Nonlinear free surface flows past a semi-infinite flat plate in water of finitedepthM. Maleewong and R. H. Grimshaw Citation: Phys. Fluids 20, 062102 (2008); doi: 10.1063/1.2930675 View online: http://dx.doi.org/10.1063/1.2930675 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v20/i6 Published by the American Institute of Physics. Related ArticlesRunup and boundary layers on sloping beaches Phys. Fluids 25, 012102 (2013) Trapped disturbances and finite amplitude downstream wavetrains on the f-plane Phys. Fluids 24, 106601 (2012) Internal solitary waves with a weakly stratified critical layer Phys. Fluids 24, 056602 (2012) Numerical simulation of mass transport in internal solitary waves Phys. Fluids 24, 016602 (2012) Instability in internal solitary waves with trapped cores Phys. Fluids 24, 016601 (2012) Additional information on Phys. FluidsJournal Homepage: http://pof.aip.org/ Journal Information: http://pof.aip.org/about/about_the_journal Top downloads: http://pof.aip.org/features/most_downloaded Information for Authors: http://pof.aip.org/authors

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Nonlinear free surface flows past a semi-infinite flat platein water of finite depth

M. Maleewong1 and R. H. J. Grimshaw2,a�

1Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand2Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU,United Kingdom

�Received 8 January 2008; accepted 19 April 2008; published online 13 June 2008�

We consider the steady free surface two-dimensional flow past a semi-infinite flat plate in water ofa constant finite depth. The fluid is assumed to be inviscid, incompressible and the flow isirrotational; surface tension at the free surface is neglected. Our concern is with the periodic wavesgenerated downstream of the plate edge. These can be characterized by a depth-based Froudenumber F and the depth d �draft� of the depressed plate. Previous analytical studies have beenrestricted to small values of d and subcritical flows F�1, when linearized theory can be used. Here,our main concern here is with the critical regime, F�1, where we use a weakly nonlinearlong-wave analysis, For finite values of d, we solve the fully nonlinear problem numerically usinga boundary integral equation method. Our results confirm previous studies and extend these to thecritical regime. As d increases, our numerical results demonstrate that the downstream wavesapproach the highest possible wave. We also find some wave-free solutions, which when comparedto the weakly nonlinear results are essentially just one-half of a solitary wave. © 2008 AmericanInstitute of Physics. �DOI: 10.1063/1.2930675�

I. INTRODUCTION

Two-dimensional free surface flows past a fixed surface-piercing obstacle is a much-studied problem, presenting bothanalytical and numerical challenges, and relevant inter aliato the generation of bow and stern flows in ship hydrody-namics. Our concern here is with the waves generated down-stream, and so we assume that the obstacle length is semi-infinite. In general, there may be strong singularities at theintersection of the free surface with the rigid obstacle, caus-ing difficulties for both analytical and numerical studies.Here, however, we assume that the flow leaves the obstacletangentially, and so we are able to model the obstacle as asemi-infinite flat horizontal plate. We will assume that thefluid is inviscid and incompressible, with constant densityand zero surface tension, and that the flow is steady andirrotational.

In water of infinite depth, Schmidt1 considered the lin-earized problem for stern flow when the free surface leavesthe plate tangentially, under the assumption that V2 /gd�1,where d is the draft and V is the downstream flow speed. Anexact solution was found which showed that periodic sinewaves are found in the downstream far field. The corre-sponding nonlinear problem was considered byVanden-Broeck2 using a boundary integral equation method.For small drafts, the results agreed with the linear theory, butas the draft increases �that is the draft-based Froude numberV / �gd�1/2 decreases�, the waves steepen and eventually reachthe highest possible wave. On the other hand, Andersson andVanden-Broeck3 considered the analogous problem for bow

flows with surface tension included but gravity neglected.They found that flows with waves do not leave the platetangentially.

In water of finite depth, the linearized problem was con-sidered by McCue and Stump,4 who assumed that d /D�1,where D is the downstream mean fluid depth, but the Froudenumber V / �gD�1/2 is finite. An exact solution was found us-ing the Wiener–Hopf technique, which yielded the expectedlinear sine waves far downstream for subcritical flowsV / �gD�1/2�1. They also found wave-free supercritical solu-tions, but these had a singularity at the plate edge. Theyconjectured that wave-free solutions could only be foundwhen the Froude number and draft satisfy a certain specialrelationship, predicted by Vanden-Broeck and Keller5 using aconservation of momentum argument �see our discussion be-low�. Detaching wave-free stern flows from an inclined platehave been calculated numerically by Hocking.6 Subse-quently, McCue and Forbes7 considered free surface flow fora fluid of finite depth with constant vorticity, emerging froma flat plate. This is the same problem that we consider here,but with zero vorticity. They solved the nonlinear problemnumerically for both supercritical and subcritical flows, andfor the special case of zero vorticity, they obtained a set ofresults for subcritical flows. With nonzero vorticity, theyfound cases with wave-free surface profiles containing over-hanging portions near the plate edge for supercritical flow,while for subcritical flow periodic downstream waves werefound. Recently, Tooley and Vanden-Broeck8 considered thegeneration of capillary waves in the absence of gravity forfree surface flow past a flat plate and found a three-parameterfamily of solutions which are, in general, characterized by adiscontinuity in slope at the separation point. In related work,Binder and Vanden-Broeck9,10 considered a variety of prob-

a�Author to whom correspondence should be addressed. Telephone:44-1509-223480. FAX: 44-1509-223969. Electronic mail:r.h.j.grimshaw@lboro.ac.uk.

PHYSICS OF FLUIDS 20, 062102 �2008�

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lems for free surface flow past a plate of finite length, repre-senting a horizontal surfboard, or an inclined gate, or a sluicegate. These problems were solved analytically using aweakly nonlinear theory similar to that we shall use here,valid when the Froude number is close to unity and the in-clination angle of the plate is small, and otherwise numeri-cally in the nonlinear regime.

In this paper, we consider free surface flows past a semi-infinite horizontal plate depressed into water of finite depth.It is assumed that the plate edge is not a stagnation point, andit then follows that under mild smoothness criteria, the freesurface leaves the plate tangentially. Gravity is included butsurface tension is neglected. For subcritical flows, the linear-ized version of this problem was solved analytically byMcCue and Stump,4 and the nonlinear problem solved nu-merically by McCue and Forbes,7 again for subcritical flows.Our main concern here is to extend these results to the criti-cal regime, where we will show that a weakly nonlinearanalysis can be used to interpret our numerical results. InSec. II A, we present the problem formulation. Then in Secs.II B and II C, we revisit some aspects of the linearizedtheory, and in Sec. II D, we present a local analysis near theplate edge which reveals the weak singularity there. Whenthe Froude number approaches unity, the periodic wavesgenerated have very long wavelengths, and so in Sec. III, wepresent a weakly nonlinear long-wave analysis. In Sec. IV,we consider the fully nonlinear problem using a numericalboundary integral method described in Vanden-Broeck11 andMaleewong et al.12 Then in Sec. V, we present our numericalresults, which for small drafts are compared to the linearizedtheory or to the weakly nonlinear long-wave theory when theFroude number approaches unity. As the draft increases, sodoes the wave steepness, and we numerically calculate peri-odic waves, with narrow crests and board troughs, close tothe results for the highest possible waves obtained byCokelet,13 analogous to the results obtained by McCue andForbes.7 We conclude with Sec. VI.

II. PRELIMINARY DISCUSSION

A. Formulation

We consider two-dimensional flow past a semi-infiniteflat plate in water of finite depth. This stern flow is assumedto separate tangentially from the edge of the plate. The fluidis inviscid and incompressible and the flow is steady andirrotational. Far upstream, the flow is uniform with constantvelocity U and constant depth H �see Fig. 1�. The x-y coor-dinate is placed on the bottom where the y axis is at the pointof separation. The plate is depressed from the undisturbedlevel a distance Hd �draft, see below for a precise definition�to perturb the free surface and so generate waves down-stream.

The flow field is thus represented by a stream function��x ,y�, which satisfies Laplace’s equation in the fluid do-main, the velocity field is u=�y ,v=−�x, and the boundaryconditions are

� = − Q on y = 0, − � � x � � , �1�

� = 0 on y = H, − � � x � 0, and

�2�y = h, 0 � x � � ,

u2 + v2

2+ gh = B on y = h, 0 � x � � . �3�

Here Q=UH is the �constant� mass flux, B is the unknownBernoulli constant, y=h�x� ,x�0 is the unknown free sur-face, and u=�y→U as x→−�. Next we impose the condi-tion of tangential separation, namely that hx→0 as x→0+.Far downstream we require that the flow is bounded. Be-cause we expect a periodic wave to form far downstream, weintroduce the nondimensional draft d �which can be positiveor negative� by requiring that

h = H�1 + d� + h, where �h� = 0, as x → � . �4�

Here �¯� denotes an average over the periodic wave. But wenote that the wavelength for taking this average is still un-known. This issue will be addressed below in Sec. IV. Next,it is convenient to choose the Bernoulli constant

B =V2

2+ gH�1 + d� . �5�

Here V=JU / �1+d� is a measure of the downstream velocity,where we have used conservation of mass to ensure that V /Uscales with 1 / �1+d�. We expect the unknown constantJ→1 as d→0. But note that in the absence of waves, wemust set J=1 to conserve mass.

Next, we nondimensionalize with U ,H so that Eqs.�1�–�3� become

� = − 1 on y = 0, − � � x � � , �6�

� = 0 on y = 1, − � � x � 0, and

�7�y = h, 0 � x � � ,

F2�u2 + v2�2

+ h =J2F2

2�1 + d�2 + 1 + d

�8�on y = h, 0 � x � � .

The two input parameters are now F=U /�gH and d. Whend=0,J=1, and the solution is the uniform flow u=1,v

x

yUH

Hd

FIG. 1. �Color online� Sketch of physical plane.

062102-2 M. Maleewong and R. H. J. Grimshaw Phys. Fluids 20, 062102 �2008�

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=0,h=1. Otherwise, d�0 means that the plate at y=0,−��x�0 has been depressed a distance d relative to the flowas x→�. As noted above, if the downstream state is wave-free, then we must set J=1 to conserve mass, and there isthen an additional boundary condition as x→�, namely,h→1+d ,u→1 / �1+d� ,v→0. This extra constraint meansthat a solution can only exist for a special relationship be-tween F and d, so only one of these can be specified a priori.In Sec. II B we use conservation of momentum to determinethis relationship, following Vanden-Broeck and Keller5 andMcCue and Stump.4 Otherwise, when the downstream statecontains waves, both F and d must be specified a priori, andJ is determined as part of the solution.

B. Conservation of momentum

In dimensional variables, the expression for conservationof momentum is

0

�H,h� p

�+ u2�dy = constant

�9�for �− � � x � 0,0 � x � �� ,

wherep

�= B −

u2 + v2

2− gy . �10�

Here p is pressure and � is the �constant� density. Substitut-ing for the pressure, using our nondimensional variables andevaluating the constant upstream we get

�1,h�� J2F2

2�1 + d�2 + 1 + d −�1,h2�

2+

F2

2

0

�1,h�

�u2 − v2�dy

=J2F2

2�1 + d�2 +F2

2+ d +

1

2. �11�

First suppose that there is uniform flow downstream, so that�u ,v�→ �1 /1+d ,0�, J=1, and h→1+d. We can also evalu-ate this expression as x→� to get

�1 + d�� F2

2�1 + d�2 + 1 + d −�1 + d�2

2+

F2

2�1 + d�

=F2

2�1 + d�2 +F2

2+ d +

1

2.

This is the required relationship between F and d. Simplify-ing, it becomes

F = 1 + d . �12�

This is the same result as that found by Vanden-Broeck andKeller5 and McCue and Forbes,7 expressed in our presentnotation.

Next, we consider the general case when periodic wavesare generated downstream. That is, as x→�, there will be auniform periodic wave train. In this situation, we need todefine the constants J ,d in a different way. We now write

u =K

1 + d+ u, h = 1 + d + h , �13�

useful in x�0 and, in particular, as x→�, where u , h definethe periodic wave. In this case, we now define d by the

requirement that 1+d is the mean level, so that h has zeromean, see Eq. �4�. Similarly, K is defined so that u also haszero mean, noting that the mean flow must be a constant,independent of y. This in effect then determines the constantJ, where the relationship between J and K is discussed be-low, see Eq. �19�. Note that only one of the pair J ,d can beimposed a priori. We will choose d to be an input parameter,along with F, and then J is determined as a part of the solu-tion. In effect, the Bernoulli constant then becomes part ofthe solution.

Note that it will be useful to define a downstream Froudenumber,

G =Um

�gH�1 + d�=

KF

�1 + d�3/2 , �14�

where Um= �u� is the mean velocity in dimensional coordi-nates. In fact, G can also be expressed in terms of J, but herewe find it more useful to define it in terms of K. For awave-free solution, J=K=1 and then the condition �12� be-comes G=1 / �1+d�1/2. Since we expect wave-free solutionsto arise only for downstream supercritical flows, G�1, itfollows that we must then have d�0, see Sec. V C.

It is now useful to consider the conservation of mass,

0

h

udy − 1 = K − 1 +Kh

1 + d+

0

h

udy = 0, �15�

that is valid in x�0. As before, let �¯� denote an averageover the periodic wave as x→�, so that we get

K − 1 = −�1+d

1+d+hudy� . �16�

Note that the right-hand side is second order in the waveamplitude, but it is not zero. To see this, now we substitutethe expressions �13� into the Bernoulli relation �8� to get

KF2

1 + du + h +

F2

2�u2 + v2� =

F2

2�1 + d�2 �J2 − K2�

on y = 1 + d + h, 0 � x � � . �17�

We can use this to evaluate u on the free surface, and soeventually we get

K − 1 =1 + d

F2 �h2� + ¯ , �18�

correct to the second order in the wave amplitude a �definedbelow�. Then we use the Bernoulli relation again to showthat

J = K + 12 �1 + d�2�u2 − v2� + ¯ , �19�

where u ,v are evaluated at the free surface before averaging.Again, this expression is corrected to second order in the

062102-3 Nonlinear free surface flows Phys. Fluids 20, 062102 �2008�

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wave amplitude. This expression confirms that for a givenpair �F2 ,d�, J must be determined as part of the solution.

We can now return to the momentum conservation law�11� to determine the wave amplitude downstream. Substitu-tion into the momentum conservation law for x�0 yields,correct to second order in wave amplitude,

−h2

2+

F2

2

0

1+d

�u2 − v2�dy =F2

2�1 + d�2 �d2 − d�J2 − 1�� −d2

2.

�20�

The uniform-flow result �when J=1� follows immediately, asthe left-hand side then goes to zero as x approaches ��.Otherwise, we must evaluate these expressions for the peri-odic wave solution as x→�. For this purpose, we now con-sider the linearized regime in Sec. II C.

C. Far-field linearized theory

The linearized boundary conditions as x→� are ob-tained by substituting Eq. �13� into the boundary conditions�7� �after first differentiating this with respect to x� and �8�,and then linearizing about the constant mean values. Thisyields

v =hx

1 + d,

F2

1 + du + h = 0 for y = 1 + d . �21�

Here we have used the result that J and K are both equal to1+O�a2� to the leading order in the wave amplitude a �de-fined below�. Strictly speaking, the wave amplitude is ex-pected to scale with d as d→0, so consistently we shouldalso set d=0 in Eq. �21�, but it is useful to avoid this step atthis point.

We then seek solutions of the form

h = a Re�exp�ik�x − x0��� . �22�

Here a is the real-valued amplitude we seek, k is the wave-number so that the wavelength =2 /k, and x0 is the phase.We will now show that the amplitude can be determinedfrom the momentum conservation, but that the phase canonly be found by matching with the solution in x�0 and, in

particular, using the constraint that h=1, h=−d at x=0. Itnow readily follows that the wavenumber k is given by thelinear dispersion relation

F2

�1 + d�2 =tanh�k�1 + d��

k. �23�

Strictly speaking, we should also put d=0 in this expression,as the linearization only holds for d�1. Also we have

u = −ak

1 + d

cosh ky

sinh k�1 + d�Re�exp�ik�x − x0��� , �24�

with a similar expression for v. We can now evaluate theleft-hand side of the momentum conservation law to get,corrected to second order in wave amplitude,

a2

4�1 −

2k�1 + d�sinh 2k�1 + d� =

d2

2�1 −

F2

�1 + d�2 . �25�

This shows, as expected, that for a fixed F, the amplitude ascales with d when 0�d�1, and, in general, for a fixed dgives the expression for the variation of a with F. The ex-pression is complicated because the wavenumber k is givenin terms of F by the dispersion relation. But note that theleft-hand side is always positive, so we must always haveF�1+d, or the downstream Froude number G�K / �1+d�1/2 as expected here for the subcritical flow. For simplic-ity, now let d→0 so that the final formula is

a2

4�1 −

2k

sinh 2k =

d2

2�1 −

tanh k

k , �26�

where F2 =tanh k

k. �27�

This result is plotted in Figs. 2 and 3 for the case d→0. Wesee that the amplitude is predicted to increase as the wave-number increases which corresponds to decreasing theFroude number.

These results agree completely with McCue and Stump,4

as we can write

A

0.4

k

0.3

0.45

0.35

0.25

10.07.55.02.5

FIG. 2. �Color online� Plot of A=a2 /4d2 against k for the case d→0.

0.8

0.4

0.4

E

1.0

A

0.45

0.35

0.6

0.3

0.25

0.2

FIG. 3. �Color online� Plot of A=a2 /4d2 against E=F2 for the case d→0.

062102-4 M. Maleewong and R. H. J. Grimshaw Phys. Fluids 20, 062102 �2008�

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a2

d2 =2F2�1 − F2�

F2 + k2F4 − 1,

which is precisely the expression found by them, by a muchmore complicated method.

There are two anomalies with this estimate. First, as F→1, k→0, and so a long-wave analysis is needed. This fol-lows below in Sec. III. Second, as F→0, the prediction isthat the wave amplitude a /d→�2 remains finite, but that thewavenumber k→�; thus the wave steepness increases, andthe linearization fails, except for very small d. Indeed, forF→0 the present nondimensional variables are not appropri-ate, and a rescaling is needed. In essence, the waves becomeshort, do not feel the bottom, and a rescaling is needed inwhich U2 /g is the relevant length scale. Thus we rescale the�nondimensional� variables as follows:

�x,y − 1� = F2�x, y�, � = F2�, h − 1 = F2h, d = F2d .

�28�

The boundary conditions now become

�x = 0 on y = −1

F2 , − � � x � � , �29�

� = 0 on y = 0, − � � x � 0, and

�30�y = h, 0 � x � � ,

u2 + v2

2+ h =

J2

2�1 + F2d�2+ d on y = h, 0 � x � � .

�31�

Letting F2→0 yields the deep-water formulation, studied byVanden-Broeck.2 The significant point that emerges from thiswork is that solutions will exist only for some finite range of

0� d�dM =0.44. This means that d /F2 is bounded asF2→0, and so solutions can only be found for increasinglysmall values of d as F→0.

A related aspect that emerges in the fully nonlinear prob-lem is that the solutions will be limited by the wave of maxi-mum amplitude aM. From the numerical results of Cokelet13

for periodic waves, 2aM /=0.141 055 in deep water, whilein shallow water aM /1+d=0.447. These can be translated tolimiting wave amplitudes as functions of the Froude number,that is, we estimate that aM =0.443F2 for deep water �F2

→0� and also that aM =0.44 for shallow water �F2→1�. Infact, it would seem that a good estimate is aM =0.44F2 for allF2�1.

D. Local analysis at the plate edge

Returning to the full problem, we now examine closelythe conditions which hold at the edge of the plate, namely,near y=1,x=0. Assuming sufficient smoothness in the eleva-tion and velocity fields, we find from the boundary condi-tions �6�–�8� that

h = 1, v = 0 = uhx,F2

2u2 =

J2F2

2�1 + d�2 + d ,

and

F2�uux + uuyhx� + hx = 0 at y = 1, x = 0, �32�

where the last condition comes from the derivative of theBernoulli condition. Since here the plate edge is not a stag-nation point, u�0 but hx=ux=0 at y=1,x=0.

Let us take local polar coordinates �r ,�� at the plateedge. Then if U0 is the velocity at the plate edge, we expandthe stream function, which must satisfy the Laplace equationas

� = U0r sin � + Cr3/2 cos3�

2+ Dr2 sin 2� + Er5/2 cos

5�

2

+ E1r5/2�ln r cos5�

2− � + ��sin

5�

2�¯ . �33�

Then the kinematic condition on the plate is satisfied since�=0 on �=−, while the free surface, where also �=0, isgiven by

sin � � �r1/2 + r3/2 + �r3/2 ln r , �34�

where

U0� + C = 0, U0 + E + 2D� = 98C�2, U0� + E1 = 0.

�35�

Next, the velocity fields are given by

�r = U0 sin � +3C

2r1/2 cos

3�

2+ 2Dr sin 2� + ¯ .

��

r= U0 cos � −

3C

2r1/2 sin

3�

2+ 2Dr cos 2� + ¯ .

The Bernoulli condition �8� then gives locally, on the freesurface,

F2

2�− 3U0Cr1/2 sin

�

2+ 4U0Dr cos � +

9

4C2r + ¯ �

+ r sin � = 0.

Substituting the free surface approximation �34� into this ex-pression, we get

F2

2�− 3U0C�

r

2+ 4U0Dr +

9

4C2r − 5U0E1r3/2 + ¯ �

+ �r3/2 + ¯ = 0, �36�

so that

U0D = − 15C2/16, � = 5F2U0E1/2. �37�

Thus, given the leading order coefficient � in Eq. �34� we seefrom Eqs �35� and �37� that � ,C ,D ,E1 are now known, butconsideration of higher-order terms is needed to find ,E.Using Eq. �35�, we get

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�

�= −

2

5F2U02 , �38�

which agrees with the expression obtained by McCue andStump4 �see their Eq. �27��, since the free surface in x�0 isgiven approximately by

h � �x3/2 + x5/2 + �x5/2 ln x .

However, this analysis gives no information on even the signof � as it is not linked to the upstream and downstreamconditions. But it does reveal the presence of is a weak sin-gularity at the plate edge.

III. WEAKLY NONLINEAR LONG-WAVE ANALYSIS

In the joint limit F→1,d→0, we will describe a weaklynonlinear long-wave analysis which will lead to a steady-state Korteweg–de Vries equation, analogous to similar stud-ies by Binder and Vanden-Broeck9,10 and Maleewonget al.12 Thus, to describe weakly nonlinear long waves, weintroduce the scaling

X = �x, h = 1 + d + �2H�X� ,

�39�F2 = �1 + d�3 + �2�, d = �2D .

Here � is a small parameter. Note that we are expandingabout the constant downstream flow in the far field X�0.The corresponding expansions for u ,v are

u =K

1 + d+ �2U�X� − �4 y2

2UXX + ¯ , �40�

v = − �3yUX + ¯ , �41�

�J,K� = 1 + �4�M,N� + ¯ . �42�

Note that Eq. �40� satisfies Laplace’s equation at leading or-der, and that Eq. �41� comes from the continuity equation atthe leading order.

It can be shown that there are no O��2� term in the ex-pansions for �J ,K�. Substitution into the integral expressionfor the conservation of mass �15� yields,

H + U�1 + d�2 + �2�UH − 16UXX� + ¯ = − �2M , �43�

while the Bernoulli condition �8� yields

H + U�1 + d�2 + �2�U +U2

2−

1

2UXX� + ¯ = �2N

2.

�44�

Subtraction yields, as expected, the steady-stateKorteweg–de Vries equation

�U +3

2U2 −

1

3UXX =

3N

2, �45�

or

− �H +3

2H2 +

1

3HXX =

3N

2. �46�

Note that U=−H to leading order. The constant N is deter-mined as before, that is, N= �H2� �so that N�0� for a peri-odic solution. But if we impose the condition that H→0 asX→�, then N=0. Later, we will also impose the boundaryconditions H=−D ,HX=0 at X=0, although we must notethat this ignores the weak singularity at the plate edge, de-scribed above in Sec. II D. Nevertheless, this boundary con-dition should hold from an “outer” perspective and ensures aunique solution. The issue is then how this behaves asX→�.

One integration gives

HX2

3− �H2 + H3 = 3NH + C , �47�

where C is a constant. First, let us consider the periodic wavesolution given by

H = A + a cn2���X − X0�� , �48�

where

a =4

3m�2, 3A = � −

a

m�2m − 1� , �49�

3N = 3A2 − 2�A +a2

m�1 − m� , �50�

and

C = − 2A3 + �A2 − Aa2

m�1 − m� . �51�

Here cn��X� is a Jacobian elliptic function of modulus m�0�m�1�. This solution contains four parameters,A ,a��0� ,� ,m, as well as the phase shift X0, and the spatialwavelength is

=2K�m�

�, �52�

where K�m� is the complete elliptic integral of the first kind.From Eq. �49�, we see that there are two free parameters, say,a ,m, for each fixed value of �. But the definition of D re-quires that we impose the condition that �H�=0, and so

A = − �a cn2���X − X0��� = −a

m�E�m�

K�m�− 1 + m , �53�

where now

a

m=

4�2

3, � =

a

m�2 − m −

3E�m�K�m� . �54�

3N =a2

m2�m − 1 +2E�m�K�m�

�2 − m� −3E�m�2

K�m�2 , �55�

and

062102-6 M. Maleewong and R. H. J. Grimshaw Phys. Fluids 20, 062102 �2008�

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C = − Aa2

m2�E�m�K�m�

−E�m�2

K�m�2 . �56�

This condition reduces the family to a one-parameter family.We note that A�0, and that as a consequence we must haveC�0 as well as N�0. It can be shown that these conditionshold in the expressions above for all 0�m�1. Further, itcan also be shown that if we write �= �a /m���m�, then��m� varies monotonically from −1 to 1 as m increases from0 to 1, where the zero occurs at m=0.96.

Finally, we impose the boundary condition that HX=0and H=−D at X=0. There are two cases to consider. First,consider the solution A=−D and �X0=K�m� so that

D =a

m�E�m�

K�m�− 1 + m . �57�

Thus, for a given D in Eq. �57� and � in Eq. �54�, we candetermine a ,m, and hence all the parameters in the periodicwave. Indeed, from Eq. �57�, we can write a /D=b, whereasm increases in the range of 0�m�1, b increases from 2 to� �see Fig. 4�. Since from Eq. �54�, a�0, it follows thatsolutions only exist in this case for D�0. Further, we seethat the ratio � /D=E and so determines the modulus m. Asm increases in the range of 0�m�1, it can be shown that Eincreases monotonically from −� to �, with a zero at m=0.96. It follows that there is a unique solution for all� ,D��0�, where for a fixed D, � increases from −� to �. InFig. 5, we plot a /D as a function of � /D. The wavenumber� is given by 4�2 /3D=b /m, and for a fixed D decreasesfrom � to 0 as m increases. Further, it then follows that thespatial wavelength �52� increases from 0 to � as m increaseswith D fixed. As m→0, a�2D, ��−2D /m, the spatialwavelength is /�, where 4�2 /3�−�, and we recover thelinear theory �note that a is here twice the amplitude used

previously in the linearized theory of Sec. II C�. At �=0,m=0.96, and a=3.13D, the spatial wavelength is 6.06 /�,where �2=2.45D. As m→1, a /D→�, ��a, and 4�2 /3��, which is outside the range of this weakly nonlineartheory.

Next, consider the alternative solution A+a=−D andX0=0. In this case Eq. �57� is replaced by

− D =a

m�1 −

E�m�K�m� . �58�

The analysis proceeds as above, but now −a /D= F�m�,where as m increases and F�m� decreases from 2 to 1. Hence,

this case exists only for D�0. The ratio −� /D= G�m�,where as m increases, G�m� increases from −� to 1 as mincreases, with a zero at m=0.96. There is a unique solutionfor all � ,D��0� such that � / �−D��1, where for a fixed D,� increases from −� to −D. The wavenumber � is given by

4�2 / �−3D�= F�m� /m, and for a fixed D, this expression de-creases from � to 1 as m increases. Further, it then followsthat the spatial wavelength �52� increases from 0 to � as mincreases with D fixed. But here, it is important to note thatthere is a limiting solution as m→1, which is a solitarywave, of amplitude �=−D. This limiting solution is a wave-free solution and is discussed in more details in the nextparagraph.

To seek wave-free solutions directly, we must impose theextra boundary condition that H→0 as X→�, so thatC=N=0, and a solution exists only if ��0. Then if we alsoimpose the boundary condition at X=0, we must have�=−D, which is just the condition obtained previously for awave-free solution in the present long-wave limit, that is,F2=1+2�2D+¯. Note that such wave-free solutions existonly if D�0. In the unscaled variables, F2= �1+d�2 andG2=1 /1+d. Since d�0, the downstream flow is supercriti-

3.5

3.0

2.0

3.75

3.25

2.75

2.5

2.25

m0.90.80.70.60.50.40.30.20.1

b

FIG. 4. �Color online� Plot of b=a /D against m.

3.1

E

2.7

2.5

−1

2.3

b

3.0

2.9

2.8

2.6

0

2.4

2.2

−2

2.1

−3−4−5−6−7−8−9

FIG. 5. �Color online� Plot of b=a /D against E=� /D.

062102-7 Nonlinear free surface flows Phys. Fluids 20, 062102 �2008�

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cal as expected, G�1, but the upstream flow is subcritical,F�1. With N=C=0, Eq. �47� has the solitary wave solution

H = � sech2���X − X0�� ,

where 4�2=3�. This can satisfy the boundary condition atthe plate edge that HX=0,H=−D if X0=0 and �=−D asexpected. As above, we see that this is a limiting solutionof a family of downstream periodic waves, obtained as�→−D from below.

IV. NUMERICAL METHOD

A. Formulation

For the fully nonlinear problem, we use a numericalboundary integral method, similar to that used byMaleewong et al.12 Here, we just present essential details.Since the fluid is incompressible and the flow is irrotational,we can introduce a potential function ��x ,y� and a streamfunction ��x ,y� so that the complex potential function is f=�+ i�, while the complex velocity by w=u− iv. We recallthat the boundary conditions are defined by Eq. �6�–�8�. Thenumerical method is now formulated in the �� ,��-plane, sothat w=w�f�. Without loss of generality, we choose �=0 atthe edge of the plate �the separation point�. Note that thekinematic condition on the bottom can be written as

v��,�� = 0 on � = − 1. �59�

We map the complex f-plane to the complex �� , � plane bythe transformation

� = � + i = ef , �60�

which maps the flow domain in the f-plane to the lower halfof the �-plane. We define the complex velocity in the �-planeby

w = u − iv = e�−i�. �61�

In particular, we have the relations �=e� cos��� and =e� sin���. In the �-plane, we choose a contour consistingof the real axis and a half-circle of arbitrarily large radius inthe lower half plane. Applying the Cauchy integral formulato the function �− i� in the complex �-plane, we get

� − i� = −1

2i� ����� − i�����

�� − �d��. �62�

The line integral on the path of half-circle vanishes as we letthe radius goes to infinity. Letting � approach the boundary =0, we obtain

� − i� = −1

i

−�

� ����� − i������� − �

d��. �63�

The real part yields

���� =1

−�

� ������� − �

d��. �64�

The kinematic condition on the bottom implies that

���� = 0 for � � 0. �65�

Substituting Eq. �65� into Eq. �64�, we have

���� =1

1

� ������� − �

d��. �66�

We obtain another relation between � and � on the free sur-face by using Eq. �61� to get u2+v2=e2�. Thus the Bernoullirelation �8� becomes

F2e2�

2+ y =

J2F2

2�1 + d�2 + 1 + d . �67�

Finally, the free surface profile is determined by integrat-ing numerically the identity

dx

d�+ i

dy

d�= w−1 =

cos��� + i sin���e� , �68�

that is,

dy

d�=

sin���e� �69�

or

dy

d�=

e−� sin����

.

Integrating Eq. �69� yields

y��� − y�0� =1

1

� e−� sin����

d�, 1 � � � � . �70�

Equations �66�, �67�, and �70� define a nonlinear integralequation for the unknown function ���� on the free surface.

B. Numerical procedure

We solve the system of integral equations �66�, �67�, and�70� numerically by placing equally spaced points �i, i=1, . . . ,M on �=0. The Cauchy principal value integral �66�is valuated at the midpoints �i+1/2 to avoid singularities inthe integral. We calculate the values of y��� in Eq. �70� atthe mesh points using the trapezoidal rule. The values ofy��� at the midpoints are obtained by a four-point interpola-tion formula. Full details of these calculations can be foundin Vanden-Broeck11 and Maleewong et al.12

We now satisfy the free surface condition �67� by sub-stituting these values of ���� and y��� at the midpoints. Thisyields M −1 nonlinear algebraic equations for the M +1 un-knowns �i, i=1, . . . ,M, and J. We require two constraintconditions. One condition is obtained by assuming that theflow is separated from the plate tangentially,

�1 = 0. �71�

The last condition is obtained by imposing the mean depthcondition, that is �see Eq. �4��,

1 + d =1

L

L+

y�x�dx, L → � , �72�

or

062102-8 M. Maleewong and R. H. J. Grimshaw Phys. Fluids 20, 062102 �2008�

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1 + d =1

�

P

P+� y���d�

u, P → � , �73�

where �=K /1+d is the wavelength in the �-coordinate,found by averaging the expression �13� for u=�x.

The value of to be used in this last condition nowneeds to be specified. Nonlinear periodic water waves arecharacterized by four parameters, the wavelength, the meandepth, the speed, and the wave amplitude. Of these three maybe specified a priori, and the fourth is then determined �seeCokelet,13 for instance�. Here, we will specify the speedthrough the Froude number F, the mean depth 1+d, and thewavenumber k=2 /. In our present notation, the nonlineardispersion relation takes the form

G2 =F2K2

�1 + d�3 =tanh�k�1 + d��

k�1 + d��1 + k2a2�2�k�1 + d��

+ O�a4�� , �74�

where

�2 =9 − 10C2 + 9C4

8, C = coth k�1 + d� . �75�

Usually, the amplitude a, the wavenumber k, and the meandepth �1+d� are specified, and Eq. �74� then determines G2,that is, the constant K. But here we must specify F ,d asinputs, and then the amplitude a and the constant K are out-puts. Thus, in order to use Eq. �74� to find the wavelength ,we must have some a priori knowledge of the wave ampli-tude. Here, we propose three approaches. The first two meth-ods are based on the limit d→0 in Eq. �74�, noting that inthis limit a�d �see Eq. �26�� and also that then

K = 1 +�1 + d�a2

2F2 + O�a4� , �76�

which is readily derived from Eq. �18�. Thus, method 1 usesthe linear approximation to Eq. �74�, that is, we get the fol-lowing.

Method 1. The linearized expression F2=tanh k /k, validin the limit a ,d→0, is used to find k=2 /, given F ,d.Note that in this case, k does not depend on d, and consis-tently we set �= since K�1,d�0.

As a possible improvement, we again use the linearizedapproximation for the wave amplitude, but retain d, so thatwe get the following.

Method 2. The expression �74� is used, truncated atO�a2�, with the amplitude a given by Eq. �25� and K givenby Eq. �76�, also truncated at O�a2�, while d is fully retained.Hence k can be found iteratively, where in the first iterationwe take the limit a→0, so that we set K=1 and �2, etc.=0, that is, �¯�=1, in Eqs. �74� and �75�; then in the nextiteration we use this value of k and the linear expression �25�for the amplitude to evaluate K and �2, and hence find acorrection to k. In practice, we find that this O�a2� correctionfactor is quite small for most values of d that we considered,provided F is not too close to 1 �in fact, F�0.85�.

Since both these methods will fail as F→1, we thenpropose a third method.

Method 3. The weakly nonlinear long-wave analysis ofSec. III is used, which gives a good approximation whenF2�1 and �d��1.

Recently, Grandison and Vanden-Broeck14 proposed analternative method to find the wavelength, which is estimateddirectly from each iteration of the numerical solution. In-deed, we also tried a procedure similar to this, but did notfind that in the present problem it gave any significant ad-vantages over the three methods proposed above. Also, wenote that our method 2 which also involves an iteration onthe wavenumber bears some similarities to the procedureproposed by Grandison and Vanden-Broeck.14

The system of M +1 nonlinear algebraic equations withM +1 unknowns is then solved numerically by Newton’smethod. The results are described in the next section.

V. NUMERICAL RESULTS

A. Results for small draft

The relationship between the wave height �distance fromcrest to trough� and F2 when d=0.001 is shown in Fig. 6. Ford�1, the numerical results agree very well with the theoret-ical results, Eq. �26� for method 1 or Eq. �25� for method 2,from the linearized theory of Sec. II C, provided that F2 isnot close to 0 or 1. Note that the linearized theory usingmethod 2 has a �spurious� dip in amplitude as F→1. This inturn affects the amplitude of the numerical solutions obtainedby method 2.

The analysis in Sec. II C suggests that the linearizationfails when F→0, k→�, and so →0. A rescaling is neededand it was found from the work of Vanden-Broeck2 that d /F2

is bounded as F2→0. Hence as F→0, solutions can only befound for increasingly small values of d. Further, in this limitthe waves become quite steep and are increasingly difficultto calculate with our present numerical method. Indeed, forthe case d=0.001 shown in Fig. 6, we see a departure from

0 0.2 0.4 0.6 0.8 11.6

1.8

2

2.2

2.4

2.6

2.8

3x 10

−3

F2

wav

ehe

ight

(cre

st−

trou

gh)

d = 0.001

Linear Method 2Numerical Method 2Linear Method 1Numerical Method 1

FIG. 6. �Color online� Relationship between wave height �crest-trough� andF2 when d=0.001.

062102-9 Nonlinear free surface flows Phys. Fluids 20, 062102 �2008�

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the linearized theory when F�0.1 As d is increased, thenonlinear effects emerge for larger F, see the discussion inthe following paragraphs.

To obtain more accuracy of the numerical solutionswhen F2�1, we used method 3 where the wavelength isfound from the long-wave analysis. The results are shown inFig. 7 for d=0.01, where we see that the long-wave analysisand the numerical results have the same trend, and they havegood agreement when F2�1, as expected. As the Froudenumber F nears unity, there are periodic wave solutions, in-cluding supercritical ones �in fact, these solutions also existfor the case d=0.001�. The wave solutions when F�1 havevery long wavelengths and large amplitudes. When F�1 thelinearized theory using either method 1 or 2 fails to predictthe periodic waves. For these flows, we obtain better numeri-cal results by using the wavelength from the long-waveanalysis �method 3�. As F increases, the wave amplitude andwavelength increase, and the amplitude increases rapidlywhen F is only slightly greater than unity. More mesh pointsare then needed in the numerical calculations. Some typicalfree surface profiles by method 3 are shown in Fig. 8.

The case when d=0.05 is shown in Fig. 9. Now thedifference between the linearized theory and the numericalresults can be clearly seen. As the F decreases to zero, theamplitude of nonlinear waves increases rapidly, especiallywhen F2→0.3. We were unable to obtain convergence of thenumerical solution when F2�0.2. When we increase F2 to-ward unity, the wave amplitudes also increase. Similarly forthe cases of smaller d, the linearized theory fails to predictperiodic waves for F2→1. Thus method 3 is used to get thecorrected wavelength when F�1. The numerically deter-mined waves can be extended continuously at F2=0.9 fromthe solutions obtained by method 1 �linearized theory� whichgives a good approximations for the range of 0.3�F2�0.8.

Finally, the case of d=0.1 is shown in Fig. 10. For thisrelatively large d, the results from the linearized theory andthe long-wave analysis do not agree with our numerical re-sults, but they still have the same trend. We found that the

numerical solutions converge only for a finite range ofFroude number, smaller than for the case of d=0.05.

Next, we investigate the case when d�0, noting that theweakly nonlinear long-wave analysis of Sec. III suggests thatwaves can then be found for F�1+d, which is always sub-critical. Fixing d�0, the wavelength of these waves in-creases, whereas the amplitude decreases as F increases. Ford=−0.01, some typical nonlinear free surface profiles areshown in Fig. 11, whereas the relationship between waveheight and F2 is shown in Fig. 12. The results from thelong-wave analysis are in good agreement with the numeri-cal results as F approaches 1+d. As expected, the discrep-ancy increases as F decreases. From the long-wave analysisof Sec. III, this periodic wave approaches a limiting solutionwhich is one-half of a solitary wave as F→1+d. This solu-tion is discussed later in Sec. V C.

We recall that McCue and Forbes7 have previously com-puted some results analogous to those presented for a casewhen the flow has constant vorticity. For zero vorticity, their

0.8 0.85 0.9 0.95 1 1.050.016

0.018

0.02

0.022

0.024

0.026

0.028

0.03

F2

wav

ehe

ight

(cre

st−

trou

gh)

d = 0.01

Linear Method 2Linear Method 1Numerical Method 2Numerical Method 1Numerical Method 3Long wave theory

FIG. 7. �Color online� Relationship between wave height �crest-trough� andF2 when d=0.01.

0 20 40 60 80 1000.995

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04Nonlinear free surface profiles when d = 0.01, Method 3

F2 0.99

F2 1.01

F2 1.017

FIG. 8. �Color online� Free surface profiles when d=0.01.

0 0.2 0.4 0.6 0.8 1 1.20.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

F2

wav

ehe

ight

(cre

st−

trou

gh)

d = 0.05

Linear Method 1Linear Method 2Numerical Method 1Numerical Method 2Numerical Method 3Long Wave Theory

FIG. 9. �Color online� Relationship between wave height �crest-trough� andF2 when d=0.05.

062102-10 M. Maleewong and R. H. J. Grimshaw Phys. Fluids 20, 062102 �2008�

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results can be compared to ours. Although for the most parttheir results are presented for slightly different parametervalue in the range of 0.3�G�0.7, there is general agree-ment with our results. That is, for both d�0 and d�0, thewave steepness increases as the Froude number decreases,for fixed d. There is an excellent agreement with lineartheory when �d� is small, while nonlinear waves with narrowcrests and broad troughs develop as �d� increases.

B. Highest wave

By using our numerical method, for each fixed F con-sidered, we increased d�0 until the iteration failed to con-verge. This yielded the wave of maximum steepness we wereable to obtain. The relationship between the maximum steep-ness 2a / and the downstream Froude number G2 is shownin Fig. 13, where we also plot the corresponding results fromCokelet.13 A typical free surface profile when G2=0.5434 is

shown in Fig. 14. The difference between Cokelet’s resultsand our results is due to our difficulty in computing verysteep waves. The accuracy of our numerical solutions is im-proved by increasing the number of mesh points per onewavelength. But note that, as G2 increases, the accuracy lossin our computation is compounded since we have to calcu-late many wavelengths, whereas Cokelet computed only onewavelength. The solution shown in Fig. 14 is obtained byusing 60 mesh points per one wavelength and we integrateover 11 wavelengths. Note that these nonlinear waves havebroad troughs and narrow crests, as the highest obtainablewave is reached. We note that the accuracy of our numericalsolutions for very steep waves could be improved by concen-trating more mesh points near the crest of the wave, seeVanden-Broeck and Schwartz,15 for instance. However, wehave not done this here as it would already seem clear thatthe outcome would be just the generation of increasinglysteep Stokes waves up to the limiting wave.

0 0.2 0.4 0.6 0.8 1 1.20.15

0.2

0.25

0.3

0.35

0.4

0.45

F2

wav

ehe

ight

(cre

st−

trou

gh)

d = 0.1

Linear Method 1Linear Method 2Numerical Method 1Numerical Method 2Numerical Method 3Long Wave Theory

FIG. 10. �Color online� Relationship between wave height �crest-trough�and F2 when d=0.1.

0 5 10 15 20 25 30 35 400.975

0.98

0.985

0.99

0.995

1

1.005

1.01Nonlinear free surface profiles when d=−0.01, Method 3

F2 = 0.8

F2 = 0.945

F2 = 0.96

FIG. 11. �Color online� Nonlinear free surface profiles when d=−0.01 forvarious values of F2.

0 0.2 0.4 0.6 0.8 10.017

0.018

0.019

0.02

0.021

0.022

0.023

0.024

F2

wav

ehe

ight

(cre

st−

trou

gh)

d = −0.01

Long Wave TheoryNumerical Method 3

FIG. 12. �Color online� Relationship between wave height �crest-trough�and F2 when d=−0.01.

0.3 0.4 0.5 0.6 0.7 0.80

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

G2

Max

imum

Ste

epne

ss(2

a/λ)

Cokelet 1977Our results

FIG. 13. �Color online� Relationship between maximum steepness and G2.

062102-11 Nonlinear free surface flows Phys. Fluids 20, 062102 �2008�

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C. Wave-free solution

In Sec. III, we showed that a wave-free solution existsonly when F2= �1+d�2 or G2=1 /1+d. Since we expect tofind wave-free solutions only for downstream supercriticalflows, this implies that such solutions can only exist ford�0, This result was confirmed by the weakly nonlinearlong-wave analysis of Sec. III, where it was shown that suchwave-free solutions are just one-half of a solitary wave, ex-cept in a small region near the plate edge, where the localanalysis of Sec. II D shows that there is a weak singularity.Our numerical results for various values of d�0 are shownin Fig. 15 and shows very good agreement with the long-wave theory when �d� is small. In the numerical calculations,we can find such wave-free solutions for a range of Fc�F�1 and corresponding dc�d�0. However, numerically wewere unable to find any such wave-free solutions whend�0. This is to be expected since these wave-free solutions

are, in fact, well approximated by one-half of a solitarywave, which being always an elevation wave, can only ac-commodate the boundary condition at the plate edge whend�0. Also, one would expect a wave-free downstream solu-tion to exist only for a supercritical downstream flow,G�1 so that d�0.

Note that for these wave-free solutions, the ratio of theamplitude to the depth is �d � /1+d. But for the solitary waveof maximum height, this ratio is 0.833, which suggests thatthe interpretation of these wave-free solutions as one-half ofa solitary wave is only valid for at most −0.454�d�0.Thus, on theoretical grounds we estimate that the cutoffvalue dc=−0.454. This is confirmed by our numerical results,where as −d increases above 0.454, we find increasing diffi-culty in satisfying the condition at the plate edge that hx=0 atx=0, y=1. In Fig. 16, we show a closeup of the wave-freeprofiles for a set of values of d around −0.454, which showsthat the slope near the plate edge increases sharply whend�−0.45. Further, we note that McCue and Forbes7 pro-posed a limiting value of dc=−0.5, when a stagnation pointarises at the plate edge and the flow leaves at an angle of2 /3 �measured from the upstream side of the plate� insteadof tangentially. Indeed for values of d�−0.5, the wave pro-file develops a completely unacceptable sharp slope at theplate edge.

VI. SUMMARY AND DISCUSSION

We have presented numerical results for the calculationof the periodic waves generated by a steady free surface flowunder a semi-infinite flat plate in water of finite depth. Grav-ity is included but surface tension is neglected in the dy-namic free surface boundary condition. After reviewing pre-vious studies by McCue and Stump4 on the linearizedproblem and by McCue and Forbes7 on subcritical flows, wehave considered the critical case F�1. Here the downstreamperiodic waves have long wavelengths, and so have used aweakly nonlinear long-wave asymptotic theory to obtainsome analytical results.

0 2 4 6 8 10 12 14 160.9

1

1.1

1.2

1.3

1.4

1.5F2 = 0.7, G2 = 0.543, d = 0.1181, λ = 3.92

FIG. 14. �Color online� Free surface profile for G2=0.5434, F2=0.7, andd=0.1181.

0 1 2 3 4 5 6 7 8 90.5

0.6

0.7

0.8

0.9

1Long Wave TheoryNumerical Solutions

d = −0.45

d = −0.3

d = −0.1

FIG. 15. �Color online� Free surface profiles �wave-free� for various valuesof d�0.

0 0.05 0.1 0.15 0.20.88

0.9

0.92

0.94

0.96

0.98

1

1.02Numerical solutions for d=−0.4, −0.45, and −0.5

d=−0.5

d=−0.45

d=−0.4

FIG. 16. �Color online� A closeup of free surface profiles �wave-free� forvarious values of d around −0.454.

062102-12 M. Maleewong and R. H. J. Grimshaw Phys. Fluids 20, 062102 �2008�

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Waves of finite amplitude have been found numerically.Our numerical method specifies the draft d by the require-ment that the downstream mean depth is specified, and thisprocedure requires us to input the wavelength. We used threemethods: Linear method in which the wavelength is detem-ined from the linear dispersion relation for �d=0�, a linearmethod based on the dispersion relation for �d�0�, and theweakly nonlinear long-wave determination of the wave-length. This flexibility enabled us to obtain nonlinear resultsfor both subcritical and supercritical flows, particular in theregime when F�1. As d gets larger, the downstream peri-odic waves become steeper and eventually reach limitingwaves with narrow crests and board troughs. We have favor-ably compared the highest waves we can obtain to previousresults by Cokelet13 for periodic waves of maximum height.We have found downstream periodic waves for both d�0and d�0. For d�0, the wave amplitude for each fixed dincreases as F→1 and as F→0. But for d�0, the waveamplitude decreases as F increases toward the limiting valueof F=1+d. At this value, downstream wave-free solutionsare predicted to occur, and indeed we found such solutionsfor the range of Fc�F�1, where the cutoff Fc is estimatedto be around the value Fc=0.546 corresponding to the esti-mated value of dc=−0.454, obtained from the known highestsolitary wave height. They have the appearance of one-halfof a solitary wave, in agreement with the weakly nonlinearlong-wave analysis, valid as F�1.

The problem we have considered here is quite simpli-fied. We have included only the effects of gravity on the freesurface, and a more complete treatment would involve alsoincluding surface tension. Also, the plate is semi-infinite inextent, and it may be interesting to consider flow under aplate of finite length. Another issue is the stability of thegenerated downstream waves. This could be addressed by anunsteady formulation, as considered by Haussling16 for thedeep-water case and by Zhu and Zhang17 for the linearizedproblem. In both cases it was found that the solution didindeed converge in the long-time limit to the appropriatesteady-state solution.

ACKNOWLEDGMENTS

The first author would like to thank Associate ProfessorJack Asavanant at the Chulalongkorn University for someuseful comments on the numerical method. We thank thereferees for the helpful comments on the first draft of thispaper. This research was financially supported by the Thai-land Research Fund �Grant No. MRG5080231� to the firstauthor.

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062102-13 Nonlinear free surface flows Phys. Fluids 20, 062102 �2008�

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