusion
It has been shown in this paper that solitary waves generated us ing a paddle law of motion derived from Rayleigh's solution (linearized or fitted) are purer and more rapidly established than with any of the shallow-water theory-based procedures. Trailing waves after solitary waves generated using this new procedure are indeed as small as in Goring's procedure, being less than 3 % of the main pulse amplitude. However, none of these generation pro cedures is perfect. Indeed, taking Byatt-Smith's (1970, [3]) nu merical solution as a reference for the solitary wave solution, the paddle stroke is better described by second-order shallow-water theory than in the other procedures, whereas the maximum paddle velocity at mid-stroke is better predicted in our procedure or in Goring's. With respect to solitary wave generation, we suggest that this maximum paddle velocity is a key parameter. For a given maximum velocity, a Froude number is selected. The correspond ing paddle stroke for this Froude number should then be pre scribed. We found that the Rayleigh solitary wave with an accu rate description of the Froude number and outskirts decay coeffi cient meet the above requirements. Hence the generation proce dures based on Rayleigh's solitary wave solution is a good com promise for obtaining quite pure and rapidly established solitary

Acknowledgements
The authors would like to thank Jean-Marc Barnoud for technical support and assistance when performing the experiments. This work has been financially supported by the MAST-III EC programme, under contract MAS3-CT95-0027. The first author is grateful to the French Ministry of Education for attributing her a PhD grant.

Notation
The following notations are used in the paper: second-order shallow water theory t derivation with respect to time x derivation with respect to space