On Gravity Waves on the Surface of Tangential Discontinuity

Transcription

1 Alied Physics Research; Vol. 6, No. ; 4 ISSN E-ISSN Published by anadian enter of Science and Education On Gravity Waves on the Surface of Tangential Discontinuity V. G. Kirtskhalia I. Vekua Sukhumi Institute of Physics and Technology (SIPT), Tbilisi, Georgia orresondence: V. G. Kirtskhalia, I. Vekua Sukhumi Institute of Physics and Technology (SIPT), 86, 7, Mindeli St, Tbilisi, Georgia. v.kirtskhalia@gmail.com Received: January 3, 4 Acceted: February 8, 4 Online Published: March 3, 4 doi:.5539/ar.v6n9 URL: htt://dx.doi.org/.5539/ar.v6n9 Abstract On the basis of critical analysis of literature it is shown that the existing theory of surface gravity waves is incorrect and contradictory. Based on the new results ublished by the author disersive equation for linear waves generated on the surface of tangential discontinuity between air and water was obtained. It is demonstrated that this equation is alicable only to caillary waves and effect of gravitational filed can be neglected in it. Thus, it is imossible to seak about caillary-gravitational waves in linear theory and consequently there is no condition restricting length of caillary wave. ontrastingly to the wide-sread oinion according to which caillary waves are generated only in dee water, it is demonstrated that they can be generated in shallow water as well where the hase seed of wave deends on the deth of water reservoir. Keywords: tangential discontinuity, surface gravity waves, caillary-gravity waves, weakly inhomogeneous medium, strongly inhomogeneous medium, otential flow. Introduction Problem of generation and roagation of waves on the surface of division of two media are solved with the hel of the theory of hydrodynamic tangential discontinuity. The roblem of gravitational waves on the surface of water reservoirs is one of the most vital among the roblems of this tye. As is known two limit cases are considered for gravity waves: () Shortwave disturbances when the wave length λ is much smaller than the fluid deth h ( kh, where k are called wave number). In this case, the water is considered infinitely dee and the influence of surface tension of fluid is taken into account. Waves generated in such conditions on the water surface are called dee water waves or caillary-gravity waves. () Long wave disturbances when the wave length λ is much greater than the fluid deth h ( kh ). In this case the surface tension influence is ignored and generated waves are called waves on shallow water. According to the existing theory, the sectrum of frequencies of caillary-gravity waves is calculated by the formula k k g thkh where α is coefficient of surface tension of water, ρ is its density while g is acceleration of gravity. If the requirement is met g / / (.) k (.) caillarity effect can be neglected. Taking into consideration that for water α =.73 N/m and ρ = 3 kg/m 3, it is easy to calculate that the wave, length of which is λ >.73 cm, is urely gravitational, frequency of which in dee water (th(kh) ) is kg (.3) In the second case, taking into consideration that α = and th(kh) kh, we will have k gh (.4) 9

2 Alied Physics Research Vol. 6, No. ; 4 These formulae form the basis for a great deal of fundamental researches and are widely used in solution of alied roblems. For examle, hase seed of tsunami waves is calculated according to formula (.4) (Kowalik, ; Ocean Waves, 3; The shallow water wave, ). orrelation of (.), (.3) and (.4) follows from the Kelvin theory who was the first to solve the roblem of surface caillary-gravity waves in 87 (Landau & Lifshitz, 988, See 6). His theory was based on the assumtion of otentiality of fluid motion in the Earth gravity field. On the other hand it is known that fluid motion can be otential only in isentroic medium when entroy is equal in any of its oints, i.e. s const (Landau & Lifshitz, 988, See 8,9). In the Earth gravity filed this condition is violated since entroy deends on vertical coordinate z and thus we assume that these correlations are doubtful and require to be revised. To verify our doubts, in the second section we rovide review of relevant aragrahs of the monograh (Landau & Lifshitz, 988) and on the basis of analysis thereof demonstrate shortcomings of modern theory of hydrodynamic tangential discontinuity. In the third section, based on the generalized equation of gravity waves obtained in linear aroximation in the aer (Kirtskhalia, a) we derive an equation for surface caillary-gravity waves and demonstrate that the linear theory describes only caillary waves. onsequently, non-linear members must be taken into account in the system of hydrodynamic equations when considering gravity waves.. Modern Understanding of Surface Gravity Waves Intensive studies of mechanisms of generation and roagation of gravity waves on the surface of water reservoirs began after L. Euler had formulated his fluid flow equation in 755. Since then this issue has been treated in a great number of scientific works analysis of which is beyond the scoe of this survey. We will refectory the reader mainly to the monograh of (Landau & Lifshitz, 988) and bring only the most tyical examles reflecting the shortcomings related to solution of this roblem and showing that modern theory of gravity waves is incorrect and needs to be revised. Dynamic rocesses occurring in fluids (gases) are described by the system of hydrodynamic equations which includes: The flow equation (Euler s equation) V P V V g (.) t The continuity equation divv (.) t The adiabatic equation s V s, (.3) t where is ressure, V is velocity and s is entroy. Exact solution of this system in general case is imossible, so scientists have to use aroximate method of small disturbances when variables included in the system of Equations (.-.3) have the form of the sum of their stationary and disturbed values f r, t f r f r, t, where f f and ~ f ( r, t) = f z exi kx t (.4) When using this method, the system (.-3) should be comlemented with two equations: The equation of equilibrium in the gravity field of the Earth P g (.5) The medium state equation (.6) where denotes the sound roagation velocity in a medium defined by means of its termodynamic characteristics. This method for solution of the roblem of stability of tangential discontinuity on the lane interface of two

3 Alied Physics Research Vol. 6, No. ; 4 semi-bounded incomressible = fluids (waves in dee water) without taking into account the influence of surface tension force and gravitation was first used by Helmholtz in 868. He showed that the solutions of disersion equation are two self-conjugate comlex numbers (Landau & Lifshitz, 988, See 9) i kv (.7) where V -is seed of relative motion of fluids along lane surface of discontinuity. For the root with a ositive imaginary art Im, the wave amlitude increases with time, which means that the tangential discontinuity under consideration is absolutely unstable. Doubtful of solution of (.7) is obvious, since from it follows that distribution of wave on the interface of two incomressible fluids at rest V is imossible. In 944 Landau solved the roblem of Helmholtz for comressible fluids with similar densities and discovered that such tangential discontinuity is stable when V 3/ (Landau & Lifshitz, 988, See 84) and consequently, in comressible fluids tangential discontinuity is stable only in case of suersonic flows. This result contradicts to the Helmholtz result, according to which arbitrarily small seed of relative motion of fluids leads to instability of tangential discontinuity. Doubtfulness of its result was admitted by Landau himself (Landau, 969). In the aer (Kirtskhalia, 994) mistake was found in the method of solution of relevant disersive equation and it was shown that such tangential discontinuity is also absolutely unstable. The Helmholtz roblem with account of forces of gravity and surface tension was solved by Kelvin in 87 (Landau & Lifshitz, 988, See 6). Assuming the flow to be otential, Kelvin introduced the velocity otential ( r, t) satisfying the Lalace equation r, t (.8) and related to the flow velocity of incomressible fluid by equation V (.9) The fluid ressure is defined by integration of the Euler s equation with alication of (.9) as well as condition of motion otentiality rotv (Landau & Lifshitz, 988, See 9): V P gz (.) t where z is the coordinate along the axis normal to the surface of discontinuity. Assuming that the uer fluid moves with resect to the lower fluid along axis X with velocity V. The velocity otentials satisfying Equation (.8) are written in the form Ae kz cos( kx t) V x, z (.) Ae kz cos( kx t), z (.) Here it is acceted that the equation of discontinuity surface is z =. The difference in the exonent signs is exlained by the requirement of decay of disturbances as the distance from discontinuity surface increases (surface waves). The constants А and А are assumed to be small values since they corresond to the disturbed values of the otentials. At the discontinuity surface the following conditions must be fulfilled: v ( z P P ) z z d V dt t x (.3) x (.4) vz (.5) z t where ξ is dislacement of discontinuity surface along the Z-axis; ν z and ν z are the z-comonents of the erturbed velocity values in areas z > and z < resectively. ondition (.3) with consideration of (.) is written as follows t x t g g V V (.6)

4 Alied Physics Research Vol. 6, No. ; 4 where V =. Reresenting ξ as a sin( kx t) (.7) and substituting (.), (.) and (.7) into the boundary conditions (.4), (.5) and (.6) Kelvin obtained a system of three linear homogeneous algebraic equations with resect to the coefficients А, А and a (the terms containing A and A are neglected). Equating the determinant of this system to zero, he finds the disersion equation whose solution is 3 kv kg k V k (.8) For α =, g =, Kelvin s solution (.8) transforms to Helmholtz solution (.7), and when V = and α = gives the disersion Equation (.3). From (.8) it follows that the stability condition of caillary-gravity waves on water surface is the non-negativity of the exression in the square brackets, i.e. k V g( ) k (.9) ( ) By solving inequality (.9) with resect to k we find that the negativity of its discriminant gives condition of stability of tangential discontinuity for any k in the form V g (.) On the other hand, having solved (.9) with resect to V, we find the stability condition in the form V g( ) k ( ) (.) k It can be easily demonstrated that minimum value of the right art of inequality (.) is reached when k k g and equals to the right art of inequality (.), however these conditions contradict to each other. Indeed, from (.) it follows that if V the tangential discontinuity can be stable only in resence of both factors the surface tension and the gravity field while according to (.) each of the arameters g and α makes its own contribution to the stability of the tangential discontinuity as it may be stable if one of them is absent. These contradictions which were first highlighted in the work (Kirtshalia & Rukhadze, 8) are quite sufficient to get sure that the Kelvin theory is erroneous; however correlation (.) is considered classic u to now. The basic reason of erroneousness of the Kelvin theory is violation of condition of otentiality of fluid motion, according to which motion can be otential only in isentroic medium. Indeed, in the Earth gravity filed z comonent of disturbed velocity must deend on gravity acceleration g, which must be taken into account in otentials of velocities (.) and (.). However, in this case the condition of otentiality of fluid motion ν x / z = ν z / x is imossible to be met and consequently, the roblem in such setting is unsolvable. That is why the constant term V is artificially introduced in the right side of the Equation (.6). The other threshold case long gravity waves (waves on shallow water) is considered in monograh (Landau & Lifshitz, 988, See ). The authors solve the roblem of wave roagation on water surface in gravity filed of Earth along the canal (along the axis X) with deth h and width b when V =. Having alied the system of Equations (.) and (.) the authors obtain hase seed of wave on the shallow water in the form U gh (.) k which coincides with the exression (.4). The Equation (.3) is not used at all, assuming that it is satisfied identically. As is shown in the aers (Kirtskhalia, a; b). this is the fundamental error leading to incorrect solution of the roblem on the whole. Indeed, the authors assume that ν z is so small, that ν z / t =. On this assumtion, they write down the x and z comonents of Euler s Equation (.) in the form v x (.3) t x g (.4) z

5 Alied Physics Research Vol. 6, No. ; 4 Thereafter they are integrated the Equation (.4) subject to z (.5) and find g z (.6) Without going into further details, the said is quite sufficient to get sure that the ultimate result (.) here is also obtained by means of incorrect solution of the roblem: Firstly the Equation (.4) is nothing else but the fluid equilibrium condition in the Earth gravity field (.5), which is fair only for stationary values of ressure and density and consequently, subject to fulfillment of this condition oscillations are imossible. Secondly from condition (.5) it follows that ressure is constant on the disturbed liquid surface and thus it is unclear how the wave is roagated in this case. Thirdly- the requirement ν z / t = equals to the requirement ν z = meaning that liquid surface is not dislaced along the axis Z. Notwithstanding this the authors mark this dislacement through ξ and obtain wave equation for it. Solution of the task No. (Landau & Lifshitz, 988, See ) seems to be the most correct. Assuming that α = and V =, and considering motion to be otential, the authors find the disersive equation for gravity waves on interface of two liquids, density and deth of which equal to ρ, h and ρ, h resectively. Disersive equation is obtained in the form kg( ) (.7) cth( kh ) cth( kh ) Three cases are considered:. Both liquids are infinitesimally dee (kh >> and kh >> ) and then the Equation (.7) gives kg (.8). Deth of both liquids is small (kh << and kh << ), and then we have g( ) hh k (.9) h h 3. The lower liquid is shallow while the uer liquid is dee (kh << and kh >> ) which results in k gh (.3) For ρ >> ρ, (.8) and (.3) give the results (.3) and (.4), resectively. Mathematical correctness of solution of this task is conditioned by absence of relative motion of liquids (V = ). However, incorrectness related to hysics remains in force and consequently it can be claimed that solution (.7) is erroneous. As was mentioned above, this roblem is dealt with in a great number of scientific works which give the analogous results and therefore it is senseless to consider them here. 3. aillary Waves In the aer (Kirtskhalia, a) it was demonstrated that the known system of hydrodynamic (gas dynamic) equations is fair only for homogeneous media which do not exist in the nature. This system needs to be generalized for real inhomogeneous media which qualitatively changes a number of existing notions on dynamic rocesses occurring in them (Kirtskhalia, 3). In the aer (Kirtskhalia, a) a generalized equation of gravity wave in inhomogeneous medium is obtained in linear aroximation, which is given by g g g (3.) t Here 3

6 Alied Physics Research Vol. 6, No. ; 4 (3.) is squared seed of sound in the medium which is reduced from squares of adiabatic s and isobaric of sound seeds (Kirtskhalia, a, b). Existence of isobaric seed of sound associated with roagation of isobaric erturbation of density leads to necessity of generalization of the equation of mass discontinuity for inhomogeneous medium. Detailed resentation of the rocedure which rovides this equation in the form of d v v v (3.3) dt t is set out in the aer (Kirtskhalia, a). From (3.3) it is obvious that the equation of mass discontinuity acceted in the existing theory is fair only for homogeneous medium when =. onsequently, homogeneous medium cannot be considered incomressible v since in such case ρ = const and generation of mechanic wave in it is imossible. Thus, the Equation (3.3) rovides qualitatively new definition of criteria of comressibility and incomressibility. This issue is extensively considered in the aer (Kirtskhalia, 3), where it is roved that weakly inhomogeneous medium is always comressible while strongly inhomogeneous medium is always incomressible. Absence of such understanding of criteria of comressibility and incomressibility is one of the main reasons of incorrectness of wave theory in gas and hydrodynamic in whole (e.g. internal gravity waves (Kirtskhalia, 3)), and in the theory of surface gravity waves in articular. Let us now consider the correct linear theory of surface waves with consideration of surface tension force and gravity. Assuming that the surface of tangential discontinuity between water and air is lane z =. Having ignored the third summand in Equation (3.) due to its obvious smallness and taking into consideration that, it can be rewritten as s g (3.4) s t Since in the Earth atmoshere at sea level = (Kirtskhalia, a; b), air should be considered comressible ( = s ) and for it the Equation (3.4) will be as follows: Let us resent in the form of s s g (3.5) t s s x z, t ~ zexi kx t, (3.6) following which the Equation (3.5) will be as follows: z z ~ ~ d g d ~ k z (3.7) dt s dz s Solution of the Equation (3.7) with consideration of decay of waves when z (surface wave) is where z Aex kz ~ (3.8) U U k - is the hase seed of wave and s is sound seed in air at sea level. For water the Equation (3.5) will be as follows: k s s, (3.9) (3.) g g (3.) s t From determination of isobaric seed of sound (Kirtskhalia, b) we can easily find 4

7 Alied Physics Research Vol. 6, No. ; 4 / / c c (3.) T T T where β = (/V)( V/ T) is the coefficient of thermal dilatation and c is thermal caacity of unit mass of substance at constant ressure. For water β =.5-4 K -, c = J/kg K and then from formula (3.3) at temerature T = 88 K we will get = 5 m/sec. On the other hand, by of exeriment sound seed in water at the same temerature to a high accuracy equals to = 48 m/sec. Then from formula (3.) we have s / 48.6 m/sec. As we see, in water sound seed ractically equals to adiabatic sound seed i.e. = s. Similar calculations for examle for iron (β = K -, c = J/kg K, T = 88 K, = m/sec, = 53 m/sec) give s = 575 m/sec, and sound seed in iron is also adiabatic (= s ). onsequently, according to the new determination of criteria of comressibility and incomressibility (Kirtskhalia, 3), water and iron should be considered comressible media in terms of thermodynamics and incomressibility condition v must not be alied to them. Moreover, aradoxical as it may seem, water and iron are more comressible than air in uer layers of the atmoshere like, for examle at a height of km, s 3 m/sec (Kirtskhalia, b). Thus, the terms comressibility and incomressibility in thermodynamics characterize not the state of aggregation of matter, but hysical rocess of sound roagation in it. Taking into consideration that = s, solution of Equation (3.5) for amlitude of ressure disturbance in water gives ~ z B ex kz B ex kz (3.3) where U s, (3.4) U (3.5) s k s (3.6) g Thus, for disturbed values of ressure in air and water we will have x, z, t Aexkzexi kx t x z, t B ex kz B ex kz exi kx t (3.7), (3.8) After use of equilibrium condition (.5) and medium state Equation (.6) when = s the linearized Euler s equation will be as follows: dv g (3.9) dt Having reresented in (3.9) all variables in the form (.4), for amlitude of the z comonent of disturbance velocity we will get i d ~ z ~ z v~ z g (3.) z ku dz s From (3.) for air and water resectively we will have v v z z i x z, t k U z d ~ dz d ~ dz ~ s z, s i x z, t k U z ~ z, s g ex g ex i i kx t kx t (3.) (3.) As we see disturbance velocity in area z > as well as in area z < obviously deends on g, which does not 5

8 Alied Physics Research Vol. 6, No. ; 4 occur in Kelvin theory (Landau & Lifshitz, 988). Now, to boundary conditions (.3), (.4), and (.5) when V =, should be added the condition on the bottom of water reservoir and substituting values of relevant quantities from formulae a ikx t v z zh (3.3) ex (.7), (3.7), (3.8), (3.), and (3.) into these conditions, we will get the system of linear homogeneous algebraic equations regarding coefficients A, B, B, and a. Equating the determinant of this equation to zero, we shall find disersive equation of linear surface caillary-gravity waves in the form of U * kex kh ex kh ex kh ex kh (3.4) U * where U * (3.5) s Let us consider condition θ >. Taking into consideration that at sea level s 34 m/sec and g m/sec we shall find k > m - or λ <.7 5 m. Thus, this condition covers the whole range of wavelengths from caillary to tsunami and the bigger is k the better it is met. For instance, when the value of gravity wave length is λ m, for air θ.7 3 >> while for water θ.4 5 >>. It is also aarent that U s U s and then it is easy to check that δ = γ* = and δ =, following which disersive Equation (3.4) is reduced and takes the form k U thkh U (3.6) U As we see, the Equation (3.6) does not contain gravity acceleration g and does not have solution when α =, thus it can be concluded that linear theory of surface caillary-gravity waves is adequate only for caillary waves on which gravity field has no effect. Furthermore it is obvious that when α = and g = the task loses its hysical meaning and therefore the solution of Helmholtz (.7) does not contain any information. Taking into consideration that ρ << ρ = ρ, the solution of the Equation (3.6) shall be k U th(kh) (3.7) k Solution (3.7) invalidates the current oinion that caillary waves are generated only in dee water. We see that they are generated in dee (kh >, th(kh) ) as well as in shallow (kh <, th(kh) kh) water. In the first instance disersive equation is given by k (3.8) k And in the second one h k (3.9) Since condition (.) limiting the length of caillary waves no longer exists, let us consider erturbation with wavelength λ =. m (k = 6.8 m - ) for which from formulae (3.8) and (3.9) we will obtain that in dee water (h.5 m) ω =.3 sec - and U = cm/sec and in shallow water (h =.5 m) ω =.7 sec - and U = cm/sec. As we see, frequency and hase seed of caillary wave dro two times when deth lowers times. 4. onclusion The above critical analysis and calculations conclusively rove that the existing theory of surface caillary-gravity waves is incorrect and contradictory. The reason is an essential error made by Kelvin as far back in 87, excluding assumtion on otentiality of motion of liquids in the gravity field of the Earth. onsequently, roblems of surface gravity waves should be solved only on the basis of system of hydrodynamic equations. However, here again the scientists make significant mistakes, since they seek to obtain results coinciding with the results of Kelvin. They are conditioned by incorrect understanding of criteria of 6

9 Alied Physics Research Vol. 6, No. ; 4 comressibility and incomressibility of medium. The resent aer demonstrates that these contradictions were eliminated after their correct definition had been rovided in the aer (Kirtskhalia, 3). On the basis of this correct definition equations of gravity waves in linear aroximation are obtained for each of these media and by sewing (joining) their solutions on the surface of tangential discontinuity disersive equation is obtained for surface wave. It is shown that the equation is solved only if surface tension of liquid is taken into consideration i.e. it describes only caillary waves on which influence of gravity filed is small to negligible. It was to be exected, since the Equation (3.5), describes acoustic waves (Kirtskhalia, a) and consequently, is suitable only for linear waves amlitudes of which are comarable to amlitudes of acoustic waves. Agreement of result (3.8) with the result of Kelvin (.) when g =, is exlained by the fact that in such instance motion of fluid can be considered otential and Kelvin theory gives correct result. Thus, caillary-gravity waves do not exist in linear theory, since surface wave is always caillary and can be roagated in dee as well as shallow water. When considering gravity waves in the system of hydrodynamic equations nonlinear members must be taken into account. References Kirtskhalia, V. G. (994). On the stability roblem of the tangential discontinuity. Planetary and Sace Science, 4(6), htt://dx.doi.org/.6/3-633(94)83- Kirtskhalia, V. G. (a). Sound wave as a articular case of the Gravitational wave. Oen Journal of Acoustic, (3), 5-. htt://dx.doi.org/.436/oja..33 Kirtskhalia, V. G. (b). Seed of sound in atmoshere of the Earth. Oen Journal of Acoustics, (), htt://dx.doi.org/.436/oja..9 Kirtskhalia, V. G. (3). The new determination of the criteria of comressibility and incomressibility of medium. Journal of Modern Physics, 4(8), htt://dx.doi.org/.436/jm Kirtshalia, V. G., & Rukhadze, A. A. (8). On the Question of Stability of Hidrodinamic Tangential Ga. Georgian International Journal of Science and Technology, (3), Retrieved from htts:// Kowalik, Z. (). Introduction to Numerical Modeling of Tsunami Waves. Institute of Marine Science University of Alaska, Fairbank. Retrieved from htts:// Landau, L. D. (969). ollection of the Works (Vol. ). Moscow: Nauka. Landau, L. D., & Lifshitz, E. N. (988). Theoretical Physics, Hydrodynamics (Vol. 6). Moscow: Nauka. Ocean Waves. (3). Retrieved from htt://hyerhysics.hy-astr.gsu.edu/hbase/waves/watwav.html#c The shallow water wave equation and tsunami roagation (). Retrieved from htt://terrytao.wordress.com//3/3/the-shallow-water-wave-equation-and-tsunami-roagation oyrights oyright for this article is retained by the author(s), with first ublication rights granted to the journal. This is an oen-access article distributed under the terms and conditions of the reative ommons Attribution license (htt://creativecommons.org/licenses/by/3./). 7