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Evolution of Nonlinear Sloshing in a Tank Near

Half the Fundamental Resonance

By David E. Amundsen, Edward A. Cox, Michael P. Mortell,

and Sandra Reck

This article describes the evolution of shallow water waves in a tank thatis closed at one end and is periodically forced at half the fundamental fre-quency at the other end. The nonlinear response occurs at the same order asthe linear response and is governed by a forced Kortewegde Vries (K dV)equation. Unlike the corresponding problem for a gas (or the hydraulic limit),

there may be nonperiodic (beating) solutions and multiple steady solutionsfor the same frequency. The addition of a component at the fundamentalfrequency to the piston input can be used to cancel the nonlinear effects andleave only the linear response in the steady state.

1. Introduction

This article is concerned with the evolution of the motion of surface waves

in a shallow water tank subject to the periodic forcing of an idealized wavemaker. The wave maker is operating in the region of half the fundamentalfrequency of the tank. We will also be concerned with the evolution in thecase of a crank drive where the wave-maker motion is a combination ofmodes, one at half the fundamental frequency and one, at a lesser amplitude,at the fundamental.

Address for correspondence: Professor Michael P. Mortell, Department of Applied Mathematics, UniversityCollege Cork, Ireland.

STUDIES IN APPLIED MATHEMATICS107:103125 103 2001 by the Massachusetts Institute of TechnologyPublished by Blackwell Publishers, 350 Main Street, Malden, MA 02148, USA, and 108 Cowley Road,Oxford, OX4 1JF, UK.

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104 D. E. Amundsen et al.

The problem of the evolution of the surface wave in a tank when the wateris forced at the fundamental frequency is given in Cox and Mortell [1]. Inthat case, undamped linear theory breaks down and cannot predict a periodicresponse. In the situation discussed here; i.e., forcing at half the fundamen-

tal frequency, linear theory predicts a bounded periodic response. However,because of an internal resonance, first-order nonlinear effects are equally asimportant, and we show how these evolve and affect the final state.

The nonlinear problem of forcing at half the fundamental resonance seemsto have been first addressed in the case of a gas by Galiev et al. [2], who exam-ined the final periodic response using an extension of Chesters [3] theory.Mortell and Seymour [4] addressed the same periodic problem in the con-text of a finite rate theory when the input acceleration was such that a shockcould form in one travel time. They had a novel application of the standard

mapping of chaos theory. Cox and Mortell [5] then used the standard map-ping formulation, in the small rate limit (when the input acceleration, as wellas the velocity, is small), to examine the evolution of the gas motion from aninitial state of rest to the final periodic state.

In this article, we use a perturbation scheme to include the effect offrequency dispersion and generate the partial differential equation forcedKortewegde Vries (fK dV) that describes the evolution of the water surfacemotion in the tank. This problem has not been previously addressed eitherfor the periodic motions, or for the evolution. Unlike the case of forcing ofwater waves at the fundamental frequency, where Chester and Bones [6] gavethe experimental observations, we are unaware of any experimental observa-tions in the case we deal with here. There are, however, experimental obser-vations at half the fundamental frequency in the case of a gas; see Galievet al. [2], Zaripov and Ilhamov [7], Merkli quoted in Keller [8], and Althausand Thomann [9]. We use these as a check of the theory given here by reduc-ing to the special case of the hydraulic limit.

In Section 2, the problem is formulated, and the basic equation (31) isderived. This is a periodically forced K dV(fK dV) for the function H. H is anonlinear standing wave, which, when added to the linear periodic solution,

gives the evolution of the surface motion at the fixed end of the tank. Itis noted that (31) is also the long-time equation (3.8) that appears in Coxand Mortell [1] for fundamental resonance. In Section 3, we reduce to thehydraulic limit and check the theory against theory and observation in thecase of long-time periodic oscillations. We also incorporate various dampingmechanisms, see for example equation (41), and again check the output fromthe model equation with experiment.

A model dispersive equation (42), which includes boundary-layer and radi-ation damping, is given based on the results of the hydraulic limit. The evo-

lution of the surface elevation of the water at the fixed end of the tank, forvarious parameter values, is calculated from (42).

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Evolution of Nonlinear Sloshing 105

In Section 5, we introduce a crank drive that incorporates two modes in thewave maker forcing with an order of magnitude difference in the amplitudesof the modes. The interesting result is (57), under the condition (55) on theamplitudes, where the nonlinear responses cancel each other and only the

linear response remains in the periodic state. This is a very simple predictionthat could be checked experimentally.

In all of this work, although the surface elevation has period 4, the nonlin-ear standing wave Hhas period 2 in t. In a companion article, we will returnto (30) and seek solutions for which Ht has period 4 in t. This articlewill illustrate the variety and complexity of solutions of a fKdV.

The mathematical problem dealt with here is a nonlinear initial value,boundary value problem on a semi-infinite strip. Using a perturbation proce-dure, we construct a mathematical model that allows us to follow the evolu-

tion of the surface motion of the fluid from its initial state of rest until thefinal state. Unlike the case of a similarly forced gas motionwhich containsshocksthe final state for water waves may not be periodic. This is not sur-prising, because Equation (31) is equivalent to (3.8) in Cox and Mortell [1]for fundamental resonance, and the latter equation has long-time nonperi-odic solutions that are confirmed by experiment, see Chester and Bones [6].The dispersive equation (31) may have different periodic (equilibrium) solu-tions for the same frequency, which are also confirmed by experiment at thefundamental resonance.

2. Formulation and derivation of the

forced KortewegdeVries equation (fK dV)

Consider the two-dimensional irrotational motion of an inviscid, incompress-ible homogeneous fluid, with density 0, subject to a constant gravitationalforce g. The fluid of undisturbed depth h0, rests on a horizontal bed atz = h0, and has a free surface at z = x t. The fluid is contained in atank of length L, closed at the end x= 0 and with an idealized wave makeroscillating with maximum displacement located atx = L cos t. Because

the wave maker is oscillating near half the fundamental frequency, is nearto c0/2L, where c0= gh0

1/2 is the linear sound speed.The behavior of the fluid in our model is determined by the dimensionless

parameters = h0/L 1, = /L 1, and the detuning = 2 1/2,

where the dimensionless frequency = L/2c0 and = 1/4 is half the

fundamental frequency. We choose

2 =K and = (1)

where K= O1 and = O1, to ensure that nonlinearity, dispersion, and

resonant detuning all enter at the same level of approximation over the timescale t=O1.

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We express the fluid flow in dimensionless variables

x = x

L z =

z

h0 t =

2

t

=

c0L =

h0 p =

p0

0c0

(2)

where is the velocity potential, p0 is the pressure at the free surface, 0 thefluid density, and is the displacement of the free surface from its undis-turbed height z = 0. In these variables (dropping the primes), the velocitypotential xzt satisfies

22

x2 +

2

z2 =0 0< x

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Equations (3)(10) constitute a nonlinear, free boundary, initial boundaryvalue problem in the region

0 x 1 cos

2t 1 z x t t0

The fast time scale t+ = t is a measure of the time taken by a wave totraverse the length of the tank. The slow timescale = t+ is a measureof the time over which nonlinear and frequency dispersion effects becomesignificant and involves many travel times in the tank. As the surface wavestravel over and back in the tank, they are slowly steepened by nonlinearityand simultaneously dispersed by the effect of the bottom boundary whilebeing externally driven by a wave maker. We propose to set up a partialdifferential equation that describes how the surface of the fluid evolves froman initial state of rest under the periodic action of the wave maker. Theresult of the evolution may, or may not, be periodic. We want to investigatethe dependence of the end state on the detuning parameter and on adamping parameter that is introduced later.

We assume a standard multiple scale expansion for and :

= 0xzt + 21xzt + (11)

= 0xt + 21xt + (12)

Substituting (11) into (3) and (4) gives

0 =0xt (13)

1 =1xt 1

2K1+ z20 xx (14)

Equation (6) gives

0t

+ 0=0 (15)

and

1t

+ 1 = 0

1

2

0x

22 0

t + 1

2K

3

0x2t

(16)

Similarly (5) yields

20x2

+0

t =0 (17)

21x2

+1

t =

K

6

40x4

0

0

20x2

+ 2 2

0x2

0x

0x

(18)

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Equations (15) and (17), and the boundary condition (7), imply

0x

=f0t x + f

0t+ x (19)

wheref0 is an arbitrary function andf0 denotes the derivative with respect tothe first argument. Thus, the wave response consists of two noninteracting,oppositely traveling waves that travel with the linear sound speed gh0

1/2.We note that f0 is modulated on the time scale .

If we now combine (7) with (16) and (18), using (19) and (17) for 0 and0, we get

1x

= K

6

f0 t x f

0 t+ x x

f

40 t+ x + f

40 t x

f0

t x f0

t+ x xf0

t+ x + f

0 t x

2

f0t x f

0t+ x x

f0t+ x + f

0t x

3

4

f0t x

2

f0t+ x 2

2x

f0t+ x f

0t+ x + f

0t x f

0t x

+1

4

f0t x f

0t+ x f0t+ x f

0t x

1

4f1t+ x f1t x (20)

where f1 is an arbitrary function.The wave-maker boundary condition (8), with (11) and (12), implies that

on the free surface at x= 1, we have

0x

1 t =

2 sin

2t (21)

and

1

x 1 t =sin

2t+

20

x2 1 t cos

2t+

1

2K

30

x3 1 t (22)

Now (19) and (21) yield the linear difference equation

ht+1 ht 1 =

2 sin

2t (23)

where ht =f0t . The solution of (23) can be written

ht =Ht

4 cos

2t (24)

where H satisfies the periodicity condition

Ht+ 2 =Ht (25)

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We can interpret the solution ht in (24) as the superposition of the peri-odic linear forced oscillation and a nonlinear standing wave Ht , whichwill be determined at the next order. We also note from (23) that

ht+4 ht =0 (26)

i.e., ht has the same period in tas the wave maker. We note from (24)and (25) that the solutionh is split naturally into the sum ofHwith period 2and/4cos /2twith period 4, and the latter is antisymmetric about its zeros.

Using (19) in (22) and reformulating the resulting equation over oneperiod of the wave maker at x= 1 yields, on using (23),

1x

1 t +1

x 1 t+ 2 =0 (27)

The boundary condition (27) on x = 1, with the representation (20) for1/x, now gives

1

8

h1t+3 h1t1

= K

6

ht+1 + ht1

+1

2

dM

d +

ht+1 + ht 1

+ 2

ht+ 1 + ht 1

+

3

2

ht+1 ht+1 + ht 1 ht 1

1

8Mht+1 (28)

where h1t =f

1t , and

M = t+3

t1

hs ds (29)

Then M is proportional to the slowly evolving mean of the flow.We now substitute for ht from (24) into (28) noting that ht has

period 2 in t, to get

1

16

h1t+ 2 h1t2

=Ht +

2 M

16

Ht+

3

2H Ht

K

6Httt

+ 14

M 2

M128

sin2

t 33

128sin t (30)

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We determine an equation forHby ensuring secular terms do not arise in h1.BecauseHhas period 2 in t, the equation for Hmust be invariant under thetransformationtt+2. Thus, the partial differential equation for Ht is

H+2 Ht+3

2H Ht

K

6Httt=

33

128sin t (31)

We also require M = 0 to avoid secular terms in h1t . Thus, h1t satisfies

h1t+ 2 h1t2 =0 (32)

and is then determined at the next order.We now have from (29), (24), and (25) with the requirement that

M =0,

t+1t1

Hs ds= 0 (33)

i.e., the mean ofHt is zero over the evolution.If we integrate (31) over one period in t, noting that H has period 2 in

t, we find that the mean ofH is independent of . Thus, when Ht 0 haszero mean on 1 t1, condition (33) is satisfied for all . Consequently,to ensure condition (33), we choose the reference state, on noting (24), sothat the initial condition ht 0 = /4cos /2t has zero mean. Then themean of Ht remains zero throughout the evolution, and the mean of aperiodic state (when it occurs) is the reference state. Variations in pressureand velocity are usually measured from this reference state in experimentalobservations.

The evolution of the surface elevation, 00 t , at the closed end of

the tank is given by, on noting (15), (19), the definition ofht and (24),

00 t = 2ht =

2 cos

2t 2Ht (34)

where Ht is determined by (31), (25), and zero mean initial conditions.This is the measurement made by Chester and Bones [6] in the experimentsthey conducted at the fundamental resonant frequency. Equation (31) is aperiodically forced K dV equation fKdV and, interestingly, is the same

equation as (3.8) given in Cox and Mortell [1] and for the long-time evolutionof the signal in the case of forcing at the fundamental frequency.

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3. Hydraulic limit

We now consider the special case when the dispersion coefficient K = 0,so that only the nonlinear, the detuning and forcing effects are taken into

account. If we define

F=2 +3

2H (35)

then (31), with K=0, implies

F+ FFt= 93

256sin t (36)

This is equivalent to equation (3.14) in Cox and Mortell [10] for the evolution

of gas oscillations in a tube that are forced at the fundamental frequency andto Equation (106) in Cox and Mortell [5] when the oscillations are forced athalf the fundamental frequency. The steady-state periodic solution is given by

FdF

dt =

93

256sin t

20

Fsds= 4 (37)

and Ft+2 = Ft, on noting (33) and (35). The mean condition in (37)determines the position of the shock in the signal F when the applied fre-quency lies within the resonant band. This agrees with the small rate limit of

Mortell and Seymour [4], and also with Keller [8].We give here a variant of (31), when K=0 to allow for the various types

of damping that may occur, and we check that the signal evolves to matchthe experimental observations in the periodic state. We consider the equationfor Ht , where Hhas period 2 in t,

H+2Ht+3

2H Ht+ H+ Htt=

33

128sin t (38)

with the initial condition

Ht =0 =

4 cos

2t1 1 t1 (39)

In (38), introduces the effect of high-frequency damping, and is theBurgers term that structures a shock. When = 0, (38) can be written as

F+ FFt+ F=2 +93

256sin t (40)

where F is given by (35). Equation (40) can then be put in the form of a

viscously damped pendulum subject to a constant external torque 2, seeCox and Mortell [11].

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The effect of boundary-layer damping from the sides and bottom of thetank, as formulated in Chester [12] can be incorporated into (38) to give

H+ 2Ht+

3

2 H Ht+ H+ Htt

+

2

sgnr +1

Ht

t r r1/2dr= 33

128sin t (41)

We compare the solutions of (41) for with the experimental observa-tions for a gas in a tube, bearing in mind the standard waterwave analogy.

In Figure 1, we illustrate the evolution of H under (41) and (39) andin Figure 2 for Ht 0 = 0, when the parameters are = 00 = 01= 001 = 005. We note that, in both cases, the zero mean is maintained

throughout. The initial condition (39) is the steady periodic solution to (38)when = = = 0. Thus, Figure 1 shows the effect of damping on thissolution.

Figure 3 is derived from (41), using (24), and (34) to calculate the surfacedisplacement, when the initial condition is Ht 0 = 0 = 00 = 01 = 001 = 04939. This is good agreement with Merklis experimentalresult as reported in Figure 4 of Keller [8].

Figure 4 shows the fully evolved periodic state of the surface displacementfor various values of the detuning as determined by (41) and (24) for initial

conditions (39) on Hwith = 01 = 01 = 0.

Figure 1. Steady state ofH for Ht 0 = 4

cos 2t1 = 0 = 01 = 001 = 005

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Figure 2. Evolution ofHwith zero initial condition. = 0 = 01 = 001 = 005.

The evolved periodic states shown in Figure 5 agree with the theoreticalperiodic solutions given in Keller [8]. Althaus and Thomann [9] gave a care-ful comparison of their experimental results with the theoretical predicitions

of Keller [8] and found good agreement. It is more difficult to get agreement

Figure 3. Evolution forHt 0 =0 = 0 = 01 = 001 = 04939.

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Figure 4. Steady states for= 01 = 01 = 0.

with the experimental results of Zaripov and Ilhamov [7]. Althaus andThomann [9] caution that the formers experimental set-up generated stronghigher harmonics in the piston velocity. However, the periodic solutions dis-

played in Figure 4 are in good qualitative agreement with the experimentalpressure waveforms of Figure 5 in Zaripov and Ilhamov [7].It should be noted that for a small rate theory dealing only with periodic

solutions; e.g., Keller [8], the steady-state solution of the hyperbolic prob-lem [(36) or (40) with F 0] is not uniquely determined, see Mortell andSeymour [4]. Because we deal with the evolution in this paper, the hyperbolicproblem is uniquely determined.

The various comparisons made here for the hydraulic case will act as aguide when we examine the frequency dispersed case K = 0, where noexperimental observations at half the fundamental frequency are available.

4. Damped dispersive oscillations

In Section 3, we were able to compare the output from certain model hyper-bolic equations, which contained damping, with the results of experimentswith a gas in a tube. We now incorporate these damping mechanisms intothe dispersive equation (31) and examine the evolution of the motionsattributable to the initial conditions and the forcing terms. We also compare

the results for the dispersive cases with those when frequency dispersion isabsent.

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Figure 5. Steady states for= 01 = 001 = 005.

The damped, dispersive equation we take is

H+ 2Ht+3

2H Ht+ H

K

6Httt

+

2

sgnr +1

Ht

t r r1/2dr= 33

128sin t (42)

and the initial condition is given by (39). We recall that the initial conditionhas zero mean; i.e., M =0.

Equation (42) is just (41) with Httremoved (we no longer need to struc-ture a shock) and the dispersive term K

6Httt inserted. This equation will

allow us to follow the evolution of the surface elevation through to the peri-odic state from the given initial condition. In the periodic state, the meanis zero, and the detuning is measured from a resonant frequency deter-mined by having the reference state as the mean of the periodic state. In thisformulation, comparison with experiments is easier.

In Figure 6, we plot t , for various values of the detuning , using(42) and (34), where the initial condition is zero, K = 01 = 005 and= 0. This figure shows the periodic surface elevation at the closed end ofthe tank for the values of the detuning shown.

In Figure 7, we show the evolution of a beating (nonperiodic)oscillation resulting from (42) and (34) with zero initial condition and

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Figure 6. Steady states forHt 0 = 0 = 0 = 005 K= 01.

Figure 7. Evolution of t in dispersive case. Ht 0 = 0 = 0 = 005 K = 01,= 02.

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Figure 8. Evolution of t in nondispersive case (with steady state shown at bottom).

Ht 0 =0 = 0 = 005 K= 0 = 02.

K = 01, = 02 = 005 = 0. Such a solution does not occur inthe nondispersive (hyperbolic) case when K = 0; see Figure 8, where theevolution to the periodic solution is shown.

Figure 9 is an example of two equilibrium (periodic) states of (42) for thesame values of the parameter: K = 01 = 02 = 01 = 005. Suchnonuniqueness does not occur for the hyperbolic equation found by setting

K=0; see Figure 5 for = 02.The beating solution in Figure 7 settles down to the periodic solution

with one peak shown in the upper part of Figure 9 when the high-frequencydamping is increased from = 0 to = 01, and the boundary layer dampingcoefficient= 005 is held constant. A boundary in an appropriate parame-ter plane has been crossed, and we will explore this aspect of solutions in acompanion paper in preparation.

The periodic solution shown in the lower part of Figure 9, which has mul-tiple peaks, was obtained by evolving to a periodic solution for = 03 and

using that solution as the initial condition for = 02. Clearly, then, thereis a dependence of the periodic state on the initial data.

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Figure 9. Multiple steady states for = 01 = 005 K= 01 = 02.

The predictions made here at half the fundamental resonance of beatingsolutions and multiple equilibrium states for the same value of the parametersare capable of being tested by experiment. The latter are associated with aDuffing spring behavior in the amplitude/frequency diagram. We note thatsuch states do exist for excitation at fundamental resonance; see Chester andBones [6] for the experimental observations and Cox and Mortell [1] for thetheoretical results.

5. Crank drive

We now consider the response of the fluid in the tank when the idealizedwave maker is located at

x= L 1cos t 2cos2t (43)

when the bar indicates a dimensional variable, 1/L 1 2/L 1, and lies in the neighborhood of half the fundamental frequency. The displace-ment (43) has the form of a crank drive. Using the same dimensionless vari-ables as in Section 2, we find that the velocity is given by

x =2u1

1

4+

1

2

sin

2t+ 24u2

1

4+

1

2

sint (44)

on

x= 1 u1cos2t 2u2cost (45)

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Figure 10. Steady states for crank drive with (5.13). K = 01 = 0 = 01 = 001,

= 005.

where

= 14

+ 12

1L

=u1 >02L

=2u2 >0 (46)

and 0 < 1. We have set the amplitude of the second term in (44),which is oscillating at fundamental resonance, at 02 as it yields a periodicresponse at 0. It can then match the response to the first term in (44).

Following the procedure in Section 2, we find the boundary conditions onx= 1:

0

x

=u1

2 sin

2t (47)

1x

1 t =u1 sin

2t u1K

3

16sin

2t

+ u2sint + u1ht+1 + ht 1 cos

2t (48)

where

ht =f0t and ht+1 ht1 =u1

2

sin

2

t (49)

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Figure 11. Steady states in hyperbolic case for crank drive with (5.13).K= 0 = 0, = 01,

= 001, = 005.

The equivalent of (27) becomes

1x

1 t +1

x 1 t+ 2 =2u2sint (50)

On using (20), (24), and (25) in (50), we get

1

16

h1t+2 h1t2

1

2u2sint

=Ht +

2

M

16

Ht+

3

2 H Ht

K

6Httt

+1

4M u1

2M

128 sin

2t u21

33

128sin t (51)

This reduces to (30) on setting u2 =0 u1=1. Because Ht has period 2in t, and we must avoid secular terms in h1, we set the mean M =0; i.e.,(33) must hold, and we determine Ht by

Ht +2Ht+3

2H Ht

K

6Httt

= 12

u2sint + u21

33

128sin t (52)

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Figure 12. Steady states for crank drive with (5.13) and = 03.K= 01, = 01, = 001,

= 005.

This reduces to (31) on setting u1=1 u2 =0. Then

h1t+ 2 h1t2 =0 (53)

and is determined at the next order.When high-frequency and boundary-layer damping are included, (52)

becomes

Ht +2Ht+32

H Ht+ HK

6Httt

2

sgnr +1

Ht

t r r1/2dr

=u2133

128sin t

1

2u2sint (54)

We note that if, in the special case = 0, we take

u2 = 32

64 u21 (55)

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Figure 13. Damped solitons.K= 01 Ht 0 =cos t= 03 = 0 = 005.

then (54) reduces to the unforced, damped K dV

H+ 2Ht+3

2H Ht+ H

K

6Httt

+

2

sgnr +1

Ht

t r r1/2dr=0 (56)

In Figure 10, we plot the periodic output from (54) and (24) for dif-

ferent value of the phase when the special relation (55) holds betweenthe amplitude of the input modes. In this case, the initial condition is zero,K = 01 = 0 = 01 = 001, and = 005. Figure 11 is the cor-responding hyperbolic case when K = 0. The prediction that the motion islinear for = 0, given by (57), holds for both dispersive water waves andnondispersive gases and should be capable of being tested experimentally.Figure 12 is for the same parameters as Figure 10, except that = 03. Wecan see the extra solitons in the periodic state (as compared with = 0); seeFigure 6.

If the initial condition for (56) is Ht 0 = 0, then Ht 0 is thesolution of (56), and there is no standing wave. IfHt 0 = 0, but has zero

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Figure 14. Nondispersive caseK= 0. Ht 0 = cos t= 0 = 0 = 002 = 005.

mean on 0 t 2; e.g., Ht 0 = cos t, then when = 0 = 0 (56)exhibits the classical soliton solutions of Zabusky and Kruskal [13], whichare nonlinear standing waves in the tank. If, however, there is damping inthe system, = 0, or = 0, then Ht 0 as . When, eitherthrough the initial conditions or through damping, the nonlinear standingwave corresponding to (56) is absent in the long term, then the periodicsurface elevation measured at the fixed end of the tank is given simply by thelinear result, independent of the dispersion, see (34),

00 t

2u1cos

2t as (57)

This is illustrated in Figure 13 for = 03. The corresponding nondisper-sive case, K=0 = 0, is illustrated in Figure 14.

A result such as (57) was noted by Keller [8] for the periodic response ofa gas in a tube. In the case of the evolution of a gas motion, the standingwave will be subject to shock formation and will, thus, always be attenuatedout of the system as to leave the periodic motion (57). However,

some other form of damping is required to attenuate the standing wave in atank of a shallow water; e.g., boundary layer damping, as in Figure 13.

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Figure 15. Evolution of surface displacement at maximum nonlinear response = 0365

when = K= 01 = 01 = 005).

If, on the other hand, we choose = in (52), we have constructiveinterference where the two terms on the right-hand side combine to give

Ht +2Ht+3

2H Ht

K

6Httt=

1

2u2+

33

128u21

sin t (58)

In this case, by varying in (58), we can find how the crank drive is tunedto give the maximum nonlinear wave response; see Figure 15, where theinitial amplitude is amplified by a factor of 4.5 in the periodic state. This isa significant amplification, although not an order of magnitude.

Acknowledgment

MPM acknowledges the hospitality of the Courant Institute of the Mathe-matical Sciences at New York University.

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References

1. E. A. Coxand M. P. Mortell, The evolution of resonant water wave oscillations, J. Fluid

Mech.162:99116 (1986).

2. S. V. Galiev, M. A. Ilhamov, and A. V. Sadykov, Periodic shock waves in a gas, Isz.Akad. Nauk S.S.S.R. Mech. Zhid. iGaza.2:5766 (1970).

3. W. Chester, Resonant oscillations in closed tubes, J. Fluid Mech. 18:4464 (1964).

4. M. P. Mortell and B. R. Seymour, A finite rate theory of quadratic resonance in a

closed tube, J. Fluid Mech. 112:411431 (1981).

5. E. A. Coxand M. P. Mortell, The evolution of nonlinear resonant oscillations in closed

tubes: Solution of the standard mapping,CHAOS 2:201214 (1992).

6. W. Chester and J. A. Bones, Resonant oscillations of water waves. II. Experiment, Proc.

Roy Soc. Lond. A306:2339 (1968).

7. R. G. Zaripov and M. A. Ilhamov, Nonlinear gas oscillations in a pipe, J. Sound Vib.46:245257 (1976).

8. J. J. Keller, Subharmonic nonlinear acoustic resonances in closed tubes, J. App. Math.

Phys. (ZAMP)26:395405 (1975).

9. R. Althaus and H. Thomann, Oscillations of a gas in a closed tube near half the funda-

mental resonance, J. Fluid Mech. 183:147161 (1987).

10. E. A. Cox and M. P. Mortell, The evolution of resonant oscillations in closed tubes.

J. App. Math. Phys. (ZAMP)34:845866 (1983).

11. E. A. Coxand M. P. Mortell, The evolution of resonant acoustic oscillations with damp-

ing, Proc. R. Ir. Acad. 89A:147160 (1989).

12. W. Chester, Resonant oscillations of water waves, I. Theory, Proc. Roy. Soc. Lond. A

306:522 (1968).

13. J. J. Zabusky and M. D. Kruskal, Interaction of solitons in a collisionless plasma and

the recurrence of initial states,Phys. Rev. Lett. 15:240243 (1965).

California Institute of Technology

University College Dublin

University College Cork

University College Dublin

(Received October 6, 2000)

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