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Wave resistance for capillary gravity waves:Finite-size effects

M. Benzaquen, F. Chevy and E. Raphael

EPL, 96 (2011) 34003

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EPL, 96 (2011) 34003 www.epljournal.org

doi: 10.1209/0295-5075/96/34003

Wave resistance for capillary gravity waves: Finite-size effects

M. Benzaquen1, F. Chevy2 and E. Raphael1(a)

1 Laboratoire PCT - UMR Gulliver CNRS 7083, ESPCI - 10 rue Vauquelin, 75005 Paris, France, EU2 Laboratoire Kastler Brossel, ENS, Universite Paris 6, CNRS - 24 rue Lhomond, 75005 Paris, France, EU

received 21 June 2011; accepted in final form 8 September 2011published online 14 October 2011

PACS 47.35.-i – Hydrodynamic wavesPACS 68.03.-g – Gas-liquid and vacuum-liquid interfaces

Abstract – We study theoretically the capillary-gravity waves created at the water-air interfaceby an external surface pressure distribution symmetrical about a point and moving at constantvelocity along a linear trajectory. Within the framework of linear wave theory and assuming thefluid to be inviscid, we calculate the wave resistance experienced by the perturbation as a functionof its size (compared to the capillary length). In particular, we analyze how the amplitude of thejump occurring at the minimum phase speed cmin = (4 gγ/ρ)

1/4 depends on the size of the pressuredistribution (ρ is the liquid density, γ is the water-air surface tension, and g is the acceleration dueto gravity). We also show how for pressure distributions broader than a few capillary lengths, theresult obtained by Havelock for the wave resistance in the particular case of pure gravity waves(i.e., γ = 0) is progressively recovered.

Copyright c© EPLA, 2011

Introduction. – Water waves are both captivatingand of great practical significance [1–3]. They have thusattracted the attention of scientists and engineers for manydecades [4]. Water waves can, for instance, be generatedby the wind blowing over the ocean, by a moving ship ona calm lake, or simply by throwing a stone into a pond.Their propagation at the surface of water is driven bya balance between the liquid inertia and its tendency,under the action of gravity or of surface tension (or acombination of both in the case of capillary-gravity waves),to return to a state of stable equilibrium [5]. The dispersivenature of water waves is responsible for the complicatedwave pattern generated at the free surface of a still liquidby a moving disturbance such as a partially immersedobject (e.g. a boat or an insect) or an external surfacepressure source [6]. The propagating waves generated bythe moving disturbance continuously remove energy toinfinity. Consequently, the disturbance will experience adrag, R, called the wave resistance. In the case of ships (forwhich surface tension is negligible), this drag is known tobe a major source of resistance [7] and has been analyzedin detail by Havelock [8]. The case of objects that are smallcompared to the capillary length has been considered onlyrecently [9–16] and has attracted strong interest in thecontext of insect locomotion on water surfaces [17–19]. Forsuch objects, one has to take into account both gravityand surface tension. In the case of a point-like surfacepressure distribution, this leads to a discontinuity of the

(a)E-mail: elie.raphael@espci.fr

wave resistance at a critical velocity given by the minimumof the wave velocity cmin = (4gρ/γ)

1/4 [9]. For clean waterat room temperature, one has cmin ≈ 0.23m s−1. When thevelocity V of the surface pressure distribution is smallerthan cmin, no steady waves are generated and the waveresistance vanishes. Emission of steady waves becomespossible only when V > cmin, leading to the onset of afinite wave drag 1. The wave resistance discontinuity at thecritical velocity cmin has been experimentally investigatedby several groups [13,14].It is important to notice that while the analysis of [9]

was mainly concerned with a point-like surface pressuredistribution2, real perturbations (like insects) have finitesizes. The aim of the present paper is thus to analyzein detail the role played by the finite size of the surfacepressure distribution for the wave resistance3. We will seein particular that for b much larger than κ−1, the influenceof surface tension is negligible. In that case, the results ofHavelock [8] for pure gravity waves are recovered.

1As shown in [15,16], the threshold at V = cmin exists only forobjects moving with no acceleration.2Note that the particular case where the typical size b is much

smaller than the capillary length κ−1 = (γ/ρg)1/2 has also been(briefly) discussed in [9]. It has been shown that the wave resistance

displays a maximum at a velocity of the order of√

γ/(ρb) [9].However, the consequences of a pressure distribution of finite size onthe amplitude of the discontinuity of the wave resistance at V = cminhas been overlooked in [9].3Note that the effect of a pressure distribution of finite size on

the amplitude of the discontinuity of the wave resistance was alreadybriefly discussed in [14].

34003-p1

M. Benzaquen et al.

Formulation. – Consider an incompressible, inviscid,infinitely deep liquid whose free surface is unlimited. Wetake the xy-plane as the equilibrium surface of the fluidand the z-axis along the upward direction perpendicularto the equilibrium surface. We study the wave motioncreated by an external surface pressure distribution thatmoves with speed V in the negative x-direction. In theframe of that moving disturbance, physical quantities arestationary: the pressure distribution is given by P (x, y)and the displacement ζ of the free surface from itsequilibrium position is of the form ζ(x, y).In order to calculate the wave resistance experienced

by the disturbance, we use a method first introduced byHavelock [8]. According to this author, we may imaginea rigid cover fitting the surface everywhere. The assignedpressure system P (x, y) is applied to the liquid by meansof this cover; hence the wave resistance is simply the totalresolved pressure in the x direction. This leads to

R=

∫

dxdy P (x, y)∂xζ(x, y). (1)

For the sake of simplicity, let us restrict ourselves to thecase of a pressure system symmetrical around the originso that P (x, y) = g(r) with r= (x2+ y2)1/2). The Fouriertransform P (kx, ky) is then a function only of k and can

be written as P (kx, ky) =G(k), where

G(k) =

∫

∞

0

dr rg(r)

∫ 2π

0

dθ e−ikr cos θ (2)

= 2π

∫

∞

0

dr rg(r)J0(kr). (3)

J0 denotes the Bessel function of the first kind of zerothorder. It has been shown by Raphael and De Gennes in [9]that in such a case the wave resistance R (for V � cmin)reduces to

R=

∫ χ

0

dθ cos θ

πγ

{k2(θ)G[k2(θ)]}2+ {k1(θ)G[k1(θ)]}2k2(θ)− k1(θ)

,

(4)where

k1(θ) = κ

(

V

cmin

)2

{cos2 θ− (cos4 θ− cos4 χ)1/2},

k2(θ) = κ

(

V

cmin

)2

{cos2 θ+(cos4 θ− cos4 χ)1/2},(5)

where χ is defined by cosχ= cmin/V . Equation (4) is animportant result as it predicts how the wave resistancevaries as a function of the velocity for any pressuredistribution.

Finite-size effects on the wave resistance. –

Equation (4) was studied in detail in [9] in the particularcase of a point-like pressure distribution g(r) = p δ(r).We here consider the case of a pressure distribution offinite size, b. For instance, we can assume the pressuredistribution to be Gaussian

g(r) =p

2πb2exp

(

− r2

2b2

)

, (6)

Fig. 1: (Color online) Plot of the wave resistance R in unitsof p2κ(πγ)−1, as a function of V/cmin. The red (dashed) curvecorresponds to a point-like pressure source g(r) = p δ(r), theblack one (solid line) corresponds to a Gaussian pressure field(6) of size b= 0.04κ−1.

Fig. 2: (Color online) Plot of the wave resistance R in unitsof p2κ(πγ)−1, as a function of V/cmin. The red (dashed) curvecorresponds to a point-like pressure field g(r) = p δ(r), the blackone (solid line) corresponds to a gaussian pressure field (6) ofsize b= 0.4κ−1 (upper graph) and b= 0.55κ−1 (lower graph).

Equation (3) then becomes

G(k) = p exp

(

−b2k2

2

)

. (7)

By inserting the Fourier transform of the pressure fieldG(k) in eq. (4), we obtain the wave resistance as a functionof V/cmin (see figs. 1 and 2).

34003-p2

Wave resistance for capillary gravity waves: Finite-size effects

Fig. 3: Plot of the amplitude of the jump AR of the waveresistance in units of p2κ(πγ)−1, as a function of bκ.

Fig. 4: Plot of Vmax corresponding to the maximum of waveresistance in units of V/cmin, as a function of bκ. The blackdotted line (horizontal) corresponds to a situation in whichthe maximum value is reached at Vmax = cmin. For bκ≪ 1, Vmaxscales as

√

γ/(ρb), whether for bκ≫ 1, Vmax scales as√gb.

The red (dashed) in curve in fig. 1 corresponds toa point-like pressure distribution g(r) = p δ(r), the blackone (solid line) corresponds to a Gaussian pressure field(6) of size b= 0.04κ−1. We clearly observe, for bothcurves, the discontinuity (or jump) at V = cmin that wediscussed earlier. In fig. 1, the black curve (solid line)presents a maximum at Vmax ∼

√

γ/(ρb) which is the firstconsequence of the finite-size effects. This separates thebehavior of the wave resistance in two regimes: belowVmax, R increases with the disturbance velocity whereasfor V > Vmax, R decreases with V .In fig. 1, the disturbance typical size b is much smaller

than the capillary length κ−1. When b becomes of the sameorder of magnitude than κ−1 (see fig. 2), the amplitudeof the jump, denoted by AR in the following, decreases asshown in fig. 3. When the typical size b gets close enough tothe capillary length κ−1, the maximum value of the waveresistance is obtained for Vmax = cmin. Such a situationis depicted in the lower graph in fig. 2. Figures 4 and 5give quantitative information on this situation: we showthat in the particular case of a Gaussian pressure fieldit occurs for 0.5κ−1 � b� 1.65κ−1. Note that the above

Fig. 5: Plot of the maximum of wave resistance Rmax in unitsof p2κ(πγ)−1, as a function of bκ. The dotted line correspondsto a situation in which the maximum value is reached atVmax = cmin.

interval is only valid for the Gaussian pressure field (6).If, for instance, one uses a Lorentzian pressure sourcep(2π)−1b(b2+ r2)−3/2 instead of (6), the above intervalbecomes 0.3κ−1 � b� 2.2κ−1.From eq. (4) on can get the jump amplitude at V = cmin:

AR =κ

2√2

G(κ)2

γ=

(

p2

πγκ

)

π

2√2

(

G(κ)

p

)2

. (8)

Equation (8) is an important result as it gives how thejump of the wave resistance at the critical velocity cminvaries with the size of the pressure distribution. In theparticular case of the Gaussian pressure field given ineq. (6), the jump amplitude becomes

AR =

(

p2

πγκ

)

π

2√2exp(

−b2k2)

. (9)

Note that in the limit b→ 0, the result of [9] AR = π/(2√2)

for of a point-like pressure distribution is recovered. Theamplitude of the jump AR of the wave resistance in unitsof p2κ(πγ)−1, is depicted in fig. 3 as a function of bκ. Wenotably observe that AR is significantly suppressed whenthe typical size b becomes greater than 2.5κ−1. In such acase, the jump in the wave resistance might be difficult toobserve experimentally (see also fig. 6 below).One may also wonder how the maximum in the wave

resistance scales with b. The abscissa (Vmax) and ordinate(Rmax) of the maximum of wave resistance as a functionof bκ are depicted in fig. 4 and fig. 5.In both figures the dotted line corresponds to the

situation in which the maximum value is reached atVmax = cmin (see lower graph in fig. 2).In the limit bκ≫ 1, the asymptotic behavior Vmax ≃

√gb

can be simply recovered by substituting b−1 to k in thepure gravity wave dispersion relation ω(k)/k=

√

g/k. Inthe opposite limit bκ≪ 1, the asymptotic behavior Vmax ≃√

γ/(ρb) can analogously be obtained by substituting

34003-p3

M. Benzaquen et al.

Fig. 6: Plot of the quotient AR/Rmax as a function of bκ. Thedotted line corresponds to a situation in which the maximumvalue is reached at Vmax = cmin.

b−1 to k in the pure capillary wave dispersion relationω(k)/k=

√

γk/ρ (see footnote 4).We observe in fig. 5 that Rmax (and hence the whole

wave resistance) strongly decreases with the size b of thepressure source. Although this might be surprising at firstsight, one has to notice that we have kept constant themagnitude p=

∫

dxdy P (x, y) of the pressure source whilevarying b.In order to discuss the general shape of the graph of the

wave resistance, and whether or not the jump in the waveresistance can be easily detected, we plot the ratio betweenthe amplitude AR and the maximum of wave resistanceRmax (see fig. 6). One can see how the ratio AR/Rmaxstarts at zero (for κ−1→ 0, AR is finite and Rmax→∞)and increases until reaching the constant value 1 in therange 0.5κ−1 � b� 1.65κ−1 discussed earlier (see fig. 2).For larger values of b, the ratio AR/Rmax is essentiallysuppressed.

Pure gravity waves. – As we have seen above, Vmaxscales as

√gb for bκ≫ 1. One shall thus wonder if, more

generally, our results for the wave resistance are for b≫κ−1 well approximated by the result of Havelock [8] forpure gravity waves (γ = 0).If γ = 0, the wave resistance (4) reduces to

R=g2

ρV 61

π

∫ π

2

0

dθ

cos5 θ

(

G( g

V 2 cos2 θ

))2

, (10)

which, for the Gaussian pressure field (6), yields

R =p2

πρgd3

(√gb

V

)6

×∫ π

2

0

dθ

cos5 θexp

(

− 1

cos4 θ

(√gb

V

)4)

. (11)

Using eq. (11), one can plot the wave resistancefor pure gravity waves as a function of V/cmin for differentvalues of b (see the black curves (solid line) in fig. 7).

4Note however, that Vmax is not simply given by√

γ/(ρb)+ gb.

Fig. 7: (Color online) Plot of the wave resistance R in units ofp2κ(πγ)−1, as a function of V/cmin. The black curve (solid line)corresponds to the formula for pure gravity waves (11), the redone (dashed) corresponds to the initial formula for capillarygravity waves (4). On the top we have b= 2κ−1, in the middlewe have b= 5κ−1, while at the bottom we have b= 10κ−1.

Note that Vmax is now exactly given by√gb. For velocities

smaller than 0.6√gb, the wave resistance is practically

unnoticeable5.We have also plotted in fig. 7 the wave resistance for

capillary-gravity waves (red (dashed) curves).One can see how both curves come closer to each other

as b is increased. For b much larger than κ−1, one doesnot see much differences between the two curves, meaningthat in this limit the problem is essentially ruled by puregravity waves theory.

Concluding remarks. – We have shown theoreti-cally how the finite size of an external axisymmetricpressure source significantly modify the wave resistance,and in particular its singular behavior at the minimumphase speed cmin, compared to the case of a point-like

5For velocities up to 0.8√gb, on can show that eq. (11)

is well approximated by p2κ(πγ−1)√

π/2 exp(−1/x4)/(2x4) withx= V/

√gb.

34003-p4

Wave resistance for capillary gravity waves: Finite-size effects

disturbance6. Our study also provides a quantitativedescription of the crossover between capillary-gravity andpurely gravity wave resistance described in previous works.

∗ ∗ ∗

We would like to thank A. Benusiglio and C. Clanetfor very interesting discussions.

REFERENCES

[1] Lighthill J., Waves in Fluids, 6th edition (CambridgeUniversity Press, Cambridge) 1979.

[2] Lamb H., Hydrodynamics, 6th edition (CambridgeUniversity Press, Cambridge) 1993.

[3] Johnson R. S., A Modern Introduction to the Mathemat-ical Theory of Water Waves (Cambridge University Press,Cambridge) 1997.

[4] Darrigol O., Worlds of Flow (Oxford University Press,New York) 2005.

[5] Landau L. D. and Lifshitz E. M., Fluid Mechanics, 2ndedition (Pergamon Press, New York) 1987.

6In the context of animal locomotion on water surfaces, it wouldbe interesting to generalize the results of the present study to non-axisymmetric pressure sources.

[6] Acheson D. J., Elementary Fluid Dynamics (ClarendonPress, Oxford) 1990.

[7] Milgram J. H., Annu. Rev. Fluid Mech., 30 (1998) 613.[8] Havelock T. H., Proc. R. Soc. A, 95 (1918) 354.[9] Raphael E. and de Gennes P.-G., Phys. Rev. E, 53(1996) 3448.

[10] Richard D. and Raphael E., Europhys. Lett., 48 (1999)53.

[11] Sun S.-M. and Keller J., Phys. Fluids, 13 (2001) 2146.[12] Chevy F. and Raphael E., Europhys. Lett., 61 (2003)

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Fluids, 22 (2010) 052107.[17] Denny M. W., J. Exp. Biol., 207 (2004) 1601; Fish F.

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34003-p5