Horizon effects with surface waves on moving water
Germain Rousseaux, Philippe Maissa, Christian Mathis, Pierre Coullet, Thomas G. Philbin, Ulf Leonhardt
HHorizon effects with surface waves on moving water
Germain Rousseaux , Philippe Ma¨ıssa , Christian Mathis ,Pierre Coullet , Thomas G Philbin and Ulf Leonhardt Universit´e de Nice-Sophia Antipolis, Laboratoire J.-A. Dieudonn´e, UMRCNRS-UNS 6621, Parc Valrose, 06108 Nice Cedex 02, France. School of Physics and Astronomy, University of St Andrews, North Haugh, StAndrews KY16 9SS, Scotland, UK.E-mail:
Abstract.
Surface waves on a stationary flow of water are considered, in a linearmodel that includes the surface tension of the fluid. The resulting gravity-capillarywaves experience a rich array of horizon effects when propagating against the flow.In some cases three horizons (points where the group velocity of the wave reverses)exist for waves with a single laboratory frequency. Some of these effects are familiarin fluid mechanics under the name of wave blocking, but other aspects, in particularwaves with negative co-moving frequency and the Hawking effect, were overlookeduntil surface waves were investigated as examples of analogue gravity [Sch¨utzhold Rand Unruh W G 2002
Phys. Rev. D kh (cid:29)
1. Introduction
The interest in black-hole analogues has been mainly driven by the intriguing possibilityof observing Hawking radiation in the laboratory [1, 2, 3, 4, 5, 6]. In addition to theexperimental challenges, this pursuit has important theory implications because of thewell-known weakness in the derivation of the Hawking effect for real black holes [7, 8].Hawking’s semi-classical calculation [9] is based on a consideration of fields that areassumed to have no appreciable gravitational effect compared to the black hole, butthe derivation contradicts this assumption because the fields attain arbitrarily highfrequencies (and therefore energies) at the horizon. This so-called trans-Planckianproblem reveals the lack of any proper understanding of quantum-gravitational effects.Unfortunately, the question of whether black holes really radiate does not seem to beamenable to experimental investigation because such radiation would be completelyswamped by the cosmic microwave background. In black-hole analogues the trans-Planckian problem is avoided by dispersion that limits the blue-shifting of waves at a r X i v : . [ g r- q c ] O c t orizon effects with surface waves on moving water
40 cm
HydrodynamicHorizon
Crest Crest
Edge of the Waves ChannelSlopeof the BumpCounter-current Water Waves
Figure 1.
Experimental white-hole horizon in hydrodynamics. The surface wavespropagate against the flow up to the point where the flow speed matches the groupvelocity of the waves. The flow speed is higher on left than on the right because of theslope on the bottom of the tank.
Perhaps the least exotic black-hole analogue that has been proposed is that ofsurface waves on a moving fluid. Sch¨utzhold and Unruh [18] showed that long- orizon effects with surface waves on moving water x/ d t of the slope ofthe world-lines gives the speed of the crests and troughs, the phase velocity ω/k of thewave since the phase is (cid:82) ( k d x − ω d t ); the lines curve upward as the wave reaches thewhite-hole horizon, showing a decrease in the phase velocity. The wavelength in thelaboratory is revealed by drawing a horizontal line in the diagram and measuring thedistance between two crests; because of the curving upward of the crest world-lines, thewavelength is seen to decrease as the horizon is approached. This is the characteristicwavelength-shortening (“blue-shifting”) of waves at a white-hole horizon, also observedin an optical analogue [6]. The inset in Figure 2 shows the behaviour of rays at awhite-hole horizon where there is no dispersion; the rays stick at the horizon whichcorresponds to an infinite blue-shifting of the waves (the trans-Planckian problem). Itis the dispersion of the surface waves that limits the amount of blue-shifting at thehorizon; in this paper we will discuss some of the rich set of possible behaviours.Not surprisingly, the blocking of waves by a counter-flow that exceeds the wavegroup velocity is a phenomenon that is well-known in the fluid-mechanics community [23,35, 25, 29, 30]. The blue-shifting of waves at the blocking line (white-hole horizon) isalso well-known and has been investigated experimentally [27, 28, 29]. There is howeveranother possible process in the interaction of a wave with a counter-flow, one that doesnot seem to have been considered by the fluid-mechanics community, even though itis a possibility clearly visible in the dispersion relation [18, 22]: this process is theHawking effect. In a stationary flow, where the flow speed at each point in the tank isconstant in time but varies from point to point, the frequency in the laboratory frameof a surface wave on the flow is conserved. As we have seen, the wavelength in thelaboratory is not conserved, so a white-hole horizon converts the wavelength of a wavepropagating against the flow to a different wavelength while conserving its frequency inthe laboratory. Figure 2 is an example of such a wave; it is right-moving against the orizon effects with surface waves on moving water Time Space
Wave Propagation Counter Flow
Figure 2.
Space-time diagram of a plane wave encountering a white-hole horizon. Inthis diagram the wave propagates to the right on a flow moving to the left. Grey valuesdenote the water height and show clearly the world-lines of the crests and troughs. Thegreen dotted line shows the initial slope of the world-line of the incoming wave, whichis inversely proportional to the phase velocity. The red line is the world-line of a crest,the slope of which increases as the wave reaches the horizon. The phase velocity thusdecreases at the horizon, leading to a decrease of the wavelength, a blue-shifting that islimited by dispersion. The inset shows the behaviour of rays at a non-dispersive white-hole horizon, where there would be infinite blue-shifting of waves (rays originating onboth sides of the horizon are shown). flow (ingoing against the flow) with positive (angular) frequency ω and positive wavenumber k and the horizon increases k while keeping ω fixed. An important quantity isthe frequency of the wave in a frame co-moving with the flow; this co-moving frequencyis not conserved, but the ingoing wave and the blue-shifted wave with higher k bothhave positive co-moving frequencies [18, 22]. It turns out that there is often a solutionof the dispersion relation for the fixed positive input frequency ω that has a negative k and a negative co-moving frequency [18, 22]. The laboratory frequency ω of the inputwave must be conserved in the wave evolution but when there is a wave with negativeco-moving frequency at same value of ω , there exists the possibility that it could begenerated in the interaction of the input wave with the counter-flow. These waveswith negative co-moving frequency would be produced in addition to the blue-shiftedwaves with positive co-moving frequency; this process is the Hawking effect, which is atroot a classical effect, although with extraordinary quantum implications [9, 38]. Thestrangest feature of the Hawking effect is that it is an amplification of the ingoing wave,an extraction of energy from the flow (or from whatever background creates the horizon).The famous quantum Hawking radiation is a rather straightforward consequence of the orizon effects with surface waves on moving water
2. Surface waves on a stationary flow
The subjects of black holes and water waves have a historical connection throughthe figure of Pierre-Simon de Laplace. In his ”Exposition du Syst`eme du Monde”in 1796, Laplace famously introduced the term ´etoile sombre (dark star) to denote anobject whose gravitational field is strong enough to prevent light from escaping (thesame concept had been described in 1783 by the Reverend John Michell in a letter toCavendish) [32]. Laplace is also well known for having derived (in 1775) a dispersionrelation for surface waves on water [33]. When the water has a background flow (which orizon effects with surface waves on moving water ω − U k ) = (cid:18) gk + γρ k (cid:19) tanh( kh ) . (1)Here ω is the (angular) frequency of the wave in the laboratory frame and k the wave-number; U is the speed of the flow, h the depth of the fluid, g the gravitationalconstant, ρ the fluid density and γ the surface tension. For water, ρ = 1000 kg m − and γ = 0 .
073 N m − . Equation (1) is a one-dimensional dispersion relation suitablefor water-tank experiments. When the flow is stationary ( U independent of time,but gradually varying in space), ω is a constant but k varies with spatial position x .Waves described by (1) are a consequence of gravity ( g ) and surface tension ( γ ) and arecalled gravity-capillary waves. For small k the gravity term dominates, which we callthe gravity regime, whereas for large k surface tension dominates, giving the capillaryregime. The pure-gravity case corresponds to γ = 0.The quantity ω − U k is the frequency in a frame co-moving with the fluid. Hencethe positive, respectively negative, square roots of (1) ω − U k = ± (cid:115)(cid:18) gk + γρ k (cid:19) tanh( kh ) (2)correspond to positive, respectively negative, co-moving frequencies. As described inthe Introduction, the Hawking effect is the generation of a wave on the negative branchof (2) from a wave on the positive branch, through interaction with a counter-flow. Itis a remarkable fact that in the extensive fluid-mechanics literature on the waves (1),including the pure-gravity case γ = 0, there seems never to have been any considerationof the possibility of conversion of waves from the positive to the negative branch of (2)through the blocking effect [23, 34, 35, 25, 26, 36, 37, 27, 28, 29, 30, 39]. If such aconversion process had been investigated, the Hawking effect would presumably havebeen (re-)discovered in fluid mechanics.Solutions of the dispersion relation (1) are usually represented graphically. Weobtain from (2) ω = U k ± (cid:115)(cid:18) gk + γρ k (cid:19) tanh( kh ) . (3)Figure 3 plots both branches of the right-hand side of (3) as functions of k , for afixed value of U <
0; figure (a) shows the pure-gravity case ( γ = 0) and figure (b)shows the full dispersion relation with surface tension included. The positive branchof (3), corresponding to positive co-moving frequency, is shown in green while thenegative branch, corresponding to negative co-moving frequency, is shown in blue. Theintersection of these curves with a given horizontal line, such as the red line in the figures,gives the possible waves for the frequency ω given by that line. In a stationary flow ω is conserved but the plots of the right-hand side of (3) change with spatial position orizon effects with surface waves on moving water U ( x ) changes and one can trace the evolution of a given solution by following itsintersection point with the horizontal line of fixed ω . As one traces the evolution of theintersection point, the changing group velocity d ω/ d k is the slope of the tangent to thecurve at the point of intersection with the horizontal line. !
50 50 k ! m ! " ! ! Ω ! s ! " k I k B k R k H k ∗ White Horizon (a) ! ! !
200 200 400 600 k ! m ! " ! ! Ω ! s ! " k C White HorizonBlue HorizonNegative Horizon (b)
Figure 3.
Dispersion relation for surface waves propagating on a water flow with agiven velocity
U <
0; (a) shows the pure gravity case ( γ = 0) and (b) shows the gravity-capillary case. The green curves are the positive branch of the right-hand side of (3)(positive co-moving frequency) and the blue curves are the negative branch (negativeco-moving frequency). Intersections of these curves with the horizontal red line (valueof ω ) show possible waves at that ω . If U becomes more negative the green and bluecurves rotate clockwise about the origin. Local maxima and minima of the green andblue curves show the possibility of blocking waves by means of appropriate velocityprofiles U ( x ) (see Figure 4). A water depth h = 0 . orizon effects with surface waves on moving water k I , k B , k R and k H ). The solution k I is a right-moving wave in thelaboratory, having positive phase and group velocities, propagating against a left-movingcounter-flow U <
0. If the flow speed | U ( x ) | increases as the wave moves to the right ( U becomes more negative) the green and blue curves tip over clockwise about the origin(as in Figure 4), so the root k I increases—the wave is blue-shifted. When the wavereaches a point where the flow speed has increased to make the roots k I and k B coalesceat a local maximum of the green curve, the group velocity of the wave is zero—it hasbeen blocked at a white-hole horizon. The wave has been stopped by a negative “groupacceleration” that is still non-zero at the blocking point so the group velocity decreasesto negative values; the wave moves back to the left in the laboratory on the k B rootof the dispersion relation, back into the region where the counter-flow is slower thanthe blocking speed and where the wave previously had wave number k I . The ingoingwave k I has thus been blue-shifted to k B by the white-hole horizon; the blue-shiftedwave k B has positive phase velocity, so its crests move to the right in the laboratory,but it has negative group velocity. The third real root of the dispersion relation withpositive co-moving frequency (green curve) in Figure 3 (a) is k R ; this is simply a wavepropagating in the same direction as the flow, to the left with negative phase and groupvelocities. The solution k R is rather trivial and is of no interest for horizon effects.The root k H in Figure 3 (a) has negative co-moving frequency (blue curve) and isof great interest for horizon effects; the conversion of some of the input wave k I to k H is the Hawking effect. Since it has a negative wave number, the wave k H has a negativephase velocity in the laboratory; its crests move backwards relative to the direction ofthe ingoing wave, in contrast to k I and k B , which both have positive phase velocity. Itis essential to understand that the three waves k I , k B and k H are all propagating tothe right relative to the fluid , even though in the laboratory k H has a phase velocitypointing left and both k B and k H have a group velocity pointing left ( k R is the rootcorresponding to a wave moving to the left relative to the fluid). Unlike the blue-shifting of k I to k B , the existence of conversion from k I to k H cannot be deduced fromdispersion plots, which only reveal it as a possibility. The amount of conversion of k I to k H depends on the details of the dispersion and the velocity profile U ( x ). Insimple cases involving limited dispersion the Hawking effect is determined by the sloped U ( x ) / d x of U at the horizon (this slope is the analogue of the surface gravity of ablack hole, the acceleration due to gravity at the horizon), but for general dispersionand velocity profiles no analytical formula for the size of the effect has been found andone must resort to numerical simulations of the wave evolution. One aspect of thechallenge to find a good intuitive understanding of the Hawking effect is apparent fromthe description of the horizon given above: this was taken as the point where the flowspeed matched the group velocity of the blue-shifting wave. But the phase velocity ofthis wave is greater than its group velocity, so one can have a group-velocity horizon butno phase-velocity horizon. On the other hand, one can have both a group- and a phase-velocity horizon, with in principle an arbitrary distance between these two horizons and orizon effects with surface waves on moving water
9a completely different value of d U ( x ) / d x at each horizon. The size of the Hawkingeffect is influenced by these and other factors. ‡ There is also the further possibility ofhaving the maximum flow speed close to but less than that required for a group-velocityhorizon. In this case one would expect some wave tunneling into the blue-shifted root k B , and perhaps also into k H (tunneling of surface waves has been studied in [40]).Numerical simulations indicate that this method of generating k H without a group-velocity horizon is mathematically possible for a steep enough velocity profile, but itshould not be possible in practice [22]. In the experiments reported in [22] waves withnegative phase velocity were observed even in the absence of a white-hole group-velocityhorizon, but, as stated in the Introduction, the origin of those waves is not clear.The conversion of k I to k B discussed above is well known in fluid mechanics, underthe name of wave blocking [23, 34, 35, 25, 26, 36, 37, 27, 28, 29, 30, 39]. The superpositionof the k I and k B waves has been shown to be describable by an Airy interferencepattern [41, 26, 36, 37]. In contrast, the root k H in Figure 3(a) has been largelyneglected by the fluid-mechanics community. Although the graphical representation ofthe dispersion relation is standard in fluid mechanics, very few authors [25, 42, 43, 44]plot the negative- k part, in either the pure-gravity or gravity-capillary cases, and theconversion of k I to k H appears not to have been considered.Turning to the full gravity-capillary case, Figure 3 (b) shows (for fixed U <
0) howthe surface tension γ changes the dispersion relation at large wave numbers comparedto the pure-gravity case in Figure 3 (a). For the value of U plotted, the positive co-moving frequency curve (green) has a local minimum as well as a local maximum, andthis is also the case for the negative co-moving frequency curve (blue), although thelocal minimum of the latter curve always occurs at negative laboratory frequency ω .If U becomes more negative the curves tip over clockwise about the origin, so that forlarger counter-flow speed there exist roots with negative co-moving frequencies at the ω shown by the red line. Each local maximum or minimum of the green and blue curvesreveals the possibility of reversing the group velocity of a wave with an appropriatevelocity profile, as in the discussion of the local maximum in the pure-gravity case(Figure 3 (a)). The local maximum of the green curve in Figure 3 (b) allows the waveblocking and blue shifting of an incident right-moving wave as in the pure-gravity case.We refer to this blocking line as the white horizon (from white hole). But after thewhite horizon has reversed the group velocity of the incident wave so that it now movesto the left, this blue-shifting wave encounters another blocking line because of the localminimum of the green curve. We refer to this second blocking line for the blue-shiftingwave as the blue horizon. At the blue horizon the group velocity reverses once more tobecome positive so the wave moves to the right again towards the white horizon. Thistime the (still blue-shifting) wave goes right through the white horizon, so overall theincident wave undergoes a double bounce. Figure 4 shows the graphical solution of thedispersion relation at four values | U | < | U | < | U | < | U | of a velocity profile U ( x ) < ‡ An extension of the standard analytical results to the dispersive case was given in [17] but only fortwo specific velocity profiles, both of which gave a group- and a phase-velocity horizon. orizon effects with surface waves on moving water U U U U U U U U !
500 500 1000 k ! ! Ω U U U U xt Figure 4.
Graphical solution and numerical ray solution for gravity-capillary waveson a water counter-flow. The green/blue curves refer to positive/negative co-movingfrequency. Four values of the velocity profile U ( x ) < x -positions where the profile takes these four values are shownin the ray solutions (lower). The values are (in m s − ) U = − . U = − . U = − . U = − . ax to describe the variation of the velocity profile, with a typical length a = 0 . ω of the waves (red horizontal line in the dispersion plots)corresponds to a period T = 0 . h = 0 . orizon effects with surface waves on moving water x -positions where the velocity profile takes thefour values used in the dispersion plots are shown in the ray plot. Rays move at thegroup velocity and so the wave blocking is clear from the ray plots. Also shown is theray solution for the wave with negative co-moving frequency (blue curve); this wave isinitially left-moving in the laboratory but its group velocity is reversed at a blockingline we refer to as the negative horizon (this horizon does not exist in the pure-gravitycase—see [22]). Comparing the dispersion and ray plots in Figure 4 (and ignoring theco-propagating wave k R discussed above) one can see how the increasing counter-flowspeed as x increases gives, successively, one k root in the profile where | U | < | U | , threeroots in the region where U lies between U and U , five roots in the region where U liesbetween U and U , and three roots in the region where | U | > | U | . One can also seehow these roots relate to the ray behaviour. (See [22] for ray plots in the pure-gravitycase.)Figure 5 shows an example where the frequency ω and counter-flow profile U ( x ) < ∂ t + ∂ x U )( ∂ t + U ∂ x ) φ = i (cid:18) g∂ x − γρ ∂ x (cid:19) tanh( − i h∂ x ) φ. (4)The method of numerically solving equations of the form (4), for essentially arbitrarydispersion, is described in [10]; further examples of this kind of numerical solution forsurface waves appear in [18] and [22]. In the wave-packet simulation in Figure 6, thecontinuous blue-shifting that accompanies the double bounce is apparent. Because ofthe spread of frequencies in the wave-packet there is some leakage of the initial wavethrough the white horizon (first bounce) as well a spreading and separation of frequencycomponents at the blue horizon (second bounce). For extensive numerical simulationsof gravity-capillary waves in the presence of a current, see [45, 44].In practice the blue-shifting of incident gravity waves into the capillary regime, asin Figure 6, will be limited by viscosity, which is not included in the model we have beendiscussing. As a consequence, the highly blue-shifted waves produced at the blue horizonwill dissipate rapidly. An experimental investigation of these effects was first performedby Badulin et al. [34]. These authors observed the initial blocking of waves in the gravityregime (white horizon) and the subsequent conversion into waves in the capillary regime(blue horizon) which then propagated through the original blocking line and vanishedthrough viscous damping [34]. Gravity-capillary waves on a counter-flow have also beenstudied experimentally by Klinke and Long [46], who produced space-time diagrams ofthe wave evolution. Trulsen and Mei [45, 44] give a theoretical treatment that includes orizon effects with surface waves on moving water U U U k Ω U U U xt Figure 5.
Graphical solution and numerical ray solution for gravity-capillary wavewith positive co-moving frequency. The conserved frequency ω corresponds to a period T = 0 . | U | < | U | < | U | of counter-flow speeds used. The three values of the flowvelocity shown are (in m s − ) U = − . U = − . U = − . h = 0 . numerical simulations. A recent theoretical survey is given by Huang [24].We have so far discussed only certain features of the dispersion relation (1). Inthe next Section we classify in more detail how the presence or absence of the varioushorizons, and their positions, depend on the conserved frequency ω and velocity profile U ( x ).
3. Results for wavelengths less than the water depth
The influence of the water depth h in the dispersion relation (1) disappears when | k | h (cid:29)
1, so that tanh kh ≈ ± k ). This is the case of orizon effects with surface waves on moving water Figure 6.
Wave packet simulation. The packet is centred on the ray in Figure 5. orizon effects with surface waves on moving water ω − U k ) = ± (cid:18) gk + γρ k (cid:19) (+ for k > − for k <
0) (5)that is easier to handle analytically than the original (1). The waves considered inFigure 3 to Figure 6 are in fact very well described by the deep water/short wavelengthdispersion relation (5), as were the waves studied in the experiments [22] and [34].At the blocking lines or horizons discussed in the last Section, the group velocityvanishes and the dispersion curve ω ( k ) has a local extremum. Three possible horizonsfor gravity-capillary waves were identified: the white, blue and negative horizons inFigure 3(b). At each horizon two real roots of the dispersion relation coalesce into onedouble root and then disappear: in the terminology of dynamical systems it is a saddle-node or tangent bifurcation [41]. The order parameter of the bifurcation is the wavenumber whereas the two control parameters are the velocity U (the “external field”)and the frequency ω (the “internal parameter”). Following the approach in [41], we findthe horizons by looking for a double root k of the cubic dispersion relation (5):( k − k )( k − k ) = 0 , (6)where k is the remaining simple root. Comparing coefficients of k in (5) and (6) weobtain expressions for k and k , as follows. The comparison of coefficients gives threeequations for the two unknowns k and k ; two of these equations are solved for k and k , and the the third equation is then a constraint relating k to k . In the case ofpositive wave numbers (plus sign in (5)) this procedure gives k = ρU γ (cid:18) ± (cid:114) − γρU ( g + 2 ωU ) (cid:19) k = ρU γ (cid:18) ∓ (cid:114) − γρU ( g + 2 ωU ) (cid:19) (positive wave numbers) (7)with the constraint k k = ρω γ (positive wave numbers) . (8)The constraint (8), for both sign possibilities in (7), leads to ω (cid:20) U + gU ω + γω U ρg − γωU ρ − gγUρ − γg ρω − γ ω ρ g (cid:21) = 0 (cid:34) positive wavenumbers (cid:35) (9)For negative wave numbers (minus sign in (5)) k and k are k = − ρU γ (cid:18) ± (cid:114) − γρU ( g − ωU ) (cid:19) k = − ρU γ (cid:18) ∓ (cid:114) − γρU ( g − ωU ) (cid:19) (negative wave numbers) (10)with the constraint k k = − ρω γ (negative wave numbers) (11) orizon effects with surface waves on moving water ω (cid:20) U − gU ω + γω U ρg + 15 γωU ρ − gγUρ + γg ρω + 27 γ ω ρ g (cid:21) = 0 (cid:34) negative wavenumbers (cid:35) (12)The significance of the constraints (9) and (12) is clear from Figure 3(b). With aparticular choice of U , local extrema in the dispersion plot (corresponding to doubleroots k ) occur at values of ω determined by this choice of U . For a given U , theconstraint (9) or (12) is a quartic in ω whose real roots give all the frequencies at whicha blocking line (horizon) occurs for this U . Alternatively, upon fixing ω the constraintgives a quintic in U whose real roots are all the flow velocities that give a horizon atthis frequency. I U I ( c m / s ) IU I TcIUcI I IIIII IVV VI VIIIUgI=gT/8IU I= 2( g/ )
TbIUbI
Figure 7.
The flow speeds | U | at which the white, blue and negative horizons occur,as a function of the wave period T (black curves). The curve lying on or close tothe straight red line is the white horizon, the red line showing the pure-gravity whitehorizon. The curve approaching the asymptote | U | = | U γ | from below is the bluehorizon, and the curve approaching this asymptote from above is the negative horizon. As in the previous Section, we consider only positive ω and negative U , whereas k can be positive or negative (this gives no restriction in the horizon effects). Thewhite horizon in Figure 3(b) occurs at a value of k low enough for the influence of thesurface tension γ in (5) to be very small. A very good approximation for the whitehorizon, which is exact in the pure-gravity case, is therefore obtained by putting γ = 0.Neglecting γ , the constraint (9) gives ωU ( U + g ω ) (cid:39) . (13)Note that the approximation (13) is also obtained in the large U limit, so for large U it becomes the exact constraint at the white horizon for gravity-capillary waves. From orizon effects with surface waves on moving water U and ω that gives the blocking of gravity wavesat a white-hole horizon [41] U g = − g ω = − gT π , (14)where T is the period. The corresponding value of the double root k is obtained fromthe γ → k = k g = 4 ω g = g U g (15)Equations (14)–(15) show the exact relationship between the frequency, wave numberand counter-flow speed at the white horizon for pure-gravity waves. Note from (14)that the flow speed at the white horizon is proportional to the conserved period of theblocked wave. The straight line | U g | versus T is shown in red in Figure 7. Superimposedon that red line is a black curve that shows the exact relationship between flow speedand period for gravity-capillary waves. In line with the comments above, the red lineagrees very well with the gravity-capillary case except for small | U | , and therefore small T . The striking feature of the gravity-capillary curve is that it ends at the point labeled( | U c | , T c ); the white horizon thus does not exist for periods T that are below a criticalvalue T c , or for counter-flows that do not reach a critical speed | U c | . The existence ofthis threshold can be seen from the dispersion plots in Figure 4: for flow velocity U there is no local maximum of the green curve, so no frequency ω experiences a whitehorizon at this flow velocity; in contrast, the other flow velocities plotted in the figure allgive a local maximum of the green curve and therefore a white horizon for the frequencyat this maximum. Similarly, if the period T is too small (frequency ω too large), thehorizontal red line in Figure 4 will not intersect a local maximum in the dispersion plotfor any U , so there can be no white horizon for such periods.Let us look in more detail at the critical values ( | U c | , T c ). Figure 8 shows graphicallythe occurrence of the threshold for the white horizon. We see that the disappearance of alocal maximum in the green curve, as | U | decreases, is accompanied by the disappearanceof the local minimum, and at the critical value U c the two local extrema coalesce to forma point of inflection at frequency ω c . This shows that ( | U c | , T c ) is also the threshold forthe occurrence of the blue horizon, which requires a local minimum in the green curve.At ( | U c | , T c ) the two double roots k in (7) (white and blue horizons) coincide, and infact the same value is taken by the simple root k , as can also be seen from Figure 8.The values ( | U c | , T c ) can be obtained by solving for the point of inflection ∂ω∂k = ∂ ω∂k = 0in the dispersion relation (5) ( k > U c and thecritical wave number k c , and ω c then follows from the dispersion realtion. Alternatively,the point of inflection is found by demanding that the square-root expression in (7)vanishes so that all k and k coincide. The result is T c = 2 π (3 + 2 √ / (cid:18) γρg (cid:19) / = 0 .
425 s (16) orizon effects with surface waves on moving water Ω c ! U ! " ! U c ! U c ! U ! ! U c ! $
500 500 1000 k $ Ω ! U ! ! ! U c ! U c ! U ! " ! U c ! xt Figure 8.
Dispersion plots and ray solutions for a wave with period T c = 2 π/ω c . Thewave with positive co-moving frequency occurs at a point of inflection when U = U c .This means that the corresponding ray (green) has a group velocity that slows to zerowhen | U | increases to | U c | ; the group velocity does not reverse, however, and the rayresumes its propagation into regions of higher | U | . and U c = − √ √ / (cid:18) γgρ (cid:19) / = − .
178 m / s (17)with the wave number k c = 1(3 + 2 √ / (cid:18) ρgγ (cid:19) / = 144 m − . (18)In using the method (6) of searching for the system parameters at horizons, we noted thatthey correspond to saddle-node or tangent bifurcations in dynamical-systems theory [41]. orizon effects with surface waves on moving water | U c | , T c ) in the | U | vs T diagram (Figure 7) is the point where two saddle-nodelines (horizons) intersect. In the terminology of dynamical systems this corresponds toa so-called pitchfork bifurcation [47]. U ! U Γ
200 200 400 600 k Ω Figure 9.
The dispersion plot when the counter-flow velocity is such that the localminimum of the green curve and the local maximum of the blue curve lie on the k -axis ( ω = 0, T = ∞ ). This counter-flow velocity is given by (20) and has the value U γ = − .
231 m / s. Since ( | U c | , T c ) is also a threshold for the existence of the blue horizon, the curvein the | U | versus T diagram (Figure 7) relating the period to the flow speed at the bluehorizon must also end at ( | U c | , T c ); this curve is also shown in Figure 7 and it is seen toapproach an asymptotic value, labelled U γ , as T → ∞ . This is because T → ∞ means ω → ω = 0 for a finite non-zero U that we call U γ . We find the flow velocity U γ as follows. The constraint (9) relates the values of U and ω at all horizons (doubleroots of the dispersion relation) for waves with positive k . Hence by taking ω → U γ ; this limit of (9) gives gU − γg ρ = 0 , (19)so U γ = −√ (cid:18) γgρ (cid:19) / = − .
231 m / s . (20)The wave vector at the local minimum (blue horizon at ω = 0) in Figure 9 is found byinserting the velocity U γ into the expression for the double root k in (7) and taking theupper sign: k γ = (cid:18) ρgγ (cid:19) / . (21) orizon effects with surface waves on moving water k > ω givenby the horizontal red line in the figure, occurs at k = 0 . (cid:18) ρgγ (cid:19) / , (22)and the corresponding ω (red line) is ω = 0 . (cid:18) ρg γ (cid:19) / , (23)where the exact but lengthy numerical coefficients have not been reproduced. Thesimple root k for U = U γ is zero for ω = 0, while for ω given by (23) (red line Figure 9)the simple root is (intersection of red line with green curve at large k > k = 1 . (cid:18) ρgγ (cid:19) / . (24)These last three results are obtained from the constraint (9) and the expressions (7) forthe double and single roots, with U = U γ .We see from Figure 9 that in the limit ω → T → ∞ ) the negative horizon alsooccurs at the flow velocity U γ , as well as the blue horizon, and the wave vector at thenegative horizon is minus that at the blue horizon, − k γ . Figure 9 shows that, for waveswith positive laboratory frequency ω , the counter-flow velocity U γ is the threshold forthe existence of waves with negative co-moving frequency; the threshold flow velocityfor such waves is the threshold for the negative horizon. It follows that the curve in the | U | versus T diagram (Figure 7) relating the period to the flow speed at the negativehorizon must lie above the line | U | = | U γ | and asymptotically approach this line as T → ∞ . This negative-horizon curve is also plotted in Figure 7; it lies above theblue-horizon curve but shares with it the asymptote | U | = | U γ | . Unlike the white- andblue-horizon curves in Figure 7, which both end at the cusp ( | U c | , T c ) for small T , thenegative-horizon curve diverges to | U | → ∞ as T →
0. This behaviour of the negativehorizon is clear from the dispersion plots because as T → ω → ∞ ) the flow speed | U | must increase without limit in order for the local maximum in the negative- k curveto reach the horizontal frequency line (see for example Figure 4).The flow velocity U γ that appears as an asymptote in the | U | versus T diagram(Figure 7) has additional significance in fluid mechanics. Firstly, it is a well-knownproperty of gravity-capillary waves on static water ( U = 0) that the minimum phasevelocity of the waves is given (apart from the sign) by the expression (20) for U γ . Thevelocity U γ is also important in the case of shear flows , i.e. velocity profiles that changewith the fluid depth. It was shown by Caponi et al. [48] that a sufficient condition fora shear flow to become spontaneously unstable is for the flow velocity on the surface ofthe fluid to exceed U γ ; the instability leads to the generation of gravity-capillary waveson the fluid surface [48]. Another example in shear flows is the appearance of negative-energy waves at the interface of two fluid layers, which occurs when the relative velocityof the layers exceeds U γ ; this is related to the famous Kelvin-Helmholtz instability, asdiscussed by Fabrikant and Stepanyants [23]. orizon effects with surface waves on moving water I VIV !
500 500 1000 k ! ! Ω I V xt Figure 10.
Dispersion plots and ray solutions for waves with period
T < T c . TheRoman numerals I and V refer to counter-flow speeds that lie in the regions labeledby these numerals in Figure 7. The period is T = 0 .
382 s and the flow velocities are − .
159 m / s (I) and − .
277 m / s (V). Figure 7 allows the classification of the behaviour of gravity-capillary waves on astationary counter-flow. The period T is conserved in the wave evolution so by fixing avertical line in the figure one can distinguish five qualitatively different possibilities forwaves of a single frequency:1. T < T c . As one moves into regions in the velocity profile U ( x ) < orizon effects with surface waves on moving water T = T c . Here the line of constant T is the vertical green line in Figure 7 thatseparates region I from regions II and III and passes through the cusp point ( | U c | , T c ).The dispersion and ray plots are shown in Figure 8; as already discussed, this case isthe threshold for the appearance of the white and blue horizons. II IV III VIIIIVIIIVI !
500 500 1000 k ! ! ! Ω II IV III VI xt Figure 11.
Dispersion plots and ray solutions for waves with period T c < T < T b . TheRoman numerals refer to counter-flow speeds that lie in the regions labeled by thesenumerals in Figure 7. The period is T = 0 .
510 s and the flow velocities are − .
169 m / s(II), − .
194 m / s (IV), − .
231 m / s (III) and − .
267 m / s (VI). T c < T < T b . The line of constant T lies between the green and brown verticallines in Figure 7 and so passes through region III. Here increasing counter-flow speedstakes us from region II to IV to III to VI in Figure 7. The dispersion plots and raysolutions for this case are shown in Figure 11. Here there is a white and blue horizon,and the white horizon occurs at a lower counter-flow speed than the negative horizon. orizon effects with surface waves on moving water Ω b ! U ! " ! U b ! U b ! U ! ! U b ! $
500 500 1000 k $ $ $ Ω ! U ! ! ! U b ! U b ! U ! " ! U b ! xt Figure 12.
Dispersion plots and ray solutions for waves with period T = T b (seeFigure 7). Both the white and negative horizons occur at the same counter-flowvelocity U = U b , the point ( T b , | U b | ) being the intersection of the curves for thesetwo horizons in Figure 7. For water, T b = 0 .
647 s, U b = − .
255 m / s. T = T b , defined by the brown vertical line in Figure 7. The line T = T b passesthrough the point where regions III, IV, VII and VI meet, at | U | = | U b | . The significanceof the point ( | U b | , T b ) is that it is the intersection of the white-horizon curve and thenegative-horizon curve; this means that for a wave with period T b the white horizonoccurs at the same counter-flow speed as the negative horizon. Figure 12 confirms thisin the dispersion plots and ray solutions. The values ( U b , T b ) must be found numericallyand for water they are T b = 0 .
647 s, U b = − .
255 m / s.5. T > T b . Here the line of constant T passes through region VII in Figure 7.Increasing counter-flow speeds takes us from region II to IV to VII to VI. The dispersion orizon effects with surface waves on moving water II IV VII VIIIIVVIIVI !
500 500 1000 k ! ! Ω II IV VII VI xt Figure 13.
Dispersion plots and ray solutions for waves with period
T > T b . TheRoman numerals refer to counter-flow speeds that lie in the regions labeled by thesenumerals in Figure 7. The period is T = 0 .
692 s and the flow velocities are − .
192 m / s(II), − .
214 m / s (IV), − .
260 m / s (VII) and − .
281 m / s (VI). plots and ray solutions for this case are shown in Figure 13. Here there is a white andblue horizon, and the white horizon occurs at a higher counter-flow speed than thenegative horizon (compare carefully with case 3 above).Trulsen [44] inferred from his results the structure of the ( | U | , T )-diagram ofFigure 7, but without the negative-horizon curve. He also derived the existence ofthe cusp ( | U c | , T c ) as a triple-root solution of the dispersion relation [44].Recently, we observed (with continuous waves trains) the regions VI, VII and IVof Figure 7, as reported in our experimental | U | versus T diagram [22]. The distinctionbetween the regions II and IV was unclear from our data. Our wave-maker was limited orizon effects with surface waves on moving water . T c = 0 . et al. [34]). In addition, weused a rather high period (far from T c ) to get long wavelengths of the ingoing wavessince the waves with negative co-moving frequency should be produced with a drasticreduction of the wavelength according to the dispersion relation. We were surprised tofind indications of waves with negative co-moving frequency even without wave blocking(a white horizon).Badulin et al. [34] performed experiments with wave packets (three to ten wavecycles centred on periods in the range T = 0 . .
66 s) sent on a counter-flow with speed | U | between 0 .
04 and 0 . / s over a sloping bottom. Double bouncing of the input waveswas observed with a strong reduction in both wavelength (from λ = 0 . T = 0 .
52 s, of theconversion phenomenon and one measurement of the amplitude of waves as a functionof the position/velocity ( T = 0 . U/ d x = 0 . − ), but beautiful measurements ofthe effect of the velocity on the wavelength. No results for periods less than T c = 0 .
425 swere reported.
4. A thermodynamic analogy
In the previous Section we summarized the behaviour of gravity-capillary waves (with kh (cid:29)
1) on a stationary counter-flow by means of a diagram in the ( | U | , T )-parameterspace (Figure 7). This diagram allows one to visualize the evolution of an incidentwave of a single frequency (which is conserved), as was illustrated in the five cases inFigures 10, 8, 11,12 and 13. In this Section we note a similarity between Figure 7 anda phase diagram in thermodynamics, where the horizon lines in Figure 7 are analogousto the lines separating different phases (first-order phase transition). In particular, thecusp point ( | U c | , T c ) in Figure 7 looks like a critical point (second-order phase transition)in a phase diagram.Let us explore this thermodynamic analogy a little further. In thermodynamics asystem is described by an equation of state of the form f ( P, V,
Θ) = 0, where P is thepressure, V is the volume and Θ is the temperature. For example, the equation of stateof a van der Waals gas can be written [47]: V − (cid:18) nb + nR Θ P (cid:19) V + n aP V − n abP = 0 , (25)where n is the number of molecules divided by Avogadro’s number, R is the gas constant, b relates to the non-zero volume of the molecules and a is a measure of the molecularinteraction. A familiar property of the van der Waals gas is the existence of a criticalpoint in the ( P, Θ)-phase diagram; this is associated with a fold catastrophe in the orizon effects with surface waves on moving water
P, V,
Θ)-space that constitutes the state-space of the gas [47]. Now thedispersion relation (5), written as k − ρU γ k + ργ (cid:18) g + 4 πUT (cid:19) k − π ργT = 0 , (26)describes the state of a wave on a counter-flow as a surface in ( U, k, T )-space and thissurface has the same kind of fold catastrophe as the van der Waals gas—see Figure 14.The connection between wave blocking and catastrophe theory was inferred a long timeago by Peregrine and Smith [36] and more recently by Trulsen [44]. The white andblue horizons appear as curves on the state surface in Figure 14, which converge andjoin at the “critical point” ( U c , k c , T c ) given by (16)–(18). Only the positive- k part ofthe state surface is shown in Figure 14, since this contains the point analogous to thethermodynamic critical point. By projecting the surface in Figure 14 on to the ( U, T )-plane one obtains the part of the “phase diagram” Figure 7 containing the white andblue horizons, similar to the ( P, Θ)-diagram of the van der Waals gas [47].We can summarize the analogy between the van der Waals gas and waves ona counter-flow by the following table, which we stress describes only a qualitativerelationship: Van der Waals Gas Wave-Current InteractionVolume V Wave Number k Temperature Θ Frequency ω Pressure P Flow Velocity U Compressibility ∂V∂P
Susceptibility ∂k∂U
Spinodal Line Blocking LinePerfect Gas P = Nk B Θ V Pure Advection U = ωk The perfect gas is seen to correspond to very large ω and U (pure advection U = ω/k ofthe surface waves). In addition, the perfect gas is obtained by setting the parameters a and b to zero; similarly, pure advection of the surface waves corresponds to setting theparameters γ and g to zero.The fold catastrophe in Figure 14 can also be projected to the ( U, k )-plane(Figure 15). From the dispersion relation (5) these projections are given by U ( k ) = ωk − (cid:114) gk + γρ k (27)with a fixed value of ω ( T ). Figure 15 shows the curves U ( k ) for different periods T , forboth the gravity-capillary case (red lines in (a), (b) and (c)) and the pure-gravity case(blue lines in (a), (b) and (c)). Figure 15(d) shows U ( k ) for a range of periods T in thegravity-capillary case; these isoperiod curves are analogous to the Andrews isothermsfor a real gas.We have seen from dispersion plots that wave-blocking corresponds to local extremaof the function ω ( k ). It follows from the implicit function theorem that the minimum(maximum) of U ( k ) corresponds to the maximum (minimum) of ω ( k ), and therefore orizon effects with surface waves on moving water Figure 14.
The state of a wave on a counter-flow is a surface in (
U, k, T )-space givenby (26). This surface has a fold catastrophe in the positive- k sector, similar to thefold catastrophe of the van der Waals gas [47]. The red curve on the state surface isthe white horizon and the blue curve is the blue horizon. The horizon curves meetat ( U c , k c , T c ), which is analogous to the critical point of the van der Waals gas. Thewhite- and blue-horizon curves in Figure 7 are the projection of the curves on the statesurface to the ( U, T )-plane. blocking lines are given by the local extrema of U ( k ). These local extrema ∂U/∂k = 0are the analogues of the spinodal line ∂P/∂V = 0 of the van der Waals gas. Withoutsurface tension (blue lines in Figure 15(a)–(c)), U ( k ) has a single minimum that lies onthe dotted black line in these plots. With surface tension (red lines in Figure 15(a)–(c)), U ( k ) has in addition a local maximum. At T = T c (Figure 15(b)) the minimum and themaximum of U ( k ) merge and an inflection point appears; this corresponds to the cuspin the ( U, T )-plane (Figure 7). We can define a mechanical susceptibility χ m = (cid:0) ∂k∂U (cid:1) T ,analogous to the isothermal compressibility coefficient χ θ = − V (cid:0) ∂V∂P (cid:1) θ , that diverges atthe horizons, just as the compressibility of the gas diverges at the spinodal line.In the limit of infinite period T → ∞ ( ω →
0) we found that the wave is describedby a flow speed (20) and wave number (21). We recover this result from the ω = 0 case orizon effects with surface waves on moving water
500 1000 1500 2000k m s
500 1000 1500 2000k m s
500 1000 1500 2000k m s
200 400 600 800 1000k m s (a) (b)(c) (d) Figure 15.
The red curves in Figures (a), (b) and (c) show U as a function of k fordifferent periods T : Figure (a) is for T > T c , (b) for T = T c and (c) for T → ∞ .Figure (d) shows a series of plots of U ( k ) for different T ; the green curve is for T → U ( k ) as T changes. The horizontal dotted purple line is U γ ; this isthe value of U at the local maximum of U ( k ) for T → ∞ (Figure (c)). The horizontaldotted green line is U c ; this is the value of U at the point of inflection of U ( k ) for T = T c (Figure (b)). The horizontal dotted orange line is U g , which is proportionalto T (see (14)); it gives the value of U at the local minimum in the pure gravity case(blue curves in Figures (a)–(c)). of (27), which is U ( k ) = − (cid:114) gk + γρ k. (28)This is plotted in Figure 15(c) and has a maximum ∂U∂k = 0 at wave number k = (cid:114) ρgγ = 1 l c = k γ , (29)where l c is the capillary length. This wave number in (28) reproduces U γ , given by (20).The principle of corresponding states implies that the properties of real gas areuniversal functions of the state variables scaled to the critical point. For the Van der orizon effects with surface waves on moving water V c = 3 nb , θ c = a bR and P c = a b ): (cid:18) P r + 3 V r (cid:19) (3 V r −
1) = 8 θ r , (30)where the subscript r means reduced variable ( V r = VV c , θ r = θθ c and P r = PP c ). Similarly,using the scalings ( k r = √ γk √ ρg , ω r = γ / ω ( ρg ) / and U r = ρ / U ( γg ) / ), we find the universaldispersion relation:( ω r − U r k r ) = k r (1 + k r ) . (31)Whatever the fluid (surface tension, density), its wave-like behavior will be the sameclose to the cusp. The dimensionless form of the constraint (9) becomes: ω r U r + 14 U r + ω r U r − ω r U r − ω r U r − − ω r = 0 . (32)One recovers U γ = −√ (cid:16) γgρ (cid:17) / by imposing ω r = 0. By introducing another scaling U (cid:48) r = U r ω r = Uωg the constraint reads: U (cid:48) r + 14 U (cid:48) r + ω r (cid:18) U (cid:48) r − U (cid:48) r − U (cid:48) r − − ω r (cid:19) = 0 (33)One recovers U g = − gω by imposing ω r = 0.
5. Conclusions and Perspectives
We have described the interaction of linear gravity-capillary waves with a counter-flow, with emphasis on the various horizon effects (wave blocking in fluid-mechanicsterminology). The case of waves with negative co-moving frequency has been includedthroughout; these waves are crucial for the Hawking effect and they have been neglectedin the fluid-mechanics literature on wave blocking. The Hawking effect is a remarkableprocess in which an incident wave generates a wave with negative co-moving frequency,with a resulting amplification of the incident wave (this implies an extraction of energyfrom the flow). It has been shown that this process is robust in the presence ofdispersion [10, 11, 12, 13, 14, 15, 16, 17] and the linear theory of surface waves fallsinto the class of systems that exhibit the Hawking effect [18]. Experimental evidenceof the generation of waves with negative co-moving frequency was reported in [22] andfurther experiments are planned.We presented analytical results for the deep water/short wavelength case kh (cid:29) kh (cid:28) orizon effects with surface waves on moving water kh (cid:28)
1) into the kh (cid:29) kh (cid:28) kh (cid:29)
1. On the otherhand, waves in the kh (cid:29) L s (roughly the width of the arch of the Airy function) which scaleslike L s ≈ gT / (cid:0) dUdx (cid:1) − / x = x ∗ where x ∗ is the position of the white horizon, and (cid:0) dUdx (cid:1) x = x ∗ isthe “surface gravity” at the horizon. An experimental measurement of the Airy shapehas been carried out by Chawla and Kirby [27]. If surface tension is taken into account,we have seen that there is a critical point at the intersection of two saddle-node lines(white and blue horizons). We anticipate that the wave at the critical point will bedescribed by a Pearcey catastrophe integral [36, 44] due to the superposition of twoAiry catastrophe integrals for the two saddle-node lines. Acknowledgments
We are indebted to Yury Stepanyants, Erwann Aubry and Gil Jannes for fruitfuldiscussions. This research was supported by the R´egion PACA (Projet exploratoireHYDRO), the Conseil G´en´eral 06, the Scottish Government, the Royal Society ofEdinburgh and the Royal Society of London.
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