Capillary-Gravity Waves Generated by a Sudden Object Motion

0.37 MB·English

by  Fabien Closa

Capillary-Gravity Waves Generated by a Sudden Object Motion
F. Closa,1 A.D. Chepelianskii,2 and E. Rapha¨el1,
∗
1Laboratoire Physico-Chimie Th´eorique, UMR CNRS GULLIVER 7083,
ESPCI, 10 rue Vauquelin, 75005 Paris, France and
2Laboratoire de Physique des Solides, UMR CNRS 8502,
Bˆatiment 510, Universit´e 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 wa-
ter 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) = −kPˆext(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 approxi-
mation [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 Pˆext(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 mo-
tion, 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 dis-
placement (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 under-
stood 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 lin-
earized 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 im-
portant 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 andFr´ed´ericChevy 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. Rapha¨elandP.-G. deGennes, Phys.Rev.E53, 3448
[2] D.J. Acheson, Elementary Fluid Dynamics (Clarendon (1996).
Press, Oxford,1990) [6] D. Richard and E. Rapha¨el, 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. Fluids. 13, 2146 (2001).
Proc. R.Soc. A 95, 354 (1918) [8] F. Chevy and E. Rapha¨el, Europhys. Lett. 61, 796
7
(2003). Physicists (AcademicPress, New York,2000), 5th ed.
[9] J.Browaeys, J.-C.Bacri,R.Perzynski,andM.Shliomis, [21] K.F.Riley,M.P.HobsonandS.J.BenceMathematical
Europhys.Lett. 53, 209 (2001). MethodsofPhysicsandEngineering (CambridgeUniver-
[10] T.BurgheleaandV.Steinberg,Phys.Rev.Lett.86,2557 sity Press, Cambridge), 3rd ed.
(2001) [22] In the particular case of a Lorentzian pressure distur-
[11] T.BurgheleaandV.Steinberg,Phys.Rev.E66,051204 bancewith aFouriertransform Pˆext(k)=P0e−bk,where
(2002). b is theobject size (set to b=0.1κ−1), we have numeri-
[12] H. Lamb, Hydrodynamics (Cambridge University Press, callycheckedthatuptoV ≈8cminthemaincontribution
Cambridge, 1993), 6th ed to the waveresistance is due tokc.
[13] A.D.Chepelianskii,F.ChevyandE.Rapha¨el,Phys.Rev. [23] F. Moisy, M. Rabaud, and K. Salsac, Exp. Fluids 46,
L 100, 074504 (2008). 1021 (2009).
[14] P.A. Cherenkov, Doklady Akad. Nauk. SSSR 2, 451 [24] O. Buhler, J. Fluid Mech. L 573, 211 (2007).
(1934). [25] R. MacNeill Alexander, Principle of Animal Locomotion
[15] J.D. Jackson, Classical Electrodynamics (Wiley, New (Princeton University Press, Princeton, 2002).
York,1998), 3rd ed. [26] J.W. Bush and D. L. Hu, Annu. Rev. Fluid Mech. 38,
[16] TheFouriertransformfˆ(k,t)isrelatedwiththefunction 339 (2006).
f(r,t) through f(r,t)=R (2dπ2k)2eik.rfˆ(k,t) [27] MA..WNi.cDasetnron,y,JJ..EExxpp..BBioiol.l.220067,,11664091((22000034));F.Fishand
[17] Lord Rayleigh, Proc. London Math. Soc. s1-15, 69
[28] J. Voise and J. Casas, J. R. Soc. Interface, 7, 343-352
(1883).
(2010)
[18] Lord Kelvin, Proc. London Math. Soc. 15, 80 (1887).
[29] F.Dias andC.Kharif, Annu.Rev.Fluid Mech,31,301-
[19] I.S.Gradshteyn and I.M. Ryzhik,Table of Integrals, Se-
346 (1999)
ries, and Products (Academic Press, USA, 1979), Cor-
[30] A. D. Chepelianskii, M. Schindler, F. Chevy and E.
rected and enlarged edition
Raphal Phys. Rev.E 81, 016306 (2010).
[20] H.J, Weber and G.B. Arkfen Mathematical Methods for