Wave Scattering in Spatially Inhomogeneous Currents
WWave Scattering in Spatially Inhomogeneous Currents
Semyon Churilov
Institute of Solar-Terrestrial Physics of the Siberian Branch of Russian Academy of Sciences,Irkutsk-33, PO Box 291, 664033, Russia.
Andrei Ermakov
School of Agricultural, Computational and Environmental Sciences,University of Southern Queensland, QLD 4350, Australia.
Yury Stepanyants ∗ Department of Applied Mathematics, Nizhny Novgorod State Technical University,Nizhny Novgorod, 603950, Russia andSchool of Agricultural, Computational and Environmental Sciences,University of Southern Queensland, QLD 4350, Australia. (Dated: October 8, 2018)We analytically study a scattering of long linear surface waves on stationary cur-rents in a duct (canal) of constant depth and variable width. It is assumed that thebackground velocity linearly increases or decreases with the longitudinal coordinatedue to the gradual variation of duct width. Such a model admits analytical solutionof the problem in hand, and we calculate the scattering coefficients as functions of in-cident wave frequency for all possible cases of sub-, super, and trans-critical currents.For completeness we study both co-current and counter-current wave propagationin accelerating and decelerating currents. The results obtained are analysed in ap-plication to recent analog gravity experiments and shed light on the problem ofhydrodynamic modelling of Hawking radiation. The paper is published in PhysicalReview D. ∗ Corresponding author: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] A ug I. INTRODUCTION
Since 1981 when Unruh established the analogy in wave transformation occurring at thehorizon of a black hole and at a critical point of a hydrodynamic flow [1] there have beenmany attempts to calculate the transformation coefficients and find the analytical expressionfor the excitation coefficient of a negative energy mode (see, for instance, [2–5] and referencestherein). In parallel with theoretical study there were several attempts to model the wavescattering in spatially inhomogeneous currents experimentally and determine this coefficientthrough the measurement data [6, 7] (similar experiments were performed or suggested inother media, for example, in the atomic Bose-Einstein condensate – see [8] and numerousreferences therein). In particular, the dependence of amplitude of a negative energy modeon frequency of incident wave in a water tank was determined experimentally [6]; howeverseveral aspects of the results obtained in this paper were subject to criticism.The problem of water wave transformation in spatially inhomogeneous currents is ofsignificant interest itself and there is a vast number of publications devoted to theoreticaland experimental study of this problem. However in applying to the modelling of Hawking’seffect, the majority of these publications suffer a drawback which is related to the parasiticeffect of dispersion, whereas the dispersion is absent in the pure gravitational Hawking effect.Below we consider a model which describes a propagation of small-amplitude long surfacewater waves in a duct (canal) of constant depth but variable width. The dispersion is absent,and the model is relevant to the analytical study of the Hawking effect. We show that thetransformation coefficients can be found in the exact analytical forms both for co-currentand counter-current wave propagation in gradually accelerating and decelerating currents.We believe that the results obtained can be of wider interest, not only as a model ofHawking’s effect, but in application to real physical phenomena occurring in currents innon-homogenous ducts, at least at relatively small Froude numbers. We consider all pos-sible configurations of the background current and incident wave. The paper’s contents ispresented below.
CONTENTS
I. Introduction 2II. Derivation of the Governing Equation 4III. Qualitative Analysis of the Problem Based on the JWKB Approximation 7IV. Wave Scattering in Inhomogeneous Currents With a Piece-Linear Velocity Profile 10A. Wave transformation in sub-critical currents 111. Accelerating currents. Transformation of downstream propagating incidentwave 122. Accelerating currents. Transformation of upstream propagating incident wave 143. Wave transformation in a decelerating sub-critical current 17B. Wave transformation in a super-critical current 171. Transformation of a positive-energy wave in an accelerating current 192. Transformation of negative-energy wave in an accelerating current 223. Wave transformation in a decelerating super-critical current 24C. Wave transformation in trans-critical accelerating currents 0 < V < < V V > > V > II. DERIVATION OF THE GOVERNING EQUATION
Let us consider the set of equations for water waves on the surface of a perfect fluid ofa constant density ρ and depth h . Assume that the water moves along the x -axis with astationary velocity U ( x ) which can be either an increasing or a decreasing function of x .Physically such a current can be thought as a model of water flow in a horizontal ductwith a properly varying width b ( x ). We will bear in mind such a model, although we donot pretend here to consider a current in a real duct, but rather to investigate an idealizedhydrodynamic model which is described by the equation analogous to that appearing in thecontext of black hole evaporation due to Hocking radiation [1–5, 9–11].In contrast to other papers also dealing with the surface waves on a spatially varyingcurrent (see, e.g., [2–5]), we consider here the case of shallow-water waves when there is nodispersion, assuming that the wavelengths λ (cid:29) h .In the hydrostatic approximation, which is relevant to long waves in shallow water [12],the pressure can be presented in the form p = p + ρg ( η − z ), where p is the atmosphericpressure, g is the acceleration due to gravity, z is the vertical coordinate, and η ( x, t ) is theperturbation of free surface ( − h ≤ z ≤ η ). Then the linearized Euler equation for smallperturbations having also only one velocity component u ( x, t ) takes the form: ∂u∂t + ∂ ( U u ) ∂x = − g ∂η∂x . (1)The second equation is the continuity equation which is equivalent to the mass conserva-tion equation for shallow-water waves: ∂S∂t + ∂∂x [ S ( U + u )] = 0 , (2)where S ( x, t ) is the portion of the cross-section of a duct occupied by water, S ( x, t ) = b ( x )[ h + η ( x, t )], where b ( x ) is the width of the duct.For the background current Eq. (2) gives the mass flux conservation Q ≡ ρ U ( x ) S ( x ) = ρ U ( x ) b ( x ) h = const. Inasmuch as h = const, we have U ( x ) b ( x ) = Q/ρh = const, andEq. (2) in the linear approximation reduces to: b ( x ) ∂η∂t + ∂∂x [ b ( x ) ( U η + uh )] = 0 . (3)Thus, the complete set of equations for shallow water waves in a duct of a variable widthconsists of Eqs. (1) and (3). This set can be reduced to one equation of the second order.To this end let us divide first Eq. (3) by b ( x ) and rewrite it in the equivalent form: ∂η∂t + U ∂η∂x = − hU ∂∂x uU . (4)Expressing now the velocity component u in terms of the velocity potential ϕ , u = ∂ϕ/∂x ,and combining Eqs. (1) and (4), we derive (cid:18) ∂∂t + U ∂∂x (cid:19) (cid:18) ∂ϕ∂t + U ∂ϕ∂x (cid:19) = c U ∂∂x (cid:18) U ∂ϕ∂x (cid:19) , (5)where c = (cid:112) gh is the speed of linear long waves in shallow water without a backgroundcurrent.As this equation describes wave propagation on the stationary moving current of perfectfluid, it provides the law of wave energy conservation which can be presented in the form(its derivation is given in Appendix A): ∂ E ∂t + ∂J∂x = 0 , (6)where E = i U (cid:20) ϕ (cid:18) ∂ϕ∂t + U ∂ϕ∂x (cid:19) − ϕ (cid:18) ∂ϕ∂t + U ∂ϕ∂x (cid:19)(cid:21) , J = E U − i c U (cid:18) ϕ ∂ϕ∂x − ϕ ∂ϕ∂x (cid:19) , and the over-bar denotes complex conjugation.Solution of the linear equation (5) can be sought in the form ϕ ( x, t ) = Φ( x )e − i ωt , then itreduces to the ODE for the function Φ( x ): (cid:18) − i ω + U ddx (cid:19) (cid:18) − i ω Φ +
U d Φ dx (cid:19) = c U ddx (cid:18) U d Φ dx (cid:19) . (7)If we normalize the variables such that U/c = V , x/L = ξ , and ωL/c = ˆ ω , where L isthe characteristic spatial scale of the basic current, then we can present the main equationin the final form: V (cid:0) − V (cid:1) d Φ dξ − (cid:2)(cid:0) V (cid:1) V (cid:48) − ωV (cid:3) d Φ dξ + V ˆ ω Φ = 0 , (8)where the prime stands for here and below differentiation with respect to the entire functionargument (in this particular case with respect to ξ ).If the perturbations are monochromatic in time, as above, then the wave energy E andenergy flux J do not depend on time, therefore, as follows from Eq. (6), the energy flux doesnot depend on x too, so J = const.For the concrete calculations we chose the piece-linear velocity profile, assuming that thecurrent varies linearly within a finite interval of x and remains constant out of this interval(see Fig. 1): V a ( ξ ) = V ≡ ξ , ξ ≤ ξ ,ξ, < ξ < ξ < ξ ,V ≡ ξ , ξ ≥ ξ ; V d ( ξ ) = V ≡ − ξ , ξ ≤ ξ , − ξ, ξ < ξ < ξ < ,V ≡ − ξ , ξ ≥ ξ , (9)where V a ( ξ ) pertains to the accelerating current, and V d ( ξ ) – to the decelerating current.To simplify further calculations, we have chosen, without the loss of generality, the originof the coordinate frame such that the velocity profile is directly proportional to ± ξ in theinterval ξ ≤ ξ ≤ ξ as shown in Fig. 1). For such velocity configurations it is convenient toset L = ( x − x ) c / ( U − U ) = ( x − x ) / | V − V | .The choice of piece-linear velocity profile allows us to reduce the governing equation (8)to the analytically solvable equation and obtain exact solutions. The corresponding waterflow can be realized in a duct with a variable width, which is constant, b = b , when ξ ≤ ξ ,then gradually varies along the ξ -axis as b ( ξ ) = b ξ /ξ in the interval ξ ≤ ξ ≤ ξ , and afterthat remains constant again, b = b ξ /ξ when ξ ≥ ξ . Schematically the sketch of a ductwith gradually decreasing width that provides an accelerating current is shown in Fig. 2. a)b) FIG. 1. Sketch of accelerating (a) and decelerating (b) background currents.
Equation (8) should be augmented by the boundary conditions at ξ → ±∞ which specifythe scattering problem, as well as by the matching conditions at ξ = ξ and ξ = ξ . The FIG. 2. The sketch of a duct with the decreasing width that provides spatially accelerating back-ground current. latter conditions reduce to the continuity of the function Φ( ξ ) and its derivative Φ (cid:48) ( ξ ) (seeAppendix B for the derivation):Φ( ξ , + 0) = Φ( ξ , − , Φ (cid:48) ( ξ , + 0) = Φ (cid:48) ( ξ , − . (10)On the basis of Eq. (8) and matching conditions (10), we are able to study analyticallyall possible cases of orientation of an incident wave and a current, assuming that the currentcan be sub-critical ( V , < V > V < V < V > V , > III. QUALITATIVE ANALYSIS OF THE PROBLEM BASED ON THE JWKBAPPROXIMATION
Before the construction of an exact solution for wave scattering in currents with thepiece-linear velocity profiles, it seems reasonable to consider the problem qualitatively toreveal its specific features which will help in the interpretation of results obtained.Consider first a long sinusoidal wave propagating on a current with constant U . Assume,in accordance with the shallow-water approximation, that the wavelength λ (cid:29) h . Thedispersion relation for such waves is ( ω − kU ) = c k , (11)where k = ( k, ,
0) is a wave vector related with a wavelength λ = 2 π/ | k | .A graphic of the dispersion relation is shown in Fig. 3 for two values of the current speed,sub-critical, U < c , and super-critical, U > c . Since we consider dispersionless shallow-water waves, graphics of the dependences ω ( k ) are straight lines formally extending fromminus to plus infinity. We suppose, however, that the frequency ω is a non-negative quantitywhich is inversely proportional to the wave period; therefore, without loss of generality, wecan ignore those portions of dispersion lines which correspond to negative frequencies (inFig. 3 they are shown by inclined dashed lines). The dashed horizontal line in Fig. 3 showsa particular fixed frequency of all waves participating in the scattering process. k FIG. 3. (color online) The dispersion dependences for surface waves on uniformly moving shallowwater. Lines 1 and 2 pertain to co-current and counter-current propagating waves respectivelyin a sub-critical current (
U < c ). Lines 3 and 4 pertain to positive- and negative-energy wavesrespectively in a super-critical current ( U > c ) both propagating downstream. For co-current propagating waves with kkk ↑↑ U the dispersion relation (11) reduces to ω =( U + c ) | k | , whereas for counter-current propagating waves with kkk ↓↑ U it is ω = | U − c || k | .Thus, the dispersion lines for surface waves on a current are not symmetrical with respectto the vertical axis k = 0. When the current speed U increases, the right branch 1 turnstoward the vertical axis (cf. lines 1 and 3 in Fig. 3). The left branch 2 in this case tiltstoward the negative half-axis k ; coincides with it when U = c , and then, when U > c , itgoes to the lower half-plane and becomes negative. However, its negative portion 2 (cid:48) goes up,passes through the axis k and appears in the upper half-plane as the dispersion line 4. Thus,waves corresponding to lines 3 and 4 are downstream propagating waves, whereas there areno upstream propagating waves, if U > c . From the physical point of view this means thatthe current is so strong that it pulls downstream even counter-current propagating waves.As was shown, for instance, in Refs. [13–15], in a such strong current, waves on branch 3have positive energy, whereas waves on branch 4 have negative energy.To consider wave propagation on a spatially variable current when it accelerates or de-celerates along x -axis, let us use the JWKB method, which physically presumes that thewavelengths are much less than the characteristic scale of inhomogeneity, λ (cid:28) L (whereasstill λ (cid:29) h and the shallow-water approximation is valid). This condition can be presentedin the form L/λ = L/ ( c T ) = Lω/ (2 πc ) = ˆ ω/ π (cid:29) T = 2 π/ω is the wave period)and if it is fulfilled, the JWKB solution of Eq. (8) can be sought in the form (see, e.g.,[16, 17]): Φ( ξ ) = exp (cid:20) i ˆ ω (cid:90) q ( ξ ) dξ (cid:21) , q ( ξ ) = q ( ξ ) + ˆ ω − q ( ξ ) + ˆ ω − q ( ξ ) + . . . (12)Substitution of these expressions into Eq. (8) gives two linearly independent solutions:Φ ( ± ) ( ξ ) = (cid:112) V ( ξ ) exp (cid:20) i ˆ ω (cid:90) dξV ( ξ ) ± O (ˆ ω − ) (cid:21) , (13)and the general solution of Eq. (8) is the linear combination of these two particular solutions:Φ( x ) = A F Φ (+) ( x ) + A B Φ ( − ) ( x ) , (14)where A F and A B are amplitudes of co-current propagating F-wave and counter-currentpropagating B-wave, respectively.In the current with a spatially varying velocity V ( ξ ), wave propagation and transforma-tion has a regular character, if V ( ξ ) (cid:54) = 1 (i.e., if U ( x ) (cid:54) = c ); then Eq. (8) does not containcritical points.In sub-critical currents, when 0 < V , < V , > ξ → ∞ , waves of both types appear.In contrast to these cases, in a trans-critical current there is a critical point where V ( ξ ) =1. The existence of such a point has only a minor influence on the co-current propagatingF-wave, but exerts a crucial action on the B-wave, because its “wave number” q ( − )0 → ∞ when V ( ξ ) →
1. Due to this, an arbitrarily small but finite viscosity leads to dissipation ofa B-wave that attains a neighborhood of the critical point. As the result of this the energy0flux J does not conserve, in general, when waves pass through this critical point. However,as will be shown below, the energy flux conserves in spatially accelerating trans-criticalcurrents, but does not conserve in decelerating currents.Indeed, in an accelerating current where 0 < V < < V , an incident wave can arriveonly from the left as the F-type wave only. In the sub-critical domain ( ξ <
1) it transformsinto the B-wave that runs backwards, towards ξ = −∞ . After passing the critical point,being in the supercritical domain ( ξ >
1) it transforms into the B-wave that runs forwardtowards ξ = + ∞ . As a result, there is no B-wave that attains the critical point; hence,there is no dissipation, and energy flux conserves. On the contrary, in decelerating currents(where V > > V >
0) B-waves, no matter incident or “reflected”, run to the criticalpoint and dissipate there; therefore the energy flux does not conserve in this case.A specific situation occurs when the incident B-wave propagates from plus infinity inthe sub-critical current towards the critical point and generates an F-wave on the currentinhomogeneity. If the current is super-critical on the left of the critical point, then no onewave can penetrate into that domain. Thus, the wave energy of incident B-wave partiallyconverts into a reflected F-wave and partially absorbs in the vicinity of the critical pointdue to vanishingly small viscosity. We will come to the discussion of these issues in SectionV when we construct exact solutions of scattering problem for Eq. (8) where it is possible.A qualitative analysis presented above demonstrates that the most interesting results canbe obtained for the trans-critical currents and that the critical points play a crucial role insuch currents. However, in the vicinity of a critical point the velocity of arbitrary type U ( x )can be generally approximated by a linear function, U ( x ) ∼ x . This makes an additionalargument in favor of studying wave scattering in currents with piece-linear velocity profiles. IV. WAVE SCATTERING IN INHOMOGENEOUS CURRENTS WITH APIECE-LINEAR VELOCITY PROFILE
Consider now exact solutions of the problem on surface wave scattering in inhomogeneouscurrents with piece-linear velocity profiles described by Eqs. (9) and shown in Fig. 1. Thebasic equation (8) has constant coefficients out of the interval ξ < ξ < ξ , where the currentvelocity linearly varies with ξ (either increasing or decreasing). Therefore out of this interval,solutions to this equation can be presented in terms of exponential functions with the purely1imaginary exponents describing sinusoidal travelling waves.Within the interval ξ < ξ < ξ Eq. (8) with the help of change of variable ζ = ξ reducesto one of the hypergeometric equations: ζ (1 − ζ ) d Φ dζ − (1 ∓ i ˆ ω ) ζ d Φ dζ + ˆ ω , (15)where upper sign pertains to the case of accelerating current, and lower sign – to the caseof decelerating current.The matching conditions at ξ = ξ and ξ = ξ are given by Eqs. (10). A. Wave transformation in sub-critical currents
Assume first that an incident wave propagates from left to right parallel to the maincurrent which is sub-critical in all domains, V < V <
1. As mentioned above, in the left( ξ < ξ ) and right ( ξ > ξ ) domains Eq. (8) has constant coefficients, and in the intermediatedomain ( ξ < ξ < ξ ), where V ( ξ ) = ξ , this equation reduces to one of hypergeometricequations (15). These equations are regular in the sub-critical case, and their coefficientsdo not turn to zero. Two linearly independent solutions can be expressed in terms of Gausshypergeometric function F ( a, b ; c ; ζ ) (see § ξ ) = A e i κ ( ξ − ξ ) + A e − i κ ( ξ − ξ ) , ξ ≤ ξ , (16)Φ( ξ ) = B w ( ξ ) + B w ( ξ ) , ξ ≤ ξ ≤ ξ , (17)Φ( ξ ) = C e i κ ( ξ − ξ ) + C e − i κ ( ξ − ξ ) , ξ ≥ ξ , (18)where κ = ˆ ω/ (1 + V ), κ = ˆ ω/ (1 − V ), κ = ˆ ω/ (1 + V ), κ = ˆ ω/ (1 − V ), A , , B , , C , are arbitrary constants, and w ( ζ ) = ζ F (1 − i ˆ ω/ , − i ˆ ω/
2; 2; ζ ) , w ( ζ ) = F ( − iˆ ω/ , − i ˆ ω/
2; 1 − i ˆ ω ; 1 − ζ ) . (19)The Wronskian of these linearly independent functions is [18]: W = w (cid:48) ( ζ ) w ( ζ ) − w ( ζ ) w (cid:48) ( ζ ) = Γ(1 − i ˆ ω )Γ (1 − i ˆ ω/
2) (1 − ζ ) i ˆ ω − . (20)Similarly the general solution of Eq. (8) for the decelerating current can be presented. Inthe domains ξ < ξ and ξ > ξ solutions are the same as above, whereas in the intermediate2domain ξ < ξ < ξ the general solution is:Φ( ξ ) = B ˜ w ( ξ ) + B ˜ w ( ξ ) , (21)where the linearly independent functions are˜ w ( ζ ) = ζ F (1 + i ˆ ω/ , ω/
2; 2; ζ ) , ˜ w ( ζ ) = F (iˆ ω/ , i ˆ ω/
2; 1 + i ˆ ω ; 1 − ζ ) . (22)with the Wronskian:˜ W = ˜ w (cid:48) ( ζ ) ˜ w ( ζ ) − ˜ w ( ζ ) ˜ w (cid:48) ( ζ ) = Γ(1 + i ˆ ω )Γ (1 + i ˆ ω/
2) (1 − ζ ) − i ˆ ω − . (23)
1. Accelerating currents. Transformation of downstream propagating incident wave
Assume that the incident wave has a unit amplitude A = 1 and calculate the transfor-mation coefficients, setting C = 0 and denoting the amplitudes of the reflected wave by R ≡ A and the transmitted wave by T ≡ C ( R and T play a role of transformation coeffi-cients, as they are usually determined in hydrodynamics – see, e.g. [19, 20] and referencestherein).Using the matching conditions at the boundaries of domains (see Appendix B), we find: B w ( V ) + B w ( V ) = R + 1 , (24) B w (cid:48) ( V ) + B w (cid:48) ( V ) = i ˆ ω V (cid:18)
11 + V − R − V (cid:19) , (25) B w ( V ) + B w ( V ) = T, (26) B w (cid:48) ( V ) + B w (cid:48) ( V ) = i ˆ ω V T V . (27)From these equations we derive the transformation coefficients:3 R = 1∆ (cid:26) ˆ ω [ w ( V ) w ( V ) − w ( V ) w ( V )]4 V V (1 + V )(1 + V ) − w (cid:48) ( V ) w (cid:48) ( V ) + w (cid:48) ( V ) w (cid:48) ( V ) +i ˆ ω (cid:20) w ( V ) w (cid:48) ( V ) − w (cid:48) ( V ) w ( V ) V (1 + V ) − w ( V ) w (cid:48) ( V ) − w (cid:48) ( V ) w ( V ) V (1 + V ) (cid:21) (cid:27) , (28) T = − i ˆ ω ∆ (1 − V ) i ˆ ω − V (1 − V ) Γ(1 − i ˆ ω )Γ (1 − i ˆ ω/ , (29) B = − i ˆ ω ∆ 1 V (1 − V ) (cid:20) i ˆ ω V (1 + V ) w ( V ) − w (cid:48) ( V ) (cid:21) , (30) B = i ˆ ω ∆ 1 V (1 − V ) (cid:20) i ˆ ω V (1 + V ) w ( V ) − w (cid:48) ( V ) (cid:21) , (31)where ∆ = w (cid:48) ( V ) w (cid:48) ( V ) − w (cid:48) ( V ) w (cid:48) ( V ) + ˆ ω [ w ( V ) w ( V ) − w ( V ) w ( V )]4 V V (1 − V )(1 + V ) +i ˆ ω (cid:20) w ( V ) w (cid:48) ( V ) − w (cid:48) ( V ) w ( V ) V (1 − V ) + w ( V ) w (cid:48) ( V ) − w (cid:48) ( V ) w ( V ) V (1 + V ) (cid:21) . (32)The modules of transformation coefficients | T | and | R | , as well as modules of intermediatecoefficients of wave excitation in the transient domain, | B | and | B | , are shown in Fig. 4 asfunctions of dimensionless frequency ˆ ω for the particular values of V = 0 . V = 0 . V and V . A2 i A2 i A2 i A2 i A2 i ˆ FIG. 4. (color online) Modules of transformation coefficient as functions of dimensionless frequencyˆ ω for V = 0 . V = 0 .
9. Line 1 – | T | , line 2 – | R | , line 3 – | B | , line 4 – | B | Dashed line 5 representsthe asymptotic for the reflection coefficient R ∼ ˆ ω − . ω →
0, the hypergeometric function F ( a, b ; c ; d )degenerates (see Appendix C), then the transformation coefficients reduce to R = 1 − V /V V /V , T = 1 + R = 21 + V /V . (33)These values are purely real and agree with the transformation coefficients derived in Ref.[15] for surface waves in a duct with the stepwise change of cross-section and velocity profile,and such an agreement takes place also for other wave-current configurations consideredbelow. Notice only that here the transformation coefficients are presented in terms of velocitypotential ϕ , whereas in Ref. [15] they are presented in terms of free surface elevation η . Therelationship between these quantities is given in the end of Appendix A).In Fig. 5a) we present the graphic of | Φ( ξ ) | (see line 1) as per Eqs. (16)–(18) with A = 1and other determined transformation coefficients A = R as per Eq. (28), C = T as perEq. (29), and C = 0. Coefficients B and B are given by Eqs. (30) and (31). The plotwas generated for the particular value of ˆ ω = 1; for other values of ˆ ω the graphics arequalitatively similar.The solution obtained should be in consistency with the energy flux conservation [14, 15],which is derived in Appendix A in terms of the velocity potential ϕ : V (cid:0) − | R | (cid:1) = V | T | . (34)Substituting here the transformation coefficients R and T from Eqs. (28) and (29), weconfirm that Eq. (34) reduces to the identity.To characterize the rate of energy flux transmission, one can introduce the energy trans-mission factor K T = V V | T | ω → −→ V /V (1 + V /V ) . (35)Then one can see that although the modulus of the transmission coefficient is greaterthan one (see line 1 in Fig. 4) the total energy flux (34) through the duct cross-sectionconserves because the cross-section decreases in the transition from the left to right domain,and the transmitted energy flux is less than the incident one ( K T <
2. Accelerating currents. Transformation of upstream propagating incident wave
If the incident wave arrives from plus infinity, then we set in Eqs. (16) and (18) itsamplitude C = 1, the amplitude of reflected wave C = R , and the amplitude of transmitted5 a)b) FIG. 5. (color online) Modules of function Φ( ξ ) for wave scattering in accelerating (a) and decel-erating (b) sub-critical currents with V = 0 . V = 0 . V = 0 . V = 0 . ξ and ξ where the speed of the background current linearlychanges. wave A = T , whereas A = 0. Then from the matching conditions we obtain B w ( V ) + B w ( V ) = T, (36) B w (cid:48) ( V ) + B w (cid:48) ( V ) = − i ˆ ω V T − V , (37) B w ( V ) + B w ( V ) = 1 + R, (38) B w (cid:48) ( V ) + B w (cid:48) ( V ) = − i ˆ ω V (cid:18) − V − R V (cid:19) . (39)Solution of this set of equations is:6 R = 1∆ (cid:26) ˆ ω [ w ( V ) w ( V ) − w ( V ) w ( V )]4 V V (1 − V )(1 − V ) − w (cid:48) ( V ) w (cid:48) ( V ) + w (cid:48) ( V ) w (cid:48) ( V ) − i ˆ ω (cid:20) w ( V ) w (cid:48) ( V ) − w (cid:48) ( V ) w ( V ) V (1 − V ) − w ( V ) w (cid:48) ( V ) − w (cid:48) ( V ) w ( V ) V (1 − V ) (cid:21) (cid:27) , (40) T = − i ˆ ω ∆ (1 − V ) i ˆ ω − V (1 − V ) Γ(1 − i ˆ ω )Γ (1 − i ˆ ω/ , (41) B = i ˆ ω ∆ 1 V (1 − V ) (cid:20) i ˆ ω V (1 − V ) w ( V ) + w (cid:48) ( V ) (cid:21) , (42) B = − i ˆ ω ∆ 1 V (1 − V ) (cid:20) i ˆ ω V (1 − V ) w ( V ) + w (cid:48) ( V ) (cid:21) , (43)where ∆ is the same as in Eq. (32).In the long-wave approximation, ˆ ω →
0, we obtain the limiting values of transformationcoefficients R = 1 − V /V V /V , T = 1 + R = 21 + V /V . (44)These values again are purely real and agree with the transformation coefficients derivedin Ref. [15] for surface waves in a duct with the stepwise change of cross-section and velocityprofile.This solution is also in consistency with the energy flux conservation, which now takesthe form: V (cid:0) − | R | (cid:1) = V | T | . (45)Substituting here the expressions for the transformation coefficients, (40) and (41), weconfirm that Eq. (45) reduces to the identity. The energy transmission factor K T in thelimit ˆ ω → | Φ( ξ ) | is presented in Fig. 5a) by line 2. The plot was generated onthe basis of solution (16)–(18) with C = 1, A = 0 and other determined transformationcoefficients C = R as per Eq. (40) and A = T as per Eq. (41). Coefficients B and B aregiven by Eqs. (42) and (43).7
3. Wave transformation in a decelerating sub-critical current
The decelerating current can occur, for example, in a widening duct. To calculate thetransformation coefficients of waves in a decelerating current with a piece-linear profile it isconvenient to choose the origin of coordinate frame such as shown in Fig. 1b).The general solutions of the basic equation (8) in the left and right domains beyond theinterval ξ < ξ < ξ are the same as in Eqs. (16) and (18), whereas in the transient domainthe solution is given by Eq. (21).To calculate the transformation coefficients one can repeat the simple, but tedious calcu-lations similar to the presented above. The result shows that the expressions for the trans-formation coefficients remain the same as in Eqs. (28)–(32) for the co-current propagatingincident wave and Eqs. (40)–(43), and (32) for the counter-current propagating incidentwave, but in both these cases ˆ ω should be replaced by − ˆ ω and w i by ˜ w i . The energy fluxEq. (34) for the co-current propagating incident wave or Eq. (45) for the counter-currentpropagating incident wave conserves in these cases too.The graphics of | Φ( ξ ) | are presented in Fig. 5b) by line 1 for co-current propagatingincident wave, and by line 2 for counter-current propagating incident wave. B. Wave transformation in a super-critical current
Assume now that the main current is super-critical everywhere, V > V >
1. In thiscase, there are no upstream propagating waves. Indeed in such strong current even wavespropagating with the speed − c in the frame moving with the water are pulled downstreamby the current whose speed U > c , therefore in the immovable laboratory frame the speedof such “counter-current” propagating waves is U − c >
0. Such waves possess a negativeenergy (see, for instance, [13–15]). Thus, the problem statement can contain an incidentsinusoidal wave propagating only downstream from the ξ < ξ domain; the wave can be ofeither positive energy with ˆ ω = ( V + 1) κ or negative energy with ˆ ω = ( V − κ . Aftertransformation on the inhomogeneous current in the interval ξ < ξ < ξ these waves producetwo transmitted waves in the right domain, ξ > ξ one of positive energy and another ofnegative energy. Below we consider such transformation in detail.In the super-critical case the basic equation (8) is also regular and its coefficients do8not turn to zero. To construct its solutions in the intermediate domain ξ ≤ ξ ≤ ξ it isconvenient to re-write the equation in slightly different form: η (1 − η ) d Ψ dη + [1 − (2 ∓ i ˆ ω ) η ] d Ψ dη ± i ˆ ω (cid:18) ∓ i ˆ ω (cid:19) Ψ = 0 , (46)where η = 1 /ζ , Ψ( η ) = η ± i ˆ ω/ Φ, upper signs pertain to the accelerating current, and lowersigns – to the decelerating currents.Solutions of Eq. (8) in the domains where the current speed is constant areΦ( ξ ) = A e i κ ( ξ − ξ ) + A e i κ ( ξ − ξ ) , ξ ≤ ξ , (47)Φ( ξ ) = C e i κ ( ξ − ξ ) + C e i κ ( ξ − ξ ) , ξ ≥ ξ , (48)were κ = ˆ ω/ ( V + 1), κ = ˆ ω/ ( V − κ = ˆ ω/ ( V + 1), κ = ˆ ω/ ( V − ξ ≤ ξ ≤ ξ the solution of hypergeometric Eq. (46) in thecase of accelerating current isΦ( ξ ) = ξ i ˆ ω (cid:2) B ˘ w (cid:0) ξ − (cid:1) + B ˘ w (cid:0) ξ − (cid:1)(cid:3) , (49)where two linearly independent solutions of Eq. (46) can be chosen in the form (see § w ( η ) = F ( − i ˆ ω/ , − i ˆ ω/
2; 1; η ) , ˘ w ( η ) = F ( − iˆ ω/ , − i ˆ ω/
2; 1 − i ˆ ω ; 1 − η ) (50)with the Wronskian˘ W = ˘ w (cid:48) ( η ) ˘ w ( η ) − ˘ w ( η ) ˘ w (cid:48) ( η ) = (1 − η ) i ˆ ω − η Γ(1 − i ˆ ω )Γ( − i ˆ ω/ − i ˆ ω/ . (51)In the case of decelerating current the solution of hypergeometric Eq. (46) isΦ( ξ ) = ( − ξ ) − i ˆ ω (cid:2) B ˆ w (cid:0) ξ − (cid:1) + B ˆ w (cid:0) ξ − (cid:1)(cid:3) , (52)and linearly independent solutions can be chosen in the form:ˆ w ( η ) = F (i ˆ ω/ , ω/
2; 1; η ) , ˆ w ( η ) = F (iˆ ω/ , ω/
2; 1 + i ˆ ω ; 1 − η ) (53)with the Wronskianˆ W = ˆ w (cid:48) ( η ) ˆ w ( η ) − ˆ w ( η ) ˆ w (cid:48) ( η ) = (1 − η ) − i ˆ ω − η Γ(1 + i ˆ ω )Γ(i ˆ ω/ ω/ . (54)9
1. Transformation of a positive-energy wave in an accelerating current
Consider first transformation of a positive energy incident wave (see line 3 in Fig. 3) withthe unit amplitude ( A = 1, A = 0). Matching the solutions in different current domainsand using the chain rule d/dξ = − ξ − d/dη , we obtain at ξ = ξ : B ˘ w ( V − ) + B ˘ w ( V − ) = V − i ˆ ω , (55) B ˘ w (cid:48) ( V − ) + B ˘ w (cid:48) ( V − ) = i ˆ ω V − i ˆ ω V + 1 , (56)where prime stands for a derivative of a corresponding function with respect to its entireargument.Similarly from the matching conditions at ξ = ξ we obtain: C + C = V i ˆ ω (cid:2) B ˘ w ( V − ) + B ˘ w ( V − ) (cid:3) , (57)( V − C − ( V + 1) C = − ω V i ˆ ω − ( V − (cid:2) B ˘ w (cid:48) ( V − ) + B ˘ w (cid:48) ( V − ) (cid:3) . (58)From Eqs. (55) and (56) we find B = − Γ( − i ˆ ω/
2) Γ(1 − i ˆ ω/ − i ˆ ω ) V i ˆ ω − (cid:0) V − (cid:1) − i ˆ ω (cid:20) ˘ w (cid:48) ( V − ) V − i ˆ ω ˘ w ( V − )2( V + 1) (cid:21) , (59) B = Γ( − i ˆ ω/
2) Γ(1 − i ˆ ω/ − i ˆ ω ) V i ˆ ω − (cid:0) V − (cid:1) − i ˆ ω (cid:20) ˘ w (cid:48) ( V − ) V − i ˆ ω ˘ w ( V − )2( V + 1) (cid:21) . (60)Substituting these in Eqs. (57) and (58), we find the transmission coefficients for thepositive energy mode T p ≡ C and negative energy mode T n ≡ C : T p = − Γ ( − i ˆ ω/ − i ˆ ω ) V i ˆ ω − V i ˆ ω − (cid:0) V − (cid:1) − i ˆ ω (cid:0) V − (cid:1) × (cid:26) ˘ w (cid:48) ( V − ) ˘ w (cid:48) ( V − ) − ˘ w (cid:48) ( V − ) ˘ w (cid:48) ( V − ) V V + ˆ ω w ( V − ) ˘ w ( V − ) − ˘ w ( V − ) ˘ w ( V − )( V + 1)( V −
1) +i ˆ ω (cid:20) ˘ w (cid:48) ( V − ) ˘ w ( V − ) − ˘ w ( V − ) ˘ w (cid:48) ( V − ) V ( V − − ˘ w ( V − ) ˘ w (cid:48) ( V − ) − ˘ w (cid:48) ( V − ) ˘ w ( V − ) V ( V + 1) (cid:21)(cid:27) , (61) T n = Γ ( − i ˆ ω/ − i ˆ ω ) V i ˆ ω − V i ˆ ω − (cid:0) V − (cid:1) − i ˆ ω (cid:0) V − (cid:1) × (cid:26) ˘ w (cid:48) ( V − ) ˘ w (cid:48) ( V − ) − ˘ w (cid:48) ( V − ) ˘ w (cid:48) ( V − ) V V − ˆ ω w ( V − ) ˘ w ( V − ) − ˘ w ( V − ) ˘ w ( V − )( V + 1)( V + 1) − i ˆ ω (cid:20) ˘ w (cid:48) ( V − ) ˘ w ( V − ) − ˘ w ( V − ) ˘ w (cid:48) ( V − ) V ( V + 1) + ˘ w ( V − ) ˘ w (cid:48) ( V − ) − ˘ w (cid:48) ( V − ) ˘ w ( V − ) V ( V + 1) (cid:21)(cid:27) . (62)The modules of transformation coefficients | T p | and | T n | together with the intermediatecoefficients of wave excitation in the transient zone | B | and | B | are shown below in Fig. 6a)as functions of dimensionless frequency ˆ ω for the particular values of V = 1 . V = 1 . V and V . ˆ ˆ a)b) FIG. 6. (color online) Modules of transformation coefficient as functions of dimensionless frequencyˆ ω when a positive energy wave scatters (panel a) and negative energy wave scatters (panel b) inthe current with V = 1 . V = 1 .
9. Line 1 – | T p | , line 2 – | T n | , line 3 – | B | , line 4 – | B | . Dashedlines 5 represent the asymptotics for | T n | ∼ ˆ ω − in panel a) and for | T p | ∼ ˆ ω − in panel b). In Fig. 7a) we present graphics of | Φ( ξ ) | as per Eqs. (47)–(49) for A = 1, A = 0, C = T p as per Eq. (61), and C = T n as per Eq. (62). Coefficients B and B are given by Eqs. (59)1and (60). The plot was generated for two particular values of frequency, ˆ ω = 1 (line 1), andˆ ω = 100 (line 2). a)b) FIG. 7. (color online) Module of function Φ( ξ ) for the scattering of positive- and negative-energywaves in accelerating with V = 1 . V = 1 . V = 1 . V = 1 . ω = 1 (line 1),and ˆ ω = 100 (line 2). The transmission coefficients are in consistency with the energy flux conservation lawwhich has the following form: J = 2ˆ ωV = 2ˆ ωV (cid:0) | T p | − | T n | (cid:1) or | T p | − | T n | = V V . (63)If we introduce two energy transmission factors, for positive- and negative-energy waves, K T p = V V | T p | and K T n = V V | T n | , (64)2then we can see that both waves grow in such a manner that K T p − K T n = 1. This meansthat the positive-energy wave not only dominates in the right domain (cf. lines 1 and 2 inFig. 6a), but it also carries a greater energy flux than the incident one. Moreover, with aproper choice of V and V even the energy flux of negative-energy wave can become greaterby modulus than that of incident wave, K T n >
1. Then we have K T p > K
T n >
1. Figure8 illustrates the dependences of energy transmission factors on the frequency for relativelysmall increase of current speed ( V = 1 . V = 1 .
9) and big increase of current speed( V = 1 . V = 8 . K T p and K T n are greater than 1 in a certainrange of frequencies ˆ ω < ˆ ω c . ˆ FIG. 8. (color online) The dependences of energy transmission factors K T p and K T n on thefrequency for a relatively small increase of current speed ( V = 1 . V = 1 . V = 1 . V = 8 . K T n ∼ ˆ ω − . In the long-wave approximation, ˆ ω → T p = 1 + V /V V /V , T n = − − V /V V /V , K T p = (1 + V /V ) V /V , K T n = (1 − V /V ) V /V . (65)
2. Transformation of negative-energy wave in an accelerating current
Consider now transformation of a negative energy incident wave (see line 4 in Fig. 3)with unit amplitude ( A = 0, A = 1). From the matching conditions at ξ = ξ we obtain: B ˘ w ( V − ) + B ˘ w ( V − ) = V − i ˆ ω , (66) B ˘ w (cid:48) ( V − ) + B ˘ w (cid:48) ( V − ) = − i ˆ ω V − i ˆ ω V − . (67)3The matching conditions at ξ = ξ remain the same as in Eqs. (57) and (58).From Eqs. (66) and (67) we find B = − Γ( − i ˆ ω/
2) Γ(1 − i ˆ ω/ − i ˆ ω ) V i ˆ ω − (cid:0) V − (cid:1) − i ˆ ω (cid:20) ˘ w (cid:48) ( V − ) V + i ˆ ω ˘ w ( V − )2( V − (cid:21) , (68) B = Γ( − i ˆ ω/
2) Γ(1 − i ˆ ω/ − i ˆ ω ) V i ˆ ω − (cid:0) V − (cid:1) − i ˆ ω (cid:20) ˘ w (cid:48) ( V − ) V + i ˆ ω ˘ w ( V − )2( V − (cid:21) . (69)Substituting these in Eqs. (57) and (58), we find the transmission coefficients for thepositive energy mode T p ≡ C and negative energy mode T n ≡ C : T p = − Γ ( − i ˆ ω/ − i ˆ ω ) V i ˆ ω − V i ˆ ω − (cid:0) V − (cid:1) − i ˆ ω (cid:0) V − (cid:1) × (cid:26) ˘ w (cid:48) ( V − ) ˘ w (cid:48) ( V − ) − ˘ w (cid:48) ( V − ) ˘ w (cid:48) ( V − ) V V − ˆ ω w ( V − ) ˘ w ( V − ) − ˘ w ( V − ) ˘ w ( V − )( V − V −
1) +i ˆ ω (cid:20) ˘ w (cid:48) ( V − ) ˘ w ( V − ) − ˘ w ( V − ) ˘ w (cid:48) ( V − ) V ( V −
1) + ˘ w ( V − ) ˘ w (cid:48) ( V − ) − ˘ w (cid:48) ( V − ) ˘ w ( V − ) V ( V − (cid:21)(cid:27) , (70) T n = Γ ( − i ˆ ω/ − i ˆ ω ) V i ˆ ω − V i ˆ ω − (cid:0) V − (cid:1) − i ˆ ω (cid:0) V − (cid:1) × (cid:26) ˘ w (cid:48) ( V − ) ˘ w (cid:48) ( V − ) − ˘ w (cid:48) ( V − ) ˘ w (cid:48) ( V − ) V V + ˆ ω w ( V − ) ˘ w ( V − ) − ˘ w ( V − ) ˘ w ( V − )( V − V + 1) − i ˆ ω (cid:20) ˘ w (cid:48) ( V − ) ˘ w ( V − ) − ˘ w ( V − ) ˘ w (cid:48) ( V − ) V ( V + 1) − ˘ w ( V − ) ˘ w (cid:48) ( V − ) − ˘ w (cid:48) ( V − ) ˘ w ( V − ) V ( V − (cid:21)(cid:27) . (71)The modules of transformation coefficients | T p | and | T n | together with the intermediatecoefficients of wave excitation in the transient zone | B | and | B | are shown in Fig. 6b) asfunctions of dimensionless frequency ˆ ω for the particular values of V = 1 . V = 1 . V and V . The graphic of | Φ( ξ ) | is the same as the graphic shown in Fig. 7a) for the case of scattering of positive-energyincident wave.The transmission coefficients are again in consistency with the energy flux conservationlaw which now has the following form: J = − ωV = − ωV (cid:0) | T n | − | T p | (cid:1) or | T n | − | T p | = V V . (72)As follows from this equation, the energy flux J is negative everywhere, and the negativeenergy wave dominates in the right domain (cf. lines 1 and 2 in Fig. 6b). Both transmitted4waves grow in a such manner that the energy transmission factors (see Eq. (64)) obeythe equality K T n − K T p = 1. Thus, the negative-energy wave not only dominates in theright domain, but also carries a greater energy flux than the incident wave. At a certainrelationship between V and V the energy fluxes of positive- and negative-energy waves canbe greater on absolute value than that of incident wave, then we have K T n > K
T p > ω →
0, we obtain (see Appendix C): T p = − − V /V V /V , T n = 1 + V /V V /V , K T p = (1 − V /V ) V /V , K T n = (1 + V /V ) V /V , (73)i.e., in comparison with Eqs. (65), the energy transmission factors are interchanged. Thevalues of transmission coefficients are purely real, but now T p < T n >
0; they are inagreement with results derived in Ref. [15].
3. Wave transformation in a decelerating super-critical current
In the case of decelerating super-critical current ( V > V >
1) the configuration of theincident wave and current is the same as above in this subsection. Again there is no reflectedwave in the left domain ξ < ξ and there are two transmitted waves in the right domain ξ > ξ .The main equation describing wave propagation is the same as Eq. (46) with only formalreplacement of ˆ ω by − ˆ ω . The general solutions of the basic equation (8) in the left and rightdomains beyond the interval ξ < ξ < ξ are the same as in Eqs. (47) and (48), whereas inthe transient domain the solution is given by Eq. (52).To calculate the transformation coefficients one can repeat the simple, but tedious cal-culations similar to those presented above. The result shows that the expressions for thetransformation coefficients remain the same as in Eqs. (61) and (62) for the incident waveof positive energy and Eqs. (70) and (71) for the incident wave of negative energy, but inboth these cases ˆ ω should be replaced by − ˆ ω and ˘ w i by ˆ w i . The corresponding energy fluxesfor the incident waves of positive and negative energies conserve, and Eqs. (63) and Eq. (72)remain the same in these cases too.The graphics of | Φ( ξ ) | for the scattering of positive- and negative-energy waves are alsothe same in the decelerating currents. They are shown in Fig. 7b) in the subsubsectionIV B 1 for two particular values of frequency, ˆ ω = 1 (line 1), and ˆ ω = 100 (line 2).5 C. Wave transformation in trans-critical accelerating currents < V < < V The specific feature of a trans-critical current is the transition of the background currentspeed U ( x ) through the critical wave speed c . In this case the basic equation (8) contains asingular point where V = 1, therefore the behavior of solutions in the vicinity of this pointshould be thoroughly investigated.The general solution of Eq. (8) in different intervals of ξ -axis can be presented in theform: Φ( ξ ) = A e i κ ( ξ − ξ ) + A e − i κ ( ξ − ξ ) , ξ < ξ , (74)Φ( ξ ) = B w ( ξ ) + B w ( ξ ) , ξ < ξ < , (75)Φ( ξ ) = ξ i ˆ ω (cid:104) ˘ B ˘ w ( ξ − ) + ˘ B ˘ w ( ξ − ) (cid:105) , < ξ < ξ , (76)Φ( ξ ) = C e i κ ( ξ − ξ ) + C e − i κ ( ξ − ξ ) , ξ > ξ , (77)where κ = ˆ ω/ (1 + V ), κ = ˆ ω/ (1 − V ), κ = ˆ ω/ ( V + 1), and κ = ˆ ω/ ( V − V ( ξ ) = 1, let us consider asymptotic behaviorof solution Φ( ξ ) in the vicinity of the point ξ = 1. To this end we use the formula valid for | arg(1 − x ) | < π (see [21], formula 9.131.2.): F ( a, b ; c ; x ) = Γ( c ) Γ( c − a − b )Γ( c − a ) Γ( c − b ) F ( a, b ; a + b − c + 1; 1 − x ) +Γ( c ) Γ( a + b − c )Γ( a ) Γ( b ) (1 − x ) c − a − b F ( c − a, c − b ; c − a − b + 1; 1 − x ) . (78)With the help of this formula let us present the asymptotic expansion of functions (75)and (76), keeping only the leading terms:Φ( ξ ) = B + Γ(i ˆ ω ) B Γ (1 + i ˆ ω/
2) + Γ( − i ˆ ω ) B Γ (1 − i ˆ ω/
2) (1 − ξ ) i ˆ ω + O (1 − ξ ) , ξ → − , (79)Φ( ξ ) = ˘ B + Γ(1 + i ˆ ω ) ˘ B (1 + i ˆ ω/
2) + Γ(1 − i ˆ ω ) ˘ B (1 − i ˆ ω/
2) ( ξ − i ˆ ω + O ( ξ − , ξ → +0 . (80)As one can see from these formulae, for real ˆ ω solutions contain fast oscillating functionsfrom both sides of a singular point ξ = 1, which correspond to B-waves, propagatingagainst the current; these functions, however, remain finite. To match the solutions across6the singular point let us take into consideration a small viscosity in Eq. (1): ∂u∂t + ∂ ( U u ) ∂x = − g ∂η∂x + ν ∂ u∂x , (81)where ν is the coefficient of kinematic viscosity.Due to this correction to Eq. (1) we obtain the modified Eq. (8) for Φ( ξ ): νV d Φ dξ + V (cid:0) − V − i ν ˆ ω (cid:1) d Φ dξ − (cid:2)(cid:0) V (cid:1) V (cid:48) − ωV (cid:3) d Φ dξ + V ˆ ω Φ = 0 . (82)Introducing a new variable ζ = ξ and bearing in mind that V ( ξ ) = ξ for the acceleratingcurrent, we re-write Eq. (82):2 νζ d Φ dζ + ζ [1 − ζ + (3 − i ˆ ω ) ν ] d Φ dζ − (cid:20) i ν ˆ ω − i ˆ ω ) ζ (cid:21) d Φ dζ + ˆ ω . (83)From this equation one can see that in the vicinity of the critical point, where | ζ − | ∼ ε (cid:28)
1, the viscosity plays an important role, if ν ∼ ε . Setting ν = ε / ζ = 1 + εz , weobtain an equation containing the terms up to ε :(1 + εz ) d Φ dz − (1 + εz ) (cid:18) z − − i ˆ ω ε (cid:19) d Φ dz − (cid:20) (1 − i ˆ ω )(1 + εz ) + i ˆ ω ε (cid:21) d Φ dz + ε ˆ ω . (84)Looking for a solution to this equation in the form of asymptotic series with respect toparameter ε , Φ( z ) = Φ ( z ) + ε Φ ( z ) + . . . , we obtain in the leading order ddz (cid:18) d Φ dz − z d Φ dz + i ˆ ω Φ (cid:19) = 0 . (85)Integration of this equation gives the second order equation d Φ dz − z d Φ dz + i ˆ ω (Φ − D ) = 0 , (86)where D is a constant of integration.This equation reduces to the equation of a parabolic cylinder with the help of ansatzΦ ( z ) = e z / G ( z ) + D : d Gdz + (cid:18) i ˆ ω + 12 − z (cid:19) G = 0 . (87)7Two linearly independent solutions of this equation can be constructed from the followingfour functions D i ˆ ω ( ± z ) and D − i ˆ ω − ( ± i z ) (see [21], 9.255.1). Thus, in the vicinity of thecritical point ξ = 1 the solution can be presented in the formΦ ( z ) = D + e z / [ D D i ˆ ω ( z ) + D D i ˆ ω ( − z )] , (88)where D , D , and D are arbitrary constants.This solution should be matched with the asymptotic expansions (79) and (80) using thefollowing asymptotics of functions of the parabolic cylinder when | s | (cid:29) D p ( s ) ∼ s p e − s / F (cid:18) − p , − p − s (cid:19) , | arg s | < π , (89) D p ( s ) ∼ s p e − s / F (cid:18) − p , − p − s (cid:19) − √ π e i πp Γ( − p ) s − p − e s / F (cid:18) p , p s (cid:19) , (90) D p ( s ) ∼ s p e − s / F (cid:18) − p , − p − s (cid:19) − √ π e − i πp Γ( − p ) s − p − e s / F (cid:18) p , p s (cid:19) , (91)where Eq. (90) is valid for π/ < arg s < π/
4, and Eq. (91) is valid for − π/ < arg s < − π/ z → −∞ , and the latter, when z → + ∞ ). To removeinfinitely growing terms from the solution, we need to set D = D = 0 in Eq. (88), thenafter the matching, we obtain in Eqs. (79), (80) and (75), (76) B = ˘ B = 0 , and B = ˘ B = D . (92)Notice that from the physical point of view the former equality, B = ˘ B = 0, is just aconsequence of the fact mentioned in Sec. III that in the trans-critical accelerating currentthe B-waves (i.e., counter-current propagating waves on the left of critical point and negative-energy waves on the right of it) cannot reach the critical point.After that assuming that the incident wave arriving from minus infinity has a unit am-plitude A = 1, using matching conditions (10) and putting T p ≡ C , T n ≡ C , we obtain8 B w (cid:0) V (cid:1) = R + 1 , (93) B w (cid:48) ( V ) = − i ˆ ωR V (1 − V ) + i ˆ ω V (1 + V ) , (94) T n + T p = V i ˆ ω ˘ w (cid:0) V − (cid:1) ˘ B , (95)( V + 1) T n − ( V − T p = 2iˆ ω V i ˆ ω − (cid:0) V − (cid:1) ˘ w (cid:48) (cid:0) V − (cid:1) ˘ B . (96)This set can be readily solved yielding the following transformation coefficients: R = − w (cid:48) ( V ) − i ˆ ω w ( V )2 V (1 + V ) w (cid:48) ( V ) + i ˆ ω w ( V )2 V (1 − V ) , (97) B = ˘ B = R + 1 w ( V ) , (98) T n = iˆ ω V i ˆ ω − (cid:0) V − (cid:1) (cid:34) ˘ w (cid:48) (cid:0) V − (cid:1) V − i ˆ ω w (cid:0) V − (cid:1) V + 1 (cid:35) B , (99) T p = − iˆ ω V i ˆ ω − (cid:0) V − (cid:1) (cid:34) ˘ w (cid:48) (cid:0) V − (cid:1) V + i ˆ ω w (cid:0) V − (cid:1) V − (cid:35) B . (100)In the long-wave approximation, ˆ ω →
0, we obtain (see Appendix C): R = 1 − V V , T p = V + 1 V + 1 , T n = − V − V + 1 , K T p, n = V V (cid:18) V ± V + 1 (cid:19) , (101)where in the last formula sign plus pertains to the positive- and sign minus – to the negative-energy transmitted wave.These values are purely real, R > T p >
0, whereas T n <
0. The problem ofsurface wave transformation in a duct with the stepwise change of cross-section and velocityprofile is undetermined for such current, therefore in Ref. [15] one of the parameters, R η – the reflection coefficient in terms of free surface perturbation, was undefined. Now fromEq. (101) it follows that the transformation coefficients in terms of free surface perturbationin Ref. [15] are R η = T pη = − T nη = 1 (for the relationships between the transformationcoefficients in terms of velocity potential and free surface perturbation see Appendix A).Because of the relationships between the coefficients (92), the solution in the domain ξ < ξ < ξ is described by the same analytical function w ( ξ ) ≡ ξ i ˆ ω ˘ w ( ξ − ) (see Eqs. (9)and (11) in § ξ = 1: J = 2ˆ ωV (cid:0) − | R | (cid:1) = 2ˆ ωV (cid:0) | T p | − | T n | (cid:1) > V (cid:0) − | R | (cid:1) = V (cid:0) | T p | − | T n | (cid:1) . (102)As one can see from these expressions, the energy flux in the reflected wave by modulusis always less than in the incident wave, therefore over-reflection here is not possible. In themeantime the energy transmission factors K T p,n can be greater than 1; this implies that theover-transmission can occur with respect to both positive- and negative-energy waves.The transformation coefficients | R | , | T p | and | T n | together with the intermediate coef-ficients of wave excitation in the transient zone, | B | = | ˘ B | , are presented in Fig. 9 asfunctions of dimensionless frequency ˆ ω for the particular values of speed, V = 0 . V = 1 .
9. Qualitatively similar graphics were obtained for other values of V and V . ˆ FIG. 9. (color online) Modules of the transformation coefficient as functions of dimensionlessfrequency ˆ ω for V = 0 . V = 1 .
9. Line 1 – | T p | , line 2 – | T n | , line 3 – | R | , line 4 – | B | = | ˘ B | .Dashed line 5 represents the asymptotic for | T n | ∼ ˆ ω − . Notice that both the transmission coefficient of negative energy wave | T n | and reflectioncoefficient of positive energy wave | R | decay asymptotically with the same rate ∼ ˆ ω − .Figure 10 illustrates the dependences of energy transmission factors on the frequency fortwo cases: (i) when both K T p,n < V = 0 . V = 1 .
9) and (ii) when both K T p,n > ω < ˆ ω c ( V = 0 . V = 8 . | Φ( ξ ) | as per Eqs. (74)–(77) for A = 1, A = R as perEq. (97), D = T n as per Eq. (99), and D = T p as per Eq. (100). Coefficients B = ˘ B = 0as per Eq. (92), and B = ˘ B are given by Eq. (98). Line 1 in this figure pertains to thecase when V = 0 . V = 1 .
9, and line 2 – to the case when V = 0 . V = 8 . ˆ FIG. 10. (color online) The dependences of energy transmission factors on the frequency (i) whenboth K T p < K T n < V = 0 . V = 1 . K T p > K T n > ω < ˆ ω c (here V = 0 . V = 8 . K T n ∼ ˆ ω − . FIG. 11. (color online) Modules of function Φ( ξ ) for wave scattering in accelerating trans-criticalcurrent with V = 0 . V = 1 . V = 0 . V = 8 . V = 0 . V = 1 .
9, anddashed vertical lines 5 and 6 show the transition zone where the current accelerates from V = 0 . V = 8 .
0. The plot was generated for ˆ ω = 1. D. Wave transformation in trans-critical decelerating currents V > > V > In this subsection we consider the wave transformation in gradually decelerating back-ground current assuming that the current is super-critical in the left domain and sub-criticalin the right domain. For the sake of simplification of hypergeometric functions used below wechose again the coordinate frame such as shown in Fig. 1b). In such a current the transition1through the critical point, where V ( ξ ) = 1, occurs at ξ = − w as per Eq. (22) and ˆ w as per Eq. (53):Φ( ξ ) = A e i κ ( ξ − ξ ) + A e i κ ( ξ − ξ ) , ξ < ξ , (103)Φ( ξ ) = ( − ξ ) − i ˆ ω (cid:104) ˆ B ˆ w (cid:0) ξ − (cid:1) + ˆ B ˆ w (cid:0) ξ − (cid:1)(cid:105) , ξ < ξ < − , (104)Φ( ξ ) = B ˜ w (cid:0) ξ (cid:1) + B ˜ w (cid:0) ξ (cid:1) , − < ξ < ξ < , (105)Φ( ξ ) = C e i κ ( ξ − ξ ) + C e − i κ ( ξ − ξ ) , ξ > ξ , (106)where κ = ˆ ω/ ( V + 1) , κ = ˆ ω/ ( V − , κ = ˆ ω/ (1 + V ) , κ = ˆ ω/ (1 − V ).The matching conditions at ξ = ξ provide (cf. Eqs. (55)and (56)): A + A = V − i ˆ ω (cid:104) ˆ B ˆ w (cid:0) V − (cid:1) + ˆ B ˆ w (cid:0) V − (cid:1)(cid:105) , (107)( V − A − ( V + 1) A = 2iˆ ω V − i ˆ ω − (cid:0) V − (cid:1) (cid:104) ˆ B ˆ w (cid:48) (cid:0) V − (cid:1) + ˆ B ˆ w (cid:48) (cid:0) V − (cid:1)(cid:105) , (108)And similarly the matching conditions at ξ = ξ provide: C + C = B ˜ w (cid:0) V (cid:1) + B ˜ w (cid:0) V (cid:1) , (109)(1 − V ) C − (1 + V ) C = 2iˆ ω V (cid:0) − V (cid:1) (cid:2) B ˜ w (cid:48) (cid:0) V (cid:1) + B ˜ w (cid:48) (cid:0) V (cid:1)(cid:3) . (110)With the help of Eq. (78) we find the asymptotic expansions when ξ → − ± Φ( ξ ) = ˆ B + Γ(1 − i ˆ ω ) ˆ B (1 − i ˆ ω/
2) + Γ(1 + i ˆ ω ) ˆ B (1 + i ˆ ω/ (cid:0) ξ − (cid:1) − i ˆ ω + O (cid:0) ξ − (cid:1) , ξ → − − , (111)Φ( ξ ) = B + Γ( − i ˆ ω ) B Γ (1 − i ˆ ω/
2) + Γ (i ˆ ω ) B Γ (1 + i ˆ ω/ (cid:0) − ξ (cid:1) − i ˆ ω + O (cid:0) − ξ (cid:1) , ξ → − +0 . (112)which are similar to Eqs. (79) and (80), and contain fast oscillating terms corresponding tocounter-current propagating B-waves as well.2To match solutions in the vicinity of critical point ξ = −
1, we again take into considerationa small viscosity. Bearing in mind that V ( ξ ) = − ξ (see Fig. 1b)) and setting ζ = ξ = 1+ εz , ν = ε /
2, we arrive at the equation similar to Eq. (84):(1 + εz ) d Φ dz + (1 + εz ) (cid:18) z + 3 + i ˆ ω ε (cid:19) d Φ dz + (cid:20) (1 + i ˆ ω )(1 + εz ) + i ˆ ω ε (cid:21) d Φ dz − ε ˆ ω . (113)This equation in the leading order on the small parameter ε (cid:28) ddz (cid:18) d Φ dz + z d Φ dz + i ˆ ω Φ (cid:19) = 0 . (114)Integrating this equation and substituting Φ ( z ) = D + e − z / G ( z ), we obtain again theequation of a parabolic cylinder in the form (cf. Eq. (115)): d Gdz + (cid:18) i ˆ ω − − z (cid:19) G = 0 . (115)Thus, the general solution to Eq. (114) in the vicinity of critical point ξ = − ( z ) = D + e − z / [ D D i ˆ ω − ( z ) + D D i ˆ ω − ( − z )] , where D , D and D are arbitrary constants.The asymptotic expansions (89)–(91) show that this solution remains limited for anyarbitrary constants. Moreover, the oscillatory terms in Eqs. (111) and (112) become ex-ponentially small after transition through the critical point ξ = −
1. As was explained inSec. III, this means that the B-waves running toward the critical point both from the left(negative-energy waves) and from the right (counter-current propagating positive-energywaves) dissipate in the vicinity of the critical point. For this reason the wave energy fluxdoes not conserve in the decelerating trans-critical currents (see Eqs. (127) and (134) below).Taking this fact into account, one can match solutions (111) and (112): D = Γ( − i ˆ ω )Γ (1 − i ˆ ω/ B + B = Γ(1 − i ˆ ω )2Γ (1 − i ˆ ω/
2) ˆ B + ˆ B . (116)After that using the identity Γ( x )Γ(1 − x ) = π/ sin πx , we find for the constants D and D the following expressions: D = − i (cid:112) π/ (1 + iˆ ω/ e − iˆ ω ln ε sinh π ˆ ω B , D = ˆ ω (cid:112) π/ (1 + iˆ ω/ e − iˆ ω ln ε sinh π ˆ ω ˆ B . (117)3Using the prepared formulae we can now calculate the transformation coefficients forincident waves of either positive or negative energy travelling in the duct from the minus toplus infinity.
1. Transformation of downstream propagating positive-energy wave
Assume first that the incident wave of unit amplitude has positive energy and let us setin Eqs. (103) and (106) A = 1, A = 0, C ≡ T , and C = 0. Then from Eqs. (107) and(108) we obtain (cf. Eqs. (55) and (56)):ˆ B ˆ w (cid:0) V − (cid:1) + ˆ B ˆ w (cid:0) V − (cid:1) = V i ˆ ω , (118)ˆ B ˆ w (cid:48) (cid:0) V − (cid:1) + ˆ B ˆ w (cid:48) (cid:0) V − (cid:1) = − i ˆ ω V i ˆ ω +21 V + 1 . (119)From this set of equations using the Wronskian (54), one can findˆ B = − Γ(i ˆ ω/
2) Γ(1 + i ˆ ω/ ω ) V − i ˆ ω − (cid:0) V − (cid:1) i ˆ ω +1 (cid:34) ˆ w (cid:48) (cid:0) V − (cid:1) V + i ˆ ω w (cid:0) V − (cid:1) V + 1 (cid:35) , (120)ˆ B = Γ(i ˆ ω/
2) Γ(1 + i ˆ ω/ ω ) V − i ˆ ω − (cid:0) V − (cid:1) i ˆ ω +1 (cid:34) ˆ w (cid:48) (cid:0) V − (cid:1) V + i ˆ ω w (cid:0) V − (cid:1) V + 1 (cid:35) . (121)Similarly from the matching conditions (109) and (110) we obtain B ˜ w (cid:0) V (cid:1) + B ˜ w (cid:0) V (cid:1) = T , (122) B ˜ w (cid:48) (cid:0) V (cid:1) + B ˜ w (cid:48) (cid:0) V (cid:1) = − i ˆ ωT V (1 + V ) . (123)Using the Wronskian (23), we derive from these equations B = − Γ (1 + i ˆ ω/ ω ) (cid:0) − V (cid:1) i ˆ ω +1 (cid:20) ˜ w (cid:48) (cid:0) V (cid:1) + i ˆ ω w ( V ) V (1 + V ) (cid:21) T , (124) B = Γ (1 + i ˆ ω/ ω ) (cid:0) − V (cid:1) i ˆ ω +1 (cid:20) ˜ w (cid:48) (cid:0) V (cid:1) + i ˆ ω w ( V ) V (1 + V ) (cid:21) T . (125)Substituting B and B , as well as ˆ B and ˆ B , in Eq. (116), we obtain the transmissioncoefficient T = − ω V − i ˆ ω − (cid:18) V − − V (cid:19) i ˆ ω +1 × w (cid:48) (cid:0) V − (cid:1) V + i ˆ ω w (cid:0) V − (cid:1) V + 1 − Γ(1 − i ˆ ω )2Γ (1 − i ˆ ω/ (cid:34) ˆ w (cid:48) ( V − ) V + i ˆ ω w (cid:0) V − (cid:1) V + 1 (cid:35) ˜ w (cid:48) (cid:0) V (cid:1) + i ˆ ω w ( V ) V (1 + V ) − Γ( − i ˆ ω )Γ (1 − i ˆ ω/ (cid:20) ˜ w (cid:48) (cid:0) V (cid:1) + i ˆ ω w ( V ) V (1 + V ) (cid:21) . (126)Calculations of the energy fluxes on each side of the transient domain show that they areboth positive, but generally different, i.e., the energy flux does not conserve, J = J ( ξ < −
1) = 2ˆ ωV (cid:54) = J = J ( ξ > −
1) = 2ˆ ωV | T | . (127) ˆ ˆ a)b) FIG. 12. (color online) Modules of the transmission coefficients | T | (line 1 in panel a) and | T | (line 1 in panel b), as well as coefficients of wave excitation in the transient domain, | B | (line 2), | B | (line 3), | ˆ B | (line 4), and | ˆ B | (line 5), for the scattering of positive energy wave (panel a) andnegative energy wave (panel b) as functions of dimensionless frequency ˆ ω for V = 1 . V = 0 . | T | ∼ ˆ ω − . This interesting fact can be explained by the partial wave absorption in the critical point5due to viscosity. The detailed explanation of this is given in Section V. The difference inthe energy flux in the incident and transmitted waves is independent of the viscosity, when ν →
0: ∆ J ≡ J − J = 2ˆ ω (cid:0) /V − | T | /V (cid:1) ˆ ω → −→ − V V V (1 + V ) ( V − V ) J , (128)and it is easily seen that it can be both positive and negative.In Fig. 12a) we present the transmission coefficient | T | together with the intermediatecoefficients of wave excitation in the transient domain, | B | , | B | , | ˆ B | , and | ˆ B | , as functionsof dimensionless frequency ˆ ω for the particular values of current speed V = 1 . V = 0 . | Φ( ξ ) | is shown in Fig. 13 by lines 1 and 2. The plot was generated forˆ ω = 1 on the basis of solution Eqs. (103)–(106) with A = 1, A = 0, D = T as perEq. (126), and D = 0. Coefficients B and B are given by Eqs. (124) and (125), andcoefficients ˆ B and ˆ B are given by Eqs. (120) and (121). The module of function Φ( ξ )is discontinuous only in the critical point ξ = −
1, and the phase of function Φ( ξ ) quicklychanges in the small vicinity of this point. FIG. 13. (color online) Modules of function Φ( ξ ) for wave scattering in decelerating trans-criticalcurrent with V = 1 . V = 0 . ω = 1. Lines 1 and 2 pertain to the scattering of a positive-energy incident wave, and lines 1 and 3 pertain to the scattering of a negative-energy incident wave(line 1 is the same both for positive- and negative-energy waves).
2. Transformation of downstream propagating negative-energy wave
Assume now that the incident wave is a unit amplitude wave of negative energy andcorrespondingly set A = 0, A = 1, C = 0, and C ≡ T . Then from Eqs. (107) and (108)we obtain: ˆ B ˆ w (cid:0) V − (cid:1) + ˆ B ˆ w (cid:0) V − (cid:1) = V i ˆ ω , (129)ˆ B ˆ w (cid:48) (cid:0) V − (cid:1) + ˆ B ˆ w (cid:48) (cid:0) V − (cid:1) = i ˆ ω V i ˆ ω +21 V − . (130)From this set we findˆ B = − Γ(i ˆ ω/
2) Γ(1 + i ˆ ω/ ω ) V − i ˆ ω − (cid:0) V − (cid:1) i ˆ ω +1 (cid:34) ˆ w (cid:48) (cid:0) V − (cid:1) V − i ˆ ω w (cid:0) V − (cid:1) V − (cid:35) , (131)ˆ B = Γ(i ˆ ω/
2) Γ(1 + i ˆ ω/ ω ) V − i ˆ ω − (cid:0) V − (cid:1) i ˆ ω +1 (cid:34) ˆ w (cid:48) (cid:0) V − (cid:1) V − i ˆ ω w (cid:0) V − (cid:1) V − (cid:35) . (132)From the matching conditions at ξ = ξ (see Eqs. (109) and (110)) we obtain the similarexpressions for the coefficients B and B as in Eqs. (124) and (125) with the only replace-ment of T by T . Substituting then all four coefficients B , B , ˆ B , and ˆ B in Eq. (116),we obtain the transmission coefficient T : T = − ω V − i ˆ ω − (cid:18) V − − V (cid:19) i ˆ ω +1 × ˆ w (cid:48) (cid:0) V − (cid:1) V − i ˆ ω w (cid:0) V − (cid:1) V − − Γ(1 − i ˆ ω )2Γ (1 − i ˆ ω/ (cid:34) ˆ w (cid:48) ( V − ) V − i ˆ ω ˆ w (cid:0) V − (cid:1) V − (cid:35) ˜ w (cid:48) (cid:0) V (cid:1) + i ˆ ω w ( V ) V (1 + V ) − Γ( − i ˆ ω )Γ (1 − i ˆ ω/ (cid:20) ˜ w (cid:48) (cid:0) V (cid:1) + i ˆ ω ˜ w ( V )2 V (1 + V ) (cid:21) . (133)Calculations of the energy fluxes on each side of the transient domain show that they arenot equal again, moreover, they have opposite signs in the left and right domains: J = J ( ξ < −
1) = − ωV < , J = J ( ξ > −
1) = 2ˆ ωV | T | > . (134)The wave of negative energy in the left domain propagates to the right, its group velocity V g is positive, but because it has a negative energy E , its energy flux, J = EV g is negative.In the long-wave approximation, ˆ ω → T = V ( V + 1) V ( V + 1) , T = V ( V − V ( V + 1) , K T ,T = V V (cid:18) V ± V + 1 (cid:19) , (135)7where in the last formula sign plus pertains to the positive- and sign minus – to the negative-energy wave. As one can see, the transmission coefficients are purely real and positive, T , >
0, in both cases.The problem of surface wave transformation in a duct with the stepwise change of cross-section and velocity profile is undetermined for such current too; however from the resultsobtained it follows that in terms of free surface perturbation the transformation coefficientsare T η = V V (cid:18) V + 1 V + 1 (cid:19) , T η = V V V − V + 1) (136)(for the relationships between the transformation coefficients in terms of velocity potentialand free surface perturbation see Appendix A).In Fig. 12b) we present the transmission coefficient | T | together with the coefficientsof wave excitation in the intermediate domain, | B | , | B | , | ˆ B | , and | ˆ B | , as functions ofdimensionless frequency ˆ ω for the particular values of current speed V = 1 . V = 0 . ω <
1, then it decreases with the frequency and asymptoticallyvanishes as | T | ∼ ˆ ω − when ˆ ω → ∞ .The graphic of | Φ( ξ ) | is shown in Fig. 13 by lines 1 and 3 (the left branch of function | Φ( ξ ) | for the incident negative- and positive-energy waves are the same). The plot wasgenerated for ˆ ω = 1 on the basis of solution Eqs. (103)–(106) with A = 0, A = 1, D = 0,and D = T as per Eq. (133). Coefficients B and B are given by Eqs. (124) and (125),and coefficients ˆ B and ˆ B are given by Eqs. (131) and (132). The module of function Φ( ξ )is discontinuous only in the critical point ξ = −
1, but the phase of function Φ( ξ ) quicklychanges in the small vicinity of this point.
3. Transformation of a counter-current propagating wave
Consider now the case when the incident wave propagates against the mean current inthe spatially variable current from the right domain where the background current is sub-critical. There are no waves capable to propagate against in the ξ < − V >
1, therefore there is no transmitted wave in this case. However, the incident wave canpropagate against the current and even penetrate into the transient zone ξ < ξ < ξ up tothe critical point ξ = − ξ < −
1, we should set in Eqs. (103)–(106) A = A = ˆ B = ˆ B = 0, C ≡ R , and C = 1. Then the matching condition (116) yields B = − Γ( − i ˆ ω )Γ (1 − i ˆ ω/ B , (137)and from Eqs. (109) and (110) we obtain for the reflection coefficient R = − ˜ w (cid:48) (cid:0) V (cid:1) − i ˆ ω w ( V ) V (1 − V ) − Γ( − i ˆ ω )Γ (1 − i ˆ ω/ (cid:20) ˜ w (cid:48) ( V ) − i ˆ ω w ( V ) V (1 − V ) (cid:21) ˜ w (cid:48) (cid:0) V (cid:1) + i ˆ ω w ( V ) V (1 + V ) − Γ( − i ˆ ω )Γ (1 − i ˆ ω/ (cid:20) ˜ w (cid:48) (cid:0) V (cid:1) + i ˆ ω w ( V ) V (1 + V ) (cid:21) . (138)Then, from Eqs. (109) and (137) we find B and B ; in particular for B we obtain: B = 1 + R ˜ w (cid:0) V (cid:1) − Γ( − i ˆ ω )Γ (1 − i ˆ ω/
2) ˜ w (cid:0) V (cid:1) . (139)Graphics of modulus of reflection coefficient | R | as well as coefficients | B | and | B | areshown in Fig. 14 as functions of dimensionless frequency ˆ ω for the particular values of V = 1 . V = 0 . ˆ FIG. 14. (color online) Modulus of the reflection coefficients | R | (line 1) and coefficients of waveexcitation in the transient domain, | B | (line 2) and | B | (line 3), as functions of dimensionlessfrequency ˆ ω for V = 1 . V = 0 .
1. Dashed line 4 represents the high-frequency asymptotic for | R | ∼ ˆ ω − . In the long-wave approximation, ˆ ω →
0, using the asymptotics of hypergeometric function F ( a, b ; c ; d ) (see Appendix C), we obtain the limiting value of the reflection coefficient R = 1 − V V . (140)9In terms of free surface perturbation this value corresponds to R η = 1 (for the relation-ships between the transformation coefficients in terms of velocity potential and free surfaceperturbation see Appendix A). This formally agrees with the solution found in Ref. [15].In Fig. 15 we present graphics of | Φ( ξ ) | as per Eqs. (103)–(106) for A = A = 0, D = 1,and D = R as per Eq. (138). Coefficients B and B are given by Eqs. (139) and (137),and coefficients ˆ B = ˆ B = 0. A plot was generated for three dimensionless frequencies: line1 – for ˆ ω = 0 .
1, line 2 – for ˆ ω = 1, and line 3 – for ˆ ω = 100. The phase of function Φ( ξ )infinitely increases when the incident wave approaches the critical point ξ = − FIG. 15. (color online) Module of function Φ( ξ ) for a counter-current propagating incident wavewhich scatters in the decelerating trans-critical current with V = 1 . V = 0 . ω : line 1 – ˆ ω = 0 .
1, line 2 – ˆ ω = 1, and line 3 – ˆ ω = 100. The energy fluxes in the incident J i and reflected J r waves in the right domain ( ξ > ξ )are J i = − ωV < , J r = 2ˆ ωV | R | > . (141)Thus, the total energy flux in the right domain ∆ J ≡ J i − J r = − (2ˆ ω/V ) (1 − | R | ) < V. DISCUSSION AND CONCLUSION
In this paper we have calculated the transformation coefficients of shallow water gravitywaves propagating on a longitudinally varying quasi-one-dimensional current. Owing tothe choice of a piece-linear velocity profile U ( x ) (or, in the dimensionless variables, V ( ξ ),see Fig. 1) we were able to calculate analytically the scattering coefficients as functions of0incident wave frequency ˆ ω for accelerating and decelerating sub-, super-, and trans-criticalcurrents, as well as for all possible types of incident wave.Presented analysis pertains to the dispersionless case when the wavelengths of all wavesparticipating in the scattering process are much greater than the water depth in the canal.However, the wavelengths λ can be comparable with or even less than the characteristiclength of current inhomogeneity L . In the long-wave limit λ (cid:29) L , the scattering coeffi-cients are expressed through the simple algebraic formulae which are in agreement with theformulae derived in [15] for the case of abrupt change of canal cross-section.The most important property of scattering processes in sub-, super-, and acceleratingtrans-critical currents is that the wave energy flux conserves, J = const, (see Eq. (6) andthe text below Eq. (8)). This law provides a highly convenient and physically transparentbasis for the analysis of wave scattering.In the simplest case of sub-critical currents ( U ( x ) < c , or V ( ξ ) < | R | + K T = 1 (seeEqs. (34) and (33) for the wave running from the left, and Eqs. (45) and (44) for the waverunning from the right).In super-critical currents ( V ( ξ ) >
1) there are positive- and negative-energy waves bothpropagating downstream but carrying energy fluxes of opposite signs. Propagating throughthe inhomogeneity domain ξ < ξ < ξ , they transform into each other in a such way thatthe energy flux of each wave grows in absolute value to the greater extent the greater thevelocity ratio is. As a result, at ξ > ξ the energy flux of each transmitted wave can becomegreater (in absolute value) than that of the incident wave (see Fig. 8). Quantitativelythe increase of wave-energy fluxes can be easily estimated in the low-frequency limit usingEqs. (65) and (73).The scattering process in accelerating trans-critical currents ( V < < V ) looks likea hybrid with respect to those in sub- and super-critical currents. The incident wave canbe the only co-current propagating wave of positive energy. Initially, at ξ < ξ , its energyflux (at unit amplitude) J = 2ˆ ω/V , but in the domain ξ < ξ < ξ = 1 its energy flux is only J (1 − | R | ) (see Eq. (102)). Further, in the super-critical domain 1 < ξ < ξ , it generates1a negative-energy wave, and the energy fluxes of both waves grow in absolute value to thegreater extent the greater V is. And again this process can be better understand in thelow-frequency limit by means of Eqs. (101).The most interesting scattering processes takes place in decelerating trans-critical currents( V > > V ) where B-waves (which are either counter-current propagating positive-energywaves or downstream propagating negative energy waves, see Sec. III) run to the criticalpoint (where V ( ξ ) = 1) and become highly-oscillating in its vicinity. For this reason we areforced to give up the model of ideal fluid and to take into account an infinitesimal viscosityin the neighborhood of the critical point. As a result, the energy flux continues to conserveon the left and right of the critical point, but changes in its small vicinity. Let us illuminatethe details of this phenomenon.Consider first an incident positive-energy F-wave arriving from the left. In the transientdomain ξ < ξ < − V is). As follows fromthe qualitative consideration on the basis of JWKB method (see Section III) and from exactanalytical solutions (see Subsection IV D), near the critical point the B-wave becomes highlyoscillating in space. This causes its absorbtion due to viscosity; as will be shown below, theabsorbtion is proportional to ν/λ . In contrast to that, the wavelength of the co-currentpropagating F-wave does not change significantly in the process of transition through thecritical point (see Section III), therefore the effect of viscosity onto this wave is negligible.After transition this wave runs through the non-unform subcritical domain − < ξ < ξ andpartially transforms into another B-wave – a counter-current propagating wave of positiveenergy. This wave approaching the critical point also becomes highly oscillating and thereforeabsorbs in the vicinity of that point. The energy flux of transmitted F-wave decreasesproportional to V . The total change of energy flux in transition from the incident to thetransmitted wave is described by Eqs. (128), and in the limit ˆ ω → V V .If the incident wave arriving from minus infinity is the B-wave of negative energy carryinga negative energy flux (see J in Eq. (134)), then in the transient zone, ξ < ξ < −
1, itgenerates due to scattering on inhomogeneous current the F-wave of positive energy, so thatthe wave fluxes of both waves grow in absolute value. The B-wave absorbs due to viscosity2in the vicinity of the critical point ξ = −
1, whereas the F-wave freely passes through thispoint with an insignificant change of its wavelength. After passing through the critical pointthe F-wave generates in the domain − < ξ < ξ a new B-wave of positive energy, whichpropagates a counter-current towards the critical point and absorbs in its vicinity due toviscosity. Therefore, the energy flux of the F-wave increases first from zero at ξ = ξ up tosome maximal value at ξ = −
1, then it decreases due to transformation of wave energy intothe B-wave to some value at ξ = ξ (see J in Eq. (134)), and then remains constant. In thelong-wave approximation, ˆ ω →
0, we obtain∆ J ≡ | J | − | J | = (cid:18) V V | T | − (cid:19) | J | = V | J | V (1 + V ) (cid:20) V − V (cid:18) V + 1 V (cid:19) + 1 (cid:21) . Analysis of this expression shows that because V + 1 /V ≥
2, then ∆ J can be positive(i.e., the energy flux of transmitted wave can be greater than the energy flux of incidentwave by absolute value), if V > √ ≈ . ξ > ξ turns to be equal to the total energy flux of two incident waves at ξ < ξ ,and in such a very particular case the energy flux conserves.Finally, if an incident B-wave of positive energy arrives from plus infinity, then in theinhomogeneous zone, − < ξ < ξ , it generates a co-current propagating F-wave of positiveenergy. The energy fluxes of both these waves have opposite signs and decrease in absolutevalue as one approaches the critical point. In the critical point the energy flux of F-wavevanishes, and the remainder of the B-wave absorbs. In this case the less the V the higher the3reflection coefficient | R | is, and this is especially clear in the low-frequency approximation,see Eq. (140).The analysis presented above is based on the fact that the wavelengths of scattered wavesdrastically decrease in the vicinity of a critical point, where V ( ξ ) = 1. In such case eitherthe dispersion, or dissipation, or both these effects may enter into play. We will show herethat at certain situations the viscosity can predominate over the dispersion. Consideringthe harmonic solution ∼ e i κξ of Eq. (82) in the vicinity of a critical point and neglecting theterm ∼ V (cid:48) , we obtain the dispersion relation extending (11). In the dimensional form it is:( ω − kU ) = c k − i νk ( ω − kU ) . (142)The solution to this equation for small a viscosity νk (cid:28) c is ω = | c ± U || k | − i νk / . (143)The viscosity effect becomes significant when the imaginary and real parts of frequencybecome of the same order of magnitude. This gives | k | ∼ | c ± U | /ν . Multiplying both sidesof this relationship by h , we obtain | k | h ∼ h | c ± U | /ν . For the counter-current propagatingB-wave | c − U | →
0, therefore the product | k | h can be small despite of smallness on ν .So, the condition | k | h ∼ h | c − U | /ν (cid:28) | k | h ∼ h ( c + U ) /ν (cid:29) | R | ∼ T n ∼ ˆ ω − . This can be presented in terms of the Hawkingtemperature T H = (1 / π )( dU/dx ) (see, e.g., [3, 10]) and dimensional frequency ω as | R | ∼ T n ∼ πT H /ω . ACKNOWLEDGMENTS
This work was initiated when one of the authors (Y.S.) was the invited Visiting Professorat the Institut Pprime, Universit´e de Poitiers in August–October, 2016. Y.S. is very gratefulto the University and Region Poitou-Charentes for the invitation and financial supportduring his visit. Y.S. acknowledges also the funding of this study from the State taskprogramme in the sphere of scientific activity of the Ministry of Education and Science ofthe Russian Federation (Project No. 5.1246.2017/4.6). The research of A.E. was supportedby the Australian Government Research Training Program Scholarship.
Appendix A: Energy Flux Conservation
Let us multiply equation (5) by the complex-conjugate function ϕ and subtract from theresult complex-conjugate equation: ϕ (cid:18) ∂∂t + U ∂∂x (cid:19) (cid:18) ∂ϕ∂t + U ∂ϕ∂x (cid:19) − ϕ (cid:18) ∂∂t + U ∂∂x (cid:19) (cid:18) ∂ϕ∂t + U ∂ϕ∂x (cid:19) = c U (cid:20) ϕ ∂∂x (cid:18) U ∂ϕ∂x (cid:19) − ϕ ∂∂x (cid:18) U ∂ϕ∂x (cid:19)(cid:21) . (A1)Dividing this equation by U and rearranging the terms we present this equation in theform: ∂∂t (cid:20) ϕU (cid:18) ∂ϕ∂t + U ∂ϕ∂x (cid:19) − ϕU (cid:18) ∂ϕ∂t + U ∂ϕ∂x (cid:19)(cid:21) + ∂∂x (cid:20) ϕ (cid:18) ∂ϕ∂t + U ∂ϕ∂x (cid:19) − ϕ (cid:18) ∂ϕ∂t + U ∂ϕ∂x (cid:19) − c U (cid:18) ϕ ∂ϕ∂x − ϕ ∂ϕ∂x (cid:19)(cid:21) = 0 . (A2)5If we denote E = i U (cid:20) ¯ ϕ (cid:18) ∂ϕ∂t + U ∂ϕ∂x (cid:19) − ϕ (cid:18) ∂ ¯ ϕ∂t + U ∂ ¯ ϕ∂x (cid:19)(cid:21) , (A3) J = E U − i c U (cid:18) ¯ ϕ ∂ϕ∂x − ϕ ∂ ¯ ϕ∂x (cid:19) , (A4)then Eq. (A2) can be presented in the form of the conservation law ∂ E ∂t + ∂J∂x = 0 , (A5)For the waves harmonic in time, ϕ = Φ( x ) e − iωt , both E and J do not depend on time,and Eq. (A5) reduces to J = const. Substituting in Eq. (A4) written in the dimensionlessform solution (16) for ξ < ξ and solution (18) for ξ > ξ , after simple manipulations weobtain J = 2ˆ ωV (cid:0) − | R | (cid:1) , ξ < ξ ; (A6) J = 2ˆ ωV | T | , ξ > ξ . (A7)Equating J calculated in Eqs. (A6) and (A7), we obtain the relationship between thetransformation coefficients presented in Eq. (34).Using then solution (17) for ξ < ξ < ξ , we obtain J = 2ˆ ω (cid:12)(cid:12) B w ( ξ ) + B w ( ξ ) (cid:12)(cid:12) − i (1 − ξ ) (cid:8) | B | (cid:2) w (cid:48) ( ξ ) w ( ξ ) − w (cid:48) ( ξ ) w ( ξ ) (cid:3) + | B | (cid:2) w (cid:48) ( ξ ) w ( ξ ) − w (cid:48) ( ξ ) w ( ξ ) (cid:3) + B B (cid:2) w (cid:48) ( ξ ) w ( ξ ) − w ( ξ ) w (cid:48) ( ξ ) (cid:3) − B B (cid:2) w (cid:48) ( ξ ) w ( ξ ) − w ( ξ ) w (cid:48) ( ξ ) (cid:3)(cid:9) = const . (A8)It was confirmed by direct calculations with the solutions (16)–(18) that J is indeedindependent of ξ for given other parameters.In a similar way, for the super-critical accelerating current one can obtain in the inter-mediate interval ξ < ξ < ξ J = 2ˆ ωξ (cid:12)(cid:12) B ˘ w ( ξ − ) + B ˘ w ( ξ − ) (cid:12)(cid:12) −
2i ( ξ − ξ (cid:8) | B | (cid:2) ˘ w (cid:48) ( ξ − ) ˘ w ( ξ − ) − ˘ w (cid:48) ( ξ − ) ˘ w ( ξ − ) (cid:3) + | B | (cid:2) ˘ w (cid:48) ( ξ − ) ˘ w ( ξ − ) − ˘ w (cid:48) ( ξ − ) ˘ w ( ξ − ) (cid:3) + B B (cid:2) ˘ w (cid:48) ( ξ − ) ˘ w ( ξ − ) − ˘ w ( ξ − ) ˘ w (cid:48) ( ξ − ) (cid:3) − B B (cid:2) ˘ w (cid:48) ( ξ − ) ˘ w ( ξ − ) − ˘ w ( ξ − ) ˘ w (cid:48) ( ξ − ) (cid:3)(cid:9) = const . (A9)6Here the coefficients B and B should be taken either from Eqs. (59) and (60) for thescattering of positive energy wave or from Eqs. (68) and (69) for the scattering of negativeenergy wave.The transformation coefficients R and T were derived in this paper in terms of the velocitypotential ϕ . But they can be also presented in terms of elevation of a free surface η . UsingEq. (4) for x < x and definition of ϕ just after that equation we obtain for a wave sinusoidalin space ( ω − k · U ) η = khu = i hk ϕ. (A10)Bearing in mind that according to the dispersion relation ω − k · U = c | k | , we find fromEq. (A10) ϕ = − i c h | k | η = − i c ( c ± U ) hω η, (A11)where sign plus pertains to co-current propagating incident wave and sign minus – to counter-current propagating reflected wave.Similarly for the transmitted wave for x > x we derive ϕ = − i c ( c + U ) hω η. (A12)Substitute expressions (A11) and (A12) for incident, reflected and transmitted wavesinto Eq. (34) and bear in mind that R ≡ ϕ r /ϕ i , T ≡ ϕ t /ϕ i , and ω and c are constantparameters: V (cid:2) (1 + V ) − (1 − V ) | R η | (cid:3) = V (1 + V ) | T η | , (A13)where R η ≡ η r η i = 1 + V − V R, and T η ≡ η t η i = 1 + V V T. (A14)In such form Eq. (34) represents exactly the conservation of energy flux (see [14, 15]). Appendix B: Derivation of Matching Conditions for Equation (8)
To derive the matching conditions in the point ξ , let us present Eq. (8) in two equivalentforms: (cid:20)(cid:18) V − V (cid:19) Φ (cid:21) (cid:48)(cid:48) − (cid:20)(cid:18) V (cid:19) V (cid:48) Φ (cid:21) (cid:48) − ω Φ (cid:48) − ˆ ω V Φ = 0 . (B1) (cid:20)(cid:18) V − V (cid:19) Φ (cid:48) (cid:21) (cid:48) − ω Φ (cid:48) − ˆ ω V Φ = 0 , (B2)7Let us multiply now Eq. (B2) by ζ = ξ − ξ and integrate it by parts with respect to ζ from − ε to ε : (cid:26) ζ (cid:20)(cid:18) V − V (cid:19) Φ (cid:48) − ω Φ (cid:21) − (cid:18) V − V (cid:19) Φ (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ε − ε + ε (cid:90) − ε (cid:20)(cid:18) V (cid:19) V (cid:48) + 2 i ˆ ω − ˆ ω V ζ (cid:21) Φ dζ = 0 . (B3)In accordance with our assumption about the velocity, function V ( ζ ) is piece-linear, andits derivative is piece-constant. Assuming that function Φ( ζ ) is limited on the entire ζ -axis, | Φ | ≤ M , where M < ∞ is a constant, we see that the integral term vanishes when ε → ζ in front of the curly brackets { . . . } , also vanishes when ε →
0, and we have (cid:20)(cid:18) V − V (cid:19) Φ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ε − ε = 0 . (B4)This implies that Φ( ζ ) is a continuous function in the point ζ = ζ .If we integrate then Eq. (B1) with respect to ζ in the same limits as above, we obtain: (cid:20)(cid:18) V − V (cid:19) Φ (cid:48) − ω Φ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ε − ε − ˆ ω ε (cid:90) − ε Φ( ζ ) V ( ζ ) dζ = 0 . (B5)Under the same assumptions about functions V ( ζ ) and Φ( ζ ), the integral term herevanishes when ε → (cid:20)(cid:18) V − V (cid:19) Φ (cid:48) − ω Φ (cid:21) − ε = (cid:20)(cid:18) V − V (cid:19) Φ (cid:48) − ω Φ (cid:21) ε . (B6)Due to continuity of functions V ( ζ ) and Φ( ζ ) in the point ζ = ζ , we conclude that thederivative Φ (cid:48) ( ζ ) is a continuous function in this point too. The same matching conditionscan be derived for the point ξ as well. Appendix C: The Transformation Coefficients in the Long-Wave Limit
The long-wave approximation in the dispresionless case considered here corresponds tothe limit ω →
0. In such a case the wavelength of each wave is much greater than the lengthof the transient domain, λ (cid:29) L , so that the current speed transition from the left domain ξ < ξ to the right domain ξ > ξ can be considered as sharp and stepwise. Then using therelationships (see [18]) F ( a, b ; b ; s ) = (1 − s ) − a and s F (1 ,
1; 2; s ) = − ln (1 − s ) , ω the following asymptotic expressions (bearing in mind that ζ = ξ and η = ξ − ): w ( ζ ) = − ln (1 − ζ ) , w (cid:48) ( ζ ) = 11 − ζ , w ( ζ ) = 1 , w (cid:48) ( ζ ) = O (ˆ ω ) , ˜ w ( ζ ) = − ln (1 − ζ ) , ˜ w (cid:48) ( ζ ) = 11 − ζ , ˜ w ( ζ ) = 1 , ˜ w (cid:48) ( ζ ) = O (ˆ ω ) , ˘ w ( η ) = 1 , ˘ w (cid:48) ( η ) = − i ˆ ω − η , ˘ w ( η ) = 1 , ˘ w (cid:48) ( η ) = iˆ ω η , ˆ w ( η ) = 1 , ˆ w (cid:48) ( η ) = i ˆ ω − η , ˆ w ( η ) = 1 , ˆ w (cid:48) ( η ) = − iˆ ω η . Using these formulae, one can readily calculate the limiting values of transformationcoefficients in the long-wave approximation when ˆ ω →
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