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Capillary-Gravity Waves Generated by a Sudden Object Motion F. Closa,1 A.D. Chepelianskii,2 and E. Raphaël1, ∗ 1Laboratoire Physico-Chimie Théorique, UMR CNRS GULLIVER 7083, ESPCI, 10 rue Vauquelin, 75005 Paris, France and 2Laboratoire de Physique des Solides, UMR CNRS 8502, Bâtiment 510, Université Paris-Sud, 91405 Orsay, France (Dated: January 27, 2010) We study theoretically the capillary-gravity waves created at the water-air interface by a small object during a sudden accelerated or decelerated rectilinear motion. We analyze the wave resis- 0 tance corresponding to the transient wave pattern and show that it is nonzero even if the involved 1 velocity (the final one in the accelerated case, the initial one in the decelerated case) is smaller 0 than the minimum phase velocity cmin =23cms−1. These results might be important for a better 2 understandingofthepropulsionofwater-walkinginsectswhereacceleratedanddeceleratedmotions frequentlyoccur. n a PACSnumbers: 47.35.-i,68.03.-g J 6 2 I. INTRODUCTION erated case) is smaller than the minimum phase velocity c = 23cms 1. The physical origin of these results ] min − n is similar to the Cherenkovradiationemitted by acceler- If a body (like a boat or an insect), or an external y ated (or decelerated) charged particles [14, 15]. pressure source, moves at the free liquid-air interface, it d generates capillary-gravity waves. These are driven by a - u balance between the liquid inertia and its tendancy, un- fl dertheactionofgravityandundersurfacetensionforces, II. EQUATIONS OF MOTION . to return to a state of stable equilibrium [1]. For an in- s c viscid liquid of infinite depth, the dispersion relation of We consider an inviscid, deep liquid with an infinitely si capillary-gravitywaves,relatingthe angularfrequency ω extending free surface. To locate a point onthe free sur- y tothe wavenumberk isgivenbyω2 =gk+γk3/ρ,where face, we introduce a vector r = (x,y) in the horizontal h γ is the liquid-air surface tension, ρ the liquid density, plane associated with the equilibrium state of a flat sur- p and g the acceleration due to gravity [2]. The energy face. Themotionofthedisturbanceinthisplaneinduces [ carried away by the waves is felt by the body (or the a vertical displacement ζ(r,t) (Monge representation)of 1 pressuresource)asadragR , calledthewave resistance the free surface from its equilibrium position. w v [3]. In the case of boats or ships, the wave resistance Assuming that the liquid equations of motion can be 5 hasbeenwellstudiedinordertodesignhullsminimizing linearized (in the limit of small wave amplitudes), one 1 it [4]. The case of objects that are small compared to has [13] 7 4 the capillarylengthκ−1 = γ/(ρg)has beenconsidered . only recently [5–11]. ∂2ζˆ(k,t) kPˆ (k,t) 1 p + ω(k)2ζˆ(k,t) = ext , (1) 0 In the case of a disturbance moving at constantveloc- ∂t2 − ρ 0 ity V = V on a rectilinear trajectory, the wave resis- 1 tance R |is|zero for V < c where c = (4gρ/γ)1/4 wherePˆ (k,t)andζˆ(k,t) arethe Fouriertransformsof w min min ext : is the minimum of the wave velocity c(k) = ω(k)/k = the pressure distribution and the displacement, respec- v i g/k+γk/ρ for capillarity gravity waves [3, 5, 12]. Ef- tively[16]. Inwhatfollows,wewillassumethatthepres- X fectively,intheframemovingwiththatobject,theprob- sure distribution is axisymmetric around the point r0(t) r lpem must be stationary. That is true only if the ra- (corresponding to the disturbance trajectory). Pˆ (k,t) a ext diated waves have a phase velocity c(k) equal to the can then be written as Pˆ (k)e ik.r0(t). ext − object’s velocity V. If V < c , no solutions exist, min hence no waves and the wave resistance is zero. For water with γ = 73mNm 1 and ρ = 103kgm 3, one has A. Uniform straight motion − − c = 0.23ms 1 (at room temperature). It was re- min − cently shown by Chepelianskii et al. [13] that no such Letusfirstrecalltheresultspreviouslyobtainedinthe velocitythresholdexistsfor asteady circularmotion,for case of a uniform straight motion [3, 12, 17, 18]. Such a which, even for small velocities, a finite wave drag is ex- motion corresponds to r (t) = ( Vt,0) where V is the 0 perienced by the object. Here we consider the case of a constant velocity of the disturban−ce. sudden acceleratedor deceleratedrectilinearmotion and Equation (1) then becomes show that the transient wave pattern leads to a nonzero wave resistance even if the involved velocity (the final ∂2ζˆ(k,t) kPˆ (k)eiVkxt + ω(k)2ζˆ(k,t) = ext . (2) one for the acceleratedcase,the initial one for the decel- ∂t2 − ρ 2 0.5 0.4 κ) 2 0 P 0.3 γ)/ ( w R 0.2 ( 0.1 0 0 1 2 3 4 5 6 7 8 9 10 V/c min FIG. 1: Wave resistance Rw (in units of P02κ/γ) as a func- mtioontioonf,tsheeerEeqd.u(5ce).d TvehleocpitryessVu/rcemdinistfuorrbaancueniifsoramssusmtreadightot FIG.2: ThewaveresistanceRw (inunitsofP02κ/γ)isshown wbeheareLobreisnttzhieano,bwjeictthsaizeFo(usertietrotrba=nsf0o.1rmκ−P1ˆ)e.xt(k)=P0e−bk, amsoatiofnunwctitiohndiofffetrhenetrUed=uceVd/ctmimine(csmeeinEκqt.(f7o)r).anReascpceecletrivaetelyd, panels (a), (b), (c) and (d) correspond to a reduced velocity U =1.5, 1.1, 0.9 and 0.5. The above equation corresponds to the equation of an harmonic oscillator forced at angular frequency Vk . B. Accelerated straight motion x We can solve it by looking for solutions with a time- dependence of the form eiVkxt, leading to We now turn to the case where the disturbance – initially at rest – is suddenly set to a uniform mo- kPˆ (k) tion (characterized by a constant velocity V) at time ζˆ(k,t) = ext eiVkxt. (3) t = 0. The corresponding trajectory is given by r (t) = −ρ(ω(k)2 (Vk )2) 0 − x V tθ(t)u . As long as the perturbation does not move x − (i.e. for t < 0), the wave resistance is equal to zero. In Following Havelock [3], the wave resistance R experi- w order to calculate the wave resistance for t>0, we solve enced by the moving disturbance is given by Eq.(2) along with the initial conditions ζˆ(k,t = 0) = 0 ∂ζˆ(k,t=0) and =0, yielding dk dk ∂t R = i x y kζˆ(k,t)Pˆ (k,t). (4) w ∗ − 2π 2π ZZ t kPˆ (k) ζˆ(k,t)= dτ ext e ikr0(τ)sin(ω(k)(t τ)). − This expression for the wave resistance represents as − ω(k)ρ − Z0 the total force exerted by the external pressure on the (6) free surface R = dxdyP (r,t)▽ζ(r,t) written in Equation (4) then leads to the following expression for w ext Fourier space. Using Eq.(3) and integrating overthe an- the wave resistance RR gular variable one obtains [5] R (t)= w Rw = Z0∞d2πk kPρ02 e−2bkV2 θ(1V−−(cc((kk))/)V)2 ux, (5) Z0∞2dπk Z0tduk3|Pρˆωex(tk()k)|2 sin(ω(k)u)J1(kVu) ux, (7) p where θ(x) is the Heaviside step function and u the where J (x) is the firstBessel functions of the first kind. x 1 unit vector along the x-axis. The behavior of R as a In the long time limit, one has [19] w function of the disturbance velocity V is illustrated in Fig. 1. There we have assumed a pressure disturbance of Lorentzian form with a Fourier transform Pˆ (k) = t ω(k)θ(V c(k)) ext lim sin(ω(k)u)J (kVu)du= − 1 P e bk, where b is the object size (set to b = 0.1κ 1). t V2k2 1 (c(k)/V)2 0 − − →∞Z0 − This choice will be taken for the figures throughout this (8) p work. The wave resistance is equal to zero for V <c andthereforethewaveresistanceEq. (7)convergestothe min and presents a discontinuous behavior at V = c (see uniformstraightmotionresult,Eq. (5). The behaviorof min reference [8] and [13] for a more complete discussion). R (t) is represented on Fig.2 for different values of the w 3 disturbance velocity and will be discussed in detail in (a) 0.5 (b) 0.5 U=1.4 U=1.1 section III. 0.4 0.4 κ) 0.3 κ) 0.3 2 P0 0.2 2 P0 0.2 C. Decelerated straight motion γ )/ (w 0 .01 γ )/ (w 0 .01 R-0.1 R-0.1 ( ( -0.2 -0.2 -0.3 -0.3 Let us now consider the case of a disturbance moving 0 5 10 15 20 25 30 0 5 10 15 20 25 30 with a constant velocity V for t < 0, and suddenly set t cminκ t cminκ andmaintainedatrestfort>0. Thiscorrespondstothe (c) 0.5 (d) 00..55 U=0.98 UU==00..77 following trajectory: r0(t) = V tθ( t)ux. As long as κ) 00..34 κκ)) 0000....3434 t < 0, the wave resistance R−is giv−en by the uniform 2 P0 0.2 22 PP00 00..22 straight motion result Eq. (5)w. In order to calculate the γ )/ (w 0 .01 γγ )/ ( )/ (ww 00 ..0011 wave resistance for t>0, we solve Eq.(2) along with the (R-0.1 (R(R--00..11 -0.2 --00..22 initial conditions -0.3 --00..33 0 5 10 15 20 25 30 00 55 1100 1155 2200 2255 3300 t cminκ tt ccmmiinnκκ kPˆ (k) ζˆ(k,t=0) = ρ(ω(−k)2 ex(tVk )2) (9) FIG.3: ThewaveresistanceRw (inunitsofP02κ/γ)isshown − x as a function of the reduced time cminκt for a decelerated and motion with different reduced velocities U = V/cmin (see Eq.(12)). Respectively,panels(a),(b),(c)and(d)correspond ∂ζˆ(k,t=0) = −kPêxt(k)(iVkx) (10) to a reduced velocity U =1.4, 1.1, 0.98 and 0.7. ∂t ρ(ω(k)2 (Vk )2) x − (where we have used Eq. (3)). This leads to III. RESULTS AND DISCUSSION ζˆ(k,t>0)= A. Accelerated straight motion kPˆ (k) kPˆ (k) ext ext cos(ω(k)t) ρω(k)2 −ρ(ω(k)2 (Vkx)2)! Figures 2 and 3 represent the behavior of the wave − k(iVk )Pˆ (k) sin(ω(k)t) kPˆ (k) resistance for the accelerated and the decelerated cases, x ext ext . (11) respectively. In order to get a better understanding of − ρ(ω(k)2−(Vkx)2) ω(k) − ρω(k)2 the behavior of Rw = Rw ux, we will perform ana- · Equation (4) then leads to the following expression for lytic expansions of Eq.(7) and Eq.(12), respectively. Let the wave resistance: us start with the accelerated case, Eq.(7). Since one has the product of two oscillating functions, sin(ω(k)u) Rw(t)= and J1(kVu), one can use a stationary phase approximation [21]. The sine function oscillates with a phase Z0∞ 2dπkk|Pˆρex.Vt(2k)|2 cos(ω(k)t) θ(V −cc((kk)))2ux wφ1ith=aωp(hka)usewφh2er=eaVs kthue. BTehsesierl pfurondctuicotnhJa1s othscuisllattweos 1 oscillating terms, one with a phase φ = φ φ and s −(cid:18) V (cid:19) the other with a phase φ = φ +−φ . 1A−cco2rding + 1 2 to the stationary phase approximation, the important +Z0∞d2πkk|Pˆρex.Vt(2k)|2 sin(ω(k)t)c(Vk)−θ(c(ck(k))−2V)ux. Twahveenlautmtebreresquaarteiognivdenoesbynodtdφka−dm=it0anayndreaddlφk+solu=tio0n.  1  s V −  and the corresponding contribution to the wave resis-  (cid:18) (cid:19)  tancedecreasesexponentiallyandcanbe neglected. The   (12) dφ equation − = 0 leads to (cg(k) V)u = 0, where In the long time limit (t ), the Riemann-Lebesgue dk − lemma [20], for a Lebesgu→e in∞tegrable function f cg(k) = dωd(kk) is the group velocity of the capillary- gravity waves [3]. This equation has solutions only if V > min(c (k)) = (3√3/2 9/4)1/4c 0.77c . g min min lim f(x)eixtdx = 0 (13) Whenthisconditionissatisfie−d,twowavenum≈bersarese- t →∞Z lected: kg (mainlydominatedbygravity)andkc (mainly permitstodeterminethelimit: thewaveresistancegiven dominated by capillary forces), with k <k (see Fig.4). g c by Eq. (12) converges to 0 for t , as expected. The Ifthesetwowavenumbersaresufficientlyseparated(that behavior of R (t) is represente→d i∞n Fig.3 for different is,forvelocitiesnottoocloseto0.77c ),onethenfinds w min values of the disturbance velocity and will be discussed that in the long time limit R oscillates around its final w in detail in section the next section (Sec.III). value as 4 2 c(k)/c min c (k)/c g min U=V/c min min 1.5 c k)/ (g c or 1 n mi c c(k)/ 0.5 0 k k k k 0 g 0.5 1 1 c 1.5 2 2 k/κ FIG. 4: Graphical representation of wavenumbers k1, k2, kc FIG.5: ThewaveresistanceRw (inunitsofP02κ/γ)isshown and kg inunitsof κ. k1 andk2 arethesolutions oftheequa- as a function of the reduced time cminκt for an accelerated tionc(k)=V andcorrespondtotheintersectionbetweenthe motionwithdifferentU =V/cmin (seeEq.(7)). Respectively, curve c(k)/cmin and the line U = V/cmin. kg and kc are panels (a), (b) and (c) correspond to a reduced velocity U = the solutions of the equation cg(k) = V and correspond to 1.5, 1.1 and 0.9. The solid lines are obtained by a numerical the intersection between the curve cg(k)/cmin and the line integration of Eq.(7). The dashed lines correspond to the U =V/cmin. Analyticalexpresionsofkc andkg canbegiven asymptotic expansion given by Eq. (14). but are rather lengthy. sudden acceleration of the disturbance (taking place at t = 0), a large range of wavenumbers is emitted. Waves withwavenumbersksuchthatc (k)>V willmovefaster 1 kc5/2 Pêxt(kc)2 cos(Ωct) thanthe disturbance (whichmogveswiththe velocityV), R (t)=R ( )+ | | w w ∞ 2πρ√V d2ω Ωct whereas waveswith wavenumbersk such that cg(k)<V ω(k ) (k ) c |dk2 c | will move slower. The main interaction with the mov- r ing disturbance will therefore correspond to wavenum- 1 k5/2 Pˆ (k )2 sin(Ω t) + g | ext g | g , (14) bers satisfying cg(k) = V, that is to kc and kg (hence 2πρ√V d2ω Ωgt their appearance in Eq.(14)). Due to the Doppler effect, ω(k ) (k ) g r|dk2 g | thewaveresistanceRw (whichistheforceexertedbythe fluid on the moving disturbance) oscillates with an an- wVh<erecmRiwn()∞, Ω)cis=gi(vce(nkcb)y−Ecqg.((k5c)))(kacndanisdeΩquga=l t(oc(zkegr)o−if gaunldarΩfgreiqnuEenqc.(y14(c)(.k)−V)k, hence the appearance ofΩc c (k ))k . g g g Note that in the case V < 0.77c , the wave re- min Therefore,evenifthedisturbancevelocityV issmaller sistance, Eq.(7), is nonzero and decreases exponentially than c , there exists a transient nonzero wave resis- min with time (see Fig.2(d)). tance [22] decreasing as 1/t (for V >0.77c ). min In order to get a better physical picture of the gen- We obtain a good agreement between the numerical erated wave patterns, we have also calculated numeri- calculation of Eq.(7) and the analytical approximation cally the transient vertical displacement of the free sur- Eq.(14) as shown on Fig.5. Note that the oscillations face ζ(r,t) in the accelerated case (Eq.(6)). The corre- displayed by the wave resistance are characterized by a sponding patterns (as seen in the frame of the moving period 2π/Ω or 2π/Ω that diverges as V approaches c g object), are presented on Figs. 6, 7 and 8 for different c . In the particular case where V c , Eq.(14) min ≫ min reduced times c κt (1, 10 and 50, respectively). At reduces to min c κt = 1, the perturbation of the free surface is very min localized around the disturbance (close to the same one 1 κc2 Pˆ (k )2 c2 κt obtained by a stone’s throw). At cminκt = 10, some R (t) = R ( )+ min| ext g | sin min capillary waves can already be observed at the front of w w ∞ 29/2π ρV5t 8V (cid:18) (cid:19) the disturbance. At cminκt=50, on can see a V-shaped 23/2κ Pˆ (k )2 8V3κt pattern that prefigure the steady pattern of the uniform ext c | | cos . (15) − π c2 ρVt 27c2 straightmotion(obtainedatverylongtimes). Thesepre- min (cid:18) min(cid:19) dictionsconcerningthe wavepatternmightbecompared Let us now give a physical interpretation for the wave toexperimentaldatausingtherecenttechniqueofMoisy, resistance behavior as described by Eq.(14): during the Rabaud, and Salsac [23]. 5 B. Decelerated straight motion Let us now turn to the case of the decelerated motion, described by Eq.(12). We first will assume that V > c . In that case, the denominators appearing in min the integrals vanish for k such that c(k)=V, that is for k = κ(U2 √U4 1) (mainly dominated by gravity) 1 − − andk =κ(U2+√U4 1))(mainlydominatedbycapil- 2 − lary forces) as shown in Fig.4, where U =V/c . Both min wavenumbers contribute to the wave resistance. Using thestationaryphaseapproximationonethenfindsthatin thelongtimelimitR oscillatesascos(Vk t+π/4)/√t. w 1 More precisely, one has (for large t) FIG. 6: Accelerated straight motion: Transient vertical displacement (in units of P κ/(ρc2 )) of the free surface at 1 k3/2 Pˆ (k )2 0 min R (t)= 1 | ext 1 | cos(Vk t+π/4). tcminκ = 1 obtained by inverse Fourier transform of Eq.(6). w √πV √ργ (k k )c (k )t 1 Notethat thesurface disturbanceislocalized around theob- 2− 1 g 1 (16) jectandclosetotheoneobtainedbyastonethrowninwater. p Thewaveresistancedisplaysoscillationsthatarechar- acterizedbyaperiod2π/(Vk ). Figure9showsthegood 1 agreement between the numerical calculation of Eq.(12) andthe analyticalapproximationEq.(16)for largetimes (c κt > 10). Again one can give a simple physical min interpretation of the wave resistance, Eq.(16). The pe- riodoftheoscillationsisgivenby2π/(Vk )anddepends 1 only on the wavenumber k (mainly dominated by grav- 1 ity), even in the case of an object with a size b much smaller than the capillary length. This can be understood as follows. For t < 0, the disturbance moves with aconstantvelocityV andemits waveswithwavenumber k and waves with wavenumber k (mainly dominated 1 2 by capillaryforces). The k -waveslag behind the distur- 1 bance, while the k -waves move ahead of it. When the 2 disturbancestopsatt=0,thek -waveskeepmovingfor- 2 FIG. 7: Accelerated straight motion: Transient vertical dis- wardanddonotinteractwiththedisturbance. However, placement (in units of P0κ/(ρc2min)) of the free surface at the k1-waveswillencounter the disturbance and interact tcminκ=10 obtained byinverseFourier transform of Eq.(6). with it. Hence the period 2π/(Vk ) of the wave resis- 1 One can already see ahead of the disturbance the waves as- tance Eq.(16). The 1/√t decrease of the magnitude of sociated withk ,whiletheonesassociated tok arelesswell 1 2 wave resistance in Eq.(16) can be understood as follows. formed. Attimet>0,thedisturbance(whichisatrestatx=0) is hit by a k -wavethat has previouslybeen emitted dis- 1 tance c (k )t away from it. The vertical amplitude ζ of g 1 this wave is inversely proportional to the square root of thisdistance. Indeed,astheliquidisinviscid,theenergy has to be conserved and one thus has ζ 1/ c (k )t. g 1 ∝ Since the wave resistance R is proportional to vertical w p amplitude of the wave ζ, on recovers that R decreases w with time as 1/ c (k )t. g 1 In the particular case where V c , Eq.(16) re- min p ≫ duces to: 1 κ3/2 Pˆ (k )2c3 κc2 R (t)= | ext c | min cos mint+π/4 . w 2√π ρV11/2√t 2V (cid:18) (cid:19) (17) FIG. 8: Accelerated straight motion: Transient vertical dis- LetusnowdiscussthecaseV <c (stillconsidering placement (in units of P κ/(ρc2 )) of the free surface at min 0 min the decelerated motion Eq.(12)). In that case, the wave tcminκ=50 obtained byinverseFourier transform of Eq.(6). resistance oscillates (in the long time limit) approxima- 6 k -wavesandthedisturbancewhichleadtoamuchmore 1 slower 1/√t decay. IV. CONCLUSIONS In this article, we have shown that a disturbance un- dergoing a rectilinear accelerated or decelerated motion at a liquid-air interface emits waves even if its velocity V (the final one in the accelerated case and the initial one in the decelerated case, respectively) is smaller than c . This corroborates the results of Ref.[13]. For this min purpose, we treat the wave emission problem by a linearized theory in a Monge representation. Then, we de- rive the analytical expression of the wave resistance and FIG.9: ThewaveresistanceRw (inunitsofP02κ/γ)isshown solveitbynumericalintegration. Asymptoticexpansions as a function of the reduced time cminκt for a decelerated permit to extract the predominant behavior of the wave motion with different reduced velocities U = V/cmin (see resistance. Some vertical displacement patterns are also Eq.(12)). Respectively, panels (a), (b), (c) and (d) corre- calculated in order to shown how the waves invade the spond toa reduced velocity U =1.5, 1.01, 0.98 and 0.7. The free surface. dashedlinescorrespondtotheasymptoticexpansiongivenby Eq.(16). The results presented in this paper should be important for a better understanding of the propulsion of water-walking insects [24–27], like whirligig beetles, tivelyas e U√1 U4tκcmin/√tκc sin(U3tκc +χ). whereacceleratedanddeceleratedmotionsfrequentlyoc- − − min min curs (e.g., when hunting a prey or escaping a predator More pre(cid:16)cisely, one has (for large t)(cid:17) [28]). Even in the case where the insect motion appears as rectilinear and uniform, one has to keep in mind that therapidlegstrokesareacceleratedandmightproducea 2κ2 Pˆ (k˜)2sin(U3c κt+χ) Rw(t)=rπ |ρce2mxtin | √κmcimnint wthaevewdarvaegpeavtetnerbnselomwigchmtinb.eTcohmepparerdedicttioonesxcpoenricmerennitnagl e U√1 U4tκcmin data using the recent technique of Moisy, Rabaud, and − − , (18) Salsac [23]. √2πU1/2(1 U4)1/4(3U4+1)1/4 − It will be interesting to take in our model some non- where lineareffects[29]becausethewavesradiatedbywhirligig beetles [28] have a large amplitude. Recently Chepelian- skii et al. derived a self-consistent integral equation de- χ=(3/2)arctan 1 U4/U2 − scribingtheflowvelocityaroundthe movingdisturbance ((cid:16)1p/2)arctan (cid:17)1 U4/(2U2) (19) [30]. Itwouldbeinterestingtoincorporatethisapproach − − into the present study. (cid:16)p (cid:17) and k˜ = κ U2+i√1 U4 . Note that in the case − V <c ,thewaveresistanceEq.(18)alsodisplayssome min (cid:0) (cid:1) Acknowledgments oscillations, although no waves were emitted at t < 0. These oscillations - that have a exponential decay in time - might be due to the sudden arrest of the distur- We thank JonathanVoice andFrédéricChevy for use- bance. Suchoscillationscouldalsobe presentinthecase ful discussions and Falko Ziebert for a critical reading of V >c but are hidden by the interaction between the the manuscript. min [1] L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Perg- [4] J.H. Milgram, Annu.Rev.Fluid Mech. 30, 613 (1998). amon Press, NewYork, 1987), 2nd ed [5] E. RaphaëlandP.-G. deGennes, Phys.Rev.E53, 3448 [2] D.J. Acheson, Elementary Fluid Dynamics (Clarendon (1996). Press, Oxford,1990) [6] D. Richard and E. Raphaël, Europhys. Lett. 48, 53 [3] J. Lighthill, Waves in Fluids (Cambridge University (1999). Press, Cambridge, 1979), 6th ed.; T.H. Havelock, [7] S.-M. Sun and J. Keller, Phys. 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