Vortex - wave resonance
G. I. Basina
M. A. Basin
Scientific Research Center “Synergetics” of Saint-Petersburg Association of Scientists and Scholars.
Universitatskaya nab., 5 , Office 300
At the investigation of bodies, nonsymmetrical streamlined by a liquid of variable density we revealed new phenomenon, called by us “vortex - wave resonance”. The essence of this phenomenon consists in resonant non-linear interplay between dispersing waves and demarcations of environments , arising in continuum, and vortical patterns forming near to a body. Resonance interaction results in qualitative change of flow around the body, formation of new vortex - wave patterns and abnormal change of the hydrodynamic forces, acting on a nonsymmetrical body.
The vast literature  is dedicated to research of wave motions in continuum. In spite of that the theoretically analysis of linear models of wave currents in non uniform mediums does not indicate on presence of any resonant interaction of arising wave motions among themselves, however, last decades by a number of the scientists was opened and widely investigated resonant interplay non-linear surface and internal gravitational wave motions. By Î. M. Phillips , Ì. S. Longuet-Higgins, D. Benney, V.E. Zacharov, A. B. Shabbat, L. M.Brekhovskich and his group, G.B. Whitham and other explorers was theoretically predicted , and then experimentally investigated non-linear wave resonance between surface and internal waves . They managed to find out theoretically and to study experimentally conditions of originating of a wave resonance. These conditions, fair both for surface, and for internal waves, were following .
Here -wave vectors and frequencies of two interacting wave motions; - wave vectors and frequencies of synchronous resonant waves. The classic methods of analysis of interaction between the bodies moving in a heavy liquid, with wave systems arising on the free surface, grounded on a linear theory of sources, were designed at the end of past century and intensively developed down to last time, in particular, with reference to research of ship waves  - . The theoretically and experimental investigation of waves which are generated by the motion of a ship at values of Froude numbers, (where - is the length of a ship), close to 0.5, has allowed to find out an interference pattern of interplay of ship shear waves, when the bow wave boosts the aft one and the wave drag of a vessel grows. Thus half of length of a shear wave created by a driving vessel, appears to be equal to the length of a ship. However this phenomenon was not esteemed as resonant. Its description was executed with the linear theory, which results don’t show the resonant interaction. In this case it was possible to speak about a favorable or unfavorable interference of linear waves, as any padding strengthening of intensity of sources modeling a ship in linear theory was not received..
The picture essentially varies at motion near to a free surface or gliding lifting body. Statement and general solution of such problem were executed in a series of classic works, in particular of N. E. Kotchin , M. V. Keldysh and M. A. Lavrentjev  and L. I. Sedov . In spite of a linearization of a problem in relation to surface waves, created by the wing, the obtaining of the unambiguous solution of a problem without padding allowances appears impossible. For its solution the registration of a padding non-linear condition on a wing surface - finiteness of speed in region of an sharp trailing edge is entered. This boundary condition, saving linear dependence from an angle of attack, results to a non-linear feedback between circulation on a wing surface and wave system which is generating at its motion near to a free surface of a liquid.
The numerical calculations of hydrodynamic parameters of foils were executed by the different authors originally only for rather great values of submersions of a wing and limiting cases of large or very small values of a chord-Froude number of a wing [9-12]. It was connected with practical requirements – hydrofoil crafts have the main regime of motion corresponding to large values of a Froude number. Besides the attempts of fulfillment of calculations at small submersions and arbitrary values of Froude numbers were connected with considerable mathematical difficulties, bound with necessity of calculus of composite wave integrals and solution of singular integral equations of a wing theory. The part of these difficulties was overcoming at the end of fiftieth- by the beginning of the sixtieth years, when by T. Nishiyama , Isai , ,A. N. Panchenkov  A. B. Lukashevich ,], etc. in addition to the results of M.V. Keldysh and M. A. Lavrentyev  the separate calculations of hydrodynamic parameters of a wing driving near to a surface of a powerful liquid were executed. The generalizing of these results was shown in the monographs [15,16].
However, the calculations of hydrodynamic parameters of a foil in all range of Froude numbers and relative depths of submersion were not fulfilled. Only at the end of 70-th, begining of the 80-th years the similar calculations were executed in USA  in the process of working out of numerical methods of the solution of non-linear wave problems.
In 1982 author with Ju. I.Faddeev organized conference "Hydrodynamics of a wings driving near to a free surface of a liquid, and planing surfaces ". M. A .Basin , A. Ja. Tkatch , M. A. Makaseev , M. V . Lotfullin , A.N. Lordkipanidze  reported about new results of calculations of hydrodynamic parameters of a wing driving near to a free surface of a heavy liquid. The results of these investigations showed the existence of a range of values of Froude numbers and relative depths of submersion where is observing the abnormal behavior of hydrodynamic parameters of the hydrofoil. Further M. A. Basin, A. N. Lordkipanidze and A. Ja. Tkatch [23-30], improving the computational schemes with usage of the quadrature formulas of Korneytchuk - Mishkevich - Gur-Milner [31-33] have fulfilled the systematic computations within the framework of a linear theory for influencing of a Froude number, relative depth of submersion, form and thickness of the profile, and also relative length of a wing on its integral and distributed hydrodynamic parameters by motion near to a free surface of water. Later similar calculations under the linear and non-linear theory were fulfilled by M. V. Lotfullin, S. I. Filippov, V. G. Stchigunov [34-35]. As have shown the results of calculations for the wing with infinite length, by nearing of a wing to a free surface of water, since , at definite for each of relative depths of submersion chord - Froude number of a wing, which decreases at reduction of a diving depth, the intensive growth of a derivative (where - lift coefficient of a wing, - lift of a wing, - density of a liquid, - wing area projection) is observed. So, for example, at relative depth of submersion = 0.2 and value of a chord-Froude number the linear theory forecasts approximately tenfold growth of . With decreasing of relative depth of submersion the theoretically meaning of the curve () is sharply increasing and even aims to infinity at . It is interesting that the relation from a combined Froude number formed on the area of a liquid above a hydrofoil The position of a maximum of at almost all values practically does not change and corresponds to value of a Froude number. At value the depth of submersion practically does not influence on the hydrofoils lift (all curves of relation as a function from are intercepted practically in one point at value =6.3). Decreasing of hydrofoils relative length results in reduction of resonant maximum values, however, the values of Froude numbers, corresponding to the resonance regime, practically do not change.
With decreasing of a relative depth of submersion the resonant value of is increasing and aims to 1 for very small relative depths. In this limiting case the wing plays a role of bottom and the resonant values of a Froude number come nearer to critical values of this parameter on a shallow water.
The similar resonant changes of lift effect were found out at calculations of relation of a angle of zero lift spotted by curvature of the profile, from a Froude number and relative depth of submersion. If in range of Froude numbers from unit to infinitum for a hydrofoil with thin section as an arc of a circle the angle of zero lift practically does not vary and has negative value, approximately equal in radians, where - the maximum relative arrow of profile, then in range of Froude numbers from 0.4 up to 0.8 is watched sharp growth of angle of zero lift, which one at becomes positive. The theory, thus, forecasts paradoxical result. For a hydrofoil with thin section in the form of circular arc, owes at è to arise not lifting, as usually, but considerable drowning force.
The linear theory forecasts even more exotic resonant effect for wings with a profile having finite thickness. For these wings in a resonant regime also must change the angle of zero lift , which one at minor increase of a Froude number does a zigzag jump from large negative value up to even greater (approximately in 2 times) positive value.
These theoretically results at first, should receive the applicable physical explanation, and, in second, required experimental check. Though the having been available separate experimental data ,  qualitatively confirmed existence of abnormal effects, however, before fulfillment of calculations the systematic experimental investigations of this regime were not conducted.
With the purpose of searching theoretically explanation of the obtained data the calculations of distribution of pressure on a wing surface in this mode were executed and the padding subroutines of calculation of deformation of a free surface of water near to a wing are created. These calculations have shown, that main criteria, determining a condition of maximum idealized value of , is the coincidence of a definite part of length of an attendant wave, reshaped on a free surface, with a chord of a wing. However, as against a case of a symmetrical picture of ship motion, this effect already with the full foundation may be called resonant, while the amplification of a wave pattern, in turn, can multiply circulation current about a wing. If to count, that the considered resonant regime has the wave nature, expedient it seemed to enter some analogy to a non-linear wave resonance.
Created by the author new classifications of waves, vortical and mushroom patterns, and also transport-information systems [27,28, 38, 39, 68, 69,70] has allowed him to esteem motion of a solid body in a liquid as the special form of a non-linear surface solitary wave, which speed is equal to the running speed of a body. Main geometrical parameter of this nonlinear wave is the size of a body in a direction of motion. The main feature of a considered nonlinear wave is, that near to boundary of a solid alongside with speed and pressure fields forms the field of the area of a vorticity, which is generating in a thin boundary layer near to a wing, is reshaped also.
On boundary of a moving solid body always are available, as a minimum, two so-called "stagnation points" (or points of change of the sign of a vorticity). In these points, near to which one the speed of a liquid in the area without vorticity is close to a running speed of a body (wave velocity), the appearance of instability of vortex motion is possible. Near to these zones there is a condition for originating of double helical vortexes ("vortical shockwaves of the first kind ). Firstly the mathematical description of originating of a similar kind of waves on the boundary of a liquid was given by M.A.Basin and N.Ju.Zavadovsky in the work , where was established the analogy between formation of a double helical vortex on boundary of potential flow of a liquid and shockwave, that was a reason of introduction for the detected phenomenon of a title - " vortical shockwaves of the first kind". Due to diffusion of a vorticity the helical vortexes are transformed into concentrated vortical patterns - vortical bubbles. The formation of vortical bubbles characterizes originating particular, not investigated earlier explicitly form of energy loss in wiscous liquid, bound with interplay of a convection and diffusions of a vorticity inside a vortical bubble. In case of unsymmetrical flow around a wing the formation of a vortical bubble near to an sharp trailing edge on a suction surface of a wing results, due to non-linear interplay with the main flow, in appearance of new layers of division-vortex sheets ("vortical shockwaves of the second kind "). They arise due to sticking of vortical boundary layers from pressure and sucking surfaces of the wing.
As a result of sticking there is a separation of a vortical bubble from a body and formation of new vortical pattern called by us " as vortical neutral mushroom pattern ". It consists from an affixed vortical wave, driving together with a wing; separated from it starting vortex (vortical scockwave of the first kind) and thin vortex wake, connecting them, (vortical wave of the second kind) in a two-dimesional case. At flow near the wing of final span the vortex wake is skirted by permanently reshaped tip concentrated vortical bubbles .As a result of this process in coordinate system, bound with a wing, forms flow with circulation, determining from a condition of Chaplygin – Jukovsky - Kutta on the trailing edge.
The set up above scheme is utilized by the author together with I. G. Shaposhnikov for working out of non-linear model of non-steady flow about the wing within the framework of a frictionless liquid .
Earlier one of founders of a wing theory - Lanchester [42-44] proceeded from similar wave submissions about formation of circulation on a wing surface. However, further, due to successes reached in the local analysis of the characteristics of wings, grounded on a hypothesis Chaplygin – Jukovsky - Kutta [45-48] to the wave flow pattern of a wing was given a little attention.
Flow pattern becomes considerably more complicated at motion of a wing near to a free surface of a liquid, which is a new, padding source of wave motions. Usually gravity waves are small and for their analysis is enough the applying of a linear theory results. However, at increase of amplitude and energy of gravity waves in region of their ridges the vortex motion is intensified and the local speed in this zone becomes close to speed of wave motion. Limiting theoretically possible stationary gravity wave on a deep water is the wave of Stokes with (- wave height, - its length), in top of which, the stagnation point will be arised, the speed of a liquid in which is equal to a wave velocity. The further increase of energy of a gravitational surface wave results in appearance in its top of supercritical speeds and formation of vortical shockwaves [49-53].
The destruction of non-linear gravitational surface waves becomes in many respects equivalent to the development of vortical shockwaves at asymmetrical circulating flow around a body. Thus, vortex flow around a body with formation on its surface of circulation current and flow, created by gravity waves, arising on a surface of a liquid, bound can be esteemed from unified positions as non-linear waves, connected with deformation or motion of demarcations of environments. The indicated approach has allowed to identify a solid body, streamlined by a fluid flow, with the solitary surface wave, which speed is equal to the speed of motion, and length is proportional to length of a body,
Then it is possible to define approximately the frequency and wave vector to of a vortical wave, equivalent to driving wing : wave vector-, frequency. After such identification it is easy to determine beforehand, not deciding previously problem, range of Froude numbers, at which it is possible to expect resonant interaction.
The parameters of free waves in a heavy liquid of infinite depth, are connected by known expressions As a condition of appearance of vortex - wave resonance at steady motion of a wing we shall consider equality of speeds of a body and affixed gravity wave, and also commensurability of an affixed wave in region of a body with its length (length of a body should make a definite part of an affixed surge, approximately from 0.5 up to 1). Equating running speeds and lengths of waves of a moving body and free gravity wave in a liquid, we receive an approximated condition of originating a non-linear vortex - wave resonance
If to take into account, that the value should vary within the limits from 1.0 up to 2.0, may be received: . Just in this range of Froude numbers the main resonant phenomena are watched at motion of a wing in range of relative depths. At smaller values of relative depths the wing, being the source of disturbances, simultaneously starts to play a role of local bottom, arranged on a shallow depth under a free surface of water. In this case nature of surface waves, their behavior near to a wing become all closer to behavior of waves on a limiting shallow water, the speed which one in a limit corresponds to value, where - depth of water, in this case - depth of wings submersion. For this reason with decreasing of a depth submersion occurs the decreasing of resonant value of a chord-Froude number of a wing and increase with aiming to unit of resonant value of a Froude number on depth.
The worked out by the writer concept of non-linear vortical shockwaves allows to estimate also, in what cases and for what reasons the linear theory can give outcomes, distinct from outcomes of experimental data, and also to forecast types of patterns, which may be formed as a result of resonant interaction. So, for example, linear theory calculations showed that the height of aft wave may be much higher as by limiting shock wave. Such wave is unstable and on its top must be formed the concentrated vortex (or cascade of concentrated vortexes), which diameter at relative depths, smaller then , is commensurable with a depth of wings submersion. How follows from the analysis of formation of vortical shockwaves on a wing, the circulation can much considerably change only then, when the vortex, formed near to a trailing edge, descends from a wing surface. In a resonant regime there is a mutual amplification of growth of circulation and of a aft wave near the wing. The destruction of the aft wave on a surface of water should result to that fact that the concentrated vortexes which are generating near to a trailing edge of a wing, can not separate from an edge, and the growth of circulation, and also energy of a wave reshaped behind a wing, will be stopped. Thus, the non-linearity at a vortex - wave resonance should result in to that the very large values of , obtained on the basis of a linear theory at small relative depths of submersion, should not be realized. Thus outcome of non-linear interplay can appear new vortical pattern reshaped near to a trailing edge of a wing, including as members a vortical shockwave reshaped on a free surface of water and a trailing vortex, arising near to a trailing edge of a hydrofoil.
The concept of non-linear vortical waves and defined on its basis conditions of vortex - wave resonance appearance allow to forecast values of foils speeds applicable to a resonant flow regime of a wing and in an inhomogeneous heavy liquid (or gas).
For this purpose was found in litrature  the dispersion relation of own wave disturbances for the stratified liquid :
where, - equilibrium distribution of density for the altitude being a continuously decreasing function of an altitude, - speed of sound of non-perturbed flow at pressure and distribation of density (is equal to relation of increment of pressure to increment of density at a constant entropy); - projection of a wave vector to axis. A resonance condition, as well as earlier has the form
At large values of take place two strongly distinguished from each other frequencies:
In the first case a wave velocity is and at low speeds of motion of the body sound waves attendant to body, are not present. In the case of the bodies motion with the velocities, near to ,apparently, at the expense of non-linearity of a flow pattern the minor dispersion of sound waves and appearance of the vortex - wave resonance is possible. However some information on existence of such regime to us is not known.
In the second case the resonance condition results in a ratio
From a condition (3) follows the conclusion: the speed of a body corresponding to resonant flow about the wing, decreases up to zero point at. This predicted theoretically result may have the same principled value, as well as phenomenon a vortex - wave resonance. Pursuant to it with decreasing of a degree of liquids non-uniformity the resonant speed of wings motion aims to zero point. A vortex-wave resonance is theoretically possible even at very small heterogeneity’s of continuum and low speeds of bodies motion.
With the purpose of check of this principled theoretically conclusion within the framework of a linear theory the solution of a problem on motion of a wing near to a demarcation of environments of different densities was reviewed and the calculations of a lift distribution and integrated lift of a wing are executed at different values and relative depths. For this case the dispersion ratio is determined by the formula
And the critical Froude number lies in range
The ratio (5), though differs from (3), however characterizes the same tendency decreasing of a resonance Froude number with reduction of a degree of a non-uniformity of liquid (or gas).
The results of executed calculations have confirmed a prediction for decreasing of resonant values of a Froude number to zero point at reduction of a degree of a non-uniformity of continuum .
. Last years by the writer together with N.V. Kornev was worked out the new vortical method of the solution of three-dimensional Navier-Stokes equations, usage of which one, for our mind, will allow to explain in the future the details of this interesting and composite physical phenomenon.  - 
There were conducted in towing and cavitation tanks the tests of a hydrofoil having rectangular plane form with relative length (- wing span) with flat –circular segment profile of a longitudinal section (thickness ratio) in a broad region of relative depths of submersion and Froude numbers [23-26], [28-30]. The tests were fulfilled at the region of angles of attack . The special attention was given to measurement of hydrodynamic parameters and visualization of a flow pattern in resonant range of Froude numbers and relative depths of submersion. As well as was predicted theoretically, the considerable negative values of lift of a wing were obtained even at positive geometrical angles of attack. At and were received the values As it was predicted on the base of the concept of vortical shockwaves, experimental data for in relation with the resonant values obtained on a linear theory, on the contrary, were strongly delivered: experimental data appeared considerably less then computational results. Besides the analysis of executed photos has confirmed theoretically conclusions about formation in the indicated range of Froude numbers and relative depths of submersion at all investigated angles of attack above a trailing edge of a wing of a breaking up wave with a vortical nozzle. At small angles of attack was observed a nose breaking up wave of another shape- having flat top, boarded by non-steady "beach-comber" The bow wave decreases with increase of an angle of attack and at practically fades. The tendency to disappearance of a bow wave with increase of an angle of attack is confirmed by outcomes of theoretically calculations. Outside of the resonant range of Froude numbers at, as well as is forecast by calculations, there is a quality change of a flow pattern. The top of a aft wave displaces back and practically bow wave at once disappears. With deleting from a wing of affixed aft the value of is increasing sharply. The meaning of passes through zero point and with further increase of speed receives positive values (if). The flow near to a wing becomes smoothly form with small wave deformations on a surface of a liquid. The measured values of hydrofoils lift practically coincide the data obtained with the linear theory. The analysis of experimental data has shown, that outside of a zone a vortex - wave resonance the predictions of a linear theory agree with experimental data. The predictions for a divergence of values between results of a linear theory and experiment grounded on the concept of vortical shockwaves, were qualitatively confirmed. The experiment has shown also existence of new, not studied before a collapse form of a non-linear bow wave. The range of Froude numbers and relative depths of submersion applicable to originating of abnormal hydrodynamic phenomena in experiment practically completely has coincided with the theoretically forecast range. That once again has confirmed main positions on which one the theoretically analysis was plotted.
The detected phenomenon of a vortex - wave resonance can find analogies not only in hydrodynamics of bodies motion in an inhomogeneous liquid or gas, but also in all those cases, when any rigid or deformed system is displaced in a nonuniform medium. Such phenomena can be watched in a meteorology, oceanology, astrophysics, plasma physics, chemistry, biology, different manufacturing processes.
The designed theory and presented concept allow to forecast conditions, at which it is necessary to search similar phenomena. Such condition is the presence of a rigid or deformed body migrating by rather inhomogeneous continuum, in which arising of dispersing waves is possible. For the prediction of conditions of resonant interaction it is enough to know the main dimensions of a body and basic dispersion relations for free wave arising in the medium. The phenomenon of vortex - wave resonance can arise and in homogeneous medium, if during motion in this medium for whatever reasons there can arise the phase changes.
So, phenomenon, which can be referred to a vortex - wave resonance, are detected by us at cavitation flow near the bodies in a transient regime, when length of a cavity is close to a chord of a wing , [64-66, 69], separated flow past of bodies , and another processes.
Coming from foregoing, it is possible to expect that appearance of resonances of this sort is possible under different natural phenomenas, in which is present a unhomogenious utter ambience (field) and moving in it objects or vortex structures, and even to predict the conditions of their origin.
Predicted theoretically and discovered experimentally phenomena of nonlinear vortex-wave resonance contributes fundamental changes in existing beliefs about the behaviour of currents near the bodies, moving in the unhomogenious utter ambience. Knowledge of its nature and main criterions of origin allows to predict theoretically, to find experimental and explain not explored earlier anomalous regimes of the bodies motions in liquids, gas and other continuous ambiences.
Opening of this phenomena stimulated a creation and development of new methods of theoretical and experimental investigations of vortex motions in hydrodynamics of ship and other areas of science and technology , in connected with need of nonlinear phenomena study by arising of vortex and wave motions of continuous media.
Concept of vortex shock waves of first and second type, developed in connection with studying of the opened phenomena, and identification of field of velocities near moving bodies and connected with them traces and cavities with specific soliton nonlinear waves is using at present for the explanation of known, but not studied before the end nonlinear phenomenas, according with motion of bodies in liquids: separating of flow from the bodies, origin and development of circulation currents near the bodies , origin and development of destroying shock waves on the free surface , phenomenas, connected with cavitation of vortexes, flows near the interceptors and holes, study of breakout of air to the hydrofoils.
Discovered phenomena a curl-wave resonance can find analogies not only in hydrodynamics of bodies, moving in the unhomogenious liquid and gas, as well as in all that cases, when some hard or deformed system moves in the unomogenious ambience. Such phenomenas are to be observe in metrology, oceanology, astrophysics, physics of plasma, chemistry , biology , different technological processes. Developping theory allows to predict a main condition, under which follows to search the similar phenomenas.
Such condition is presence of any structure, moving in the unhomogenious utter ambience (field), in which is possible the origin of dispersive nonlinear waves. For the prediction of conditions of resonance interaction is sufficiently to know main gabarit sizes of structure and main dispersion correlation for free waves, appearing in the ambience.
Phenomena of vortex-wave resonance can appear and in the uniform ambience , if in the process of motion in this ambience on one or another reasons are to occur phase transitions (for instance, cavitation).
Emphases deserves that discovered when studying of the flow near the nonsymmetrical bodies theoretical result, that when reducing degrees of unhomoenity of ambience resonance phenomena, nearly not fading, move in the area very small relative velocities of moving bodies. This allows to use an open phenomena as an efficient indicator of small unhomogenities in the ambience and device of management, since at the resonance significant effect can be reached without significant expenseses of energy.
Significance of received results consists not only in discovering and studying of new unknown earlier class of resonanse processes and vortex-wave structures, but as well as in that , that developping theory allows to predict and find unknown earlier forms of vortex -wave interaction , create artificially conditions for arising this phenomena aplicable to practical problems, as well as create new ways and develop new designs, part from which is already use at present in shipbuilding, which are to find an using also in other areas of technology and scientific studies.
Broad field of activity is opened for studying the physical phenomenas also and vortex-wave structures, which can be formed at the interaction of resonanse flow with other types of wave processes, vortex structures and information-transport systems.
Vortex-wave resonance is one of the main mechanisms of origin and stabilizations of new structures - that is to say one of the reasons a structure- and systemforming, for biological objects particularly. So condition of its origin and processes, occur during this type of occasions have a particular interest for qualitative categorizations of fundamental processes in the nature. One of the possible hypothesises of origin lifes on the World is resonance interaction of solenoidal electro-magnetic structure of genome with muschroom structure of the cell
Great role may play the using of vortex-wave resonance for the artifical recognition of images.
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(26) Basin, M.A., Lordkipanidze, A. N., Tkach, A. Ja. (1991), “Vortex-wave resonance in the hydrodynamics of foil, moving near the interface of the different density media”, “Waves and vortices in the ocean and their laboratory analogues”, The Fifth Annual Workshop of the Commission on the Problems of the World Ocean. Vladivostok, September 23-29, p.15-16.
(27) . Basin, M. A. (1992), ”A foundation of classification of non-linear wave motions and vortical patterns. A phenomenon of vortex - wave resonance at motion of asymmetrical bodies in an inhomogeneous liquid”, Problems of hydromechanics in development of ocean, Materials of conference on applied hydromechanics of Kiev “IGM ANU” p.192-193.(in Russian)
(28) . Basin, M. A. (1993), “Wave formation by the motion of a surface ship hydrodynamic complex near the free boundary. Classification of nonlinear waves. Wave-vortex resonance”. Proceedings of the VI Congress of the IMAM. Edited by Peter A. Bogdanov, vol.II, 15-20 November , Varna,. Bulgaria
(29). Basin, M. A.(1995), “Vortex-wave Resonance in Hydrofoil Hydrodynamics”, International Symposium on Ship Hydrodynamics Devoted to 85-th Anniversary of Birthday of Abram M. Basin (ISSH). St-Petersburg: May, p.399-407
(30). Basin, M. A. (1998), “Vortex - wave resonance” , Synergetics and methods of science. Spb.: "Nauka", p.415-418.(in Russian)
(31). Kornejtchuk, A. A. (1964), “The quadrature formulas for singular integrals”, Numerical methods of the differential and integral equations solution and quadrature formulas, v.4, ¹ 4 p.64-74.
(32). Mishkevich, V.G. (1980),“About design of the quadrature formulas for singular integrals , meeting in flow mechanics”. Problems of shipbuilding, A serial: "Designing of vessels" 1980, ¹ 23, p.98-108 (in
(33). (1983), “Propellers. Modern computational methods”,. (Bavin V. F., Zavadovsky N. Ju., Levkovsky Ju. L., Mishkevich V.G, Leningrad: "Sudostrojenie”,1983. 236p.(in Russian)
(34). Lotfullin, M. V., Filippov S.I. (1987), “Streamlining of a profile by a heavy liquid near to demarcations of media”, Mechanics of engineering, Thesises of the reports of II republican technological conferences. Section of mechanics of a liquid, gas and plasma, Bregnev, p. 44. (in Russian)
(35). Stchigunov, V.G. (1995), ”The solution of non-linear non-steady wave problems by a vortical method”. Ph. thesis. SPb.: SPb SMTU.(in Russian).
(38). Basin, M. A. (1993), ”A Wave formation by the motion of surface ship hydrodynamic complex near the free boundary. Classification of nonlinear waves. Wave-vortex resonance”, Papers of IMAM 93 Congress, Ed. by P.A. Bogdanov, vol. II. Varna, Bulgaria, November 15-20. p.51-58.
(39). Basin, M. A. (1998).,“Foundations of classification of non-linear wave motions, vortexes and transport systems”, Synergetics and methods of science,. M.A.Basin –editor. SPb.: "Nauka" p.95-131. (on Russian)
(40). Basin, M. A., Zavadovsky, N. Ju.. (1985), “Model of a double helical vortex as the limiting form of a free surface of non-steady flow of ideal incompressible fluid” .Works of a seminar on boundary value problems. ¹.22.. Kazan: KSU. (on Russian)
(41). Basin, M. A, Shaposhnikov, I. G .(1989), “New model of non-steady flow about a wing in a frictionless liquid”, Mathematical and physical simulation in hydrodynamics of a vessel, Transactions of “NTO SP”, Issue 18, SPb: “Sudostrojenie”, p.27-38.
(42). Lanchester, F.W.(1884), “Stability of an Aerodrome”, The report on the meeting of Bimingem society, July 19.
(43). Lanchester, F. W. (1907), “Aerodynamics”, London.
(44). (1937) ”Aerodynamics”, Under general edition of R.F. Durand,. vol. I. “ONTI-NKTP. USSR”, 436 p.(in Russian)
(45). Joukovski, N. E., (1897), ”About the optimal angle of a slope of the aeroplanes”, An aerostation and research of atmosphere, ¹1. (in Russian)
(46). Kutta, W. M.(1902), “Lift in the flow of liquid”, Aerodynamic reports with illustrations, p.133. (in German)
(47). Joukovski, N. E.(1905-1906) “About adjacent vortexes”. Was reported by the author in the Moscow mathematical society (November 15 1905) and was printed out in “Transactions of branch of physical sciences “OLE”,v. XIII, ¹.2, 1906. (in Russian))
(48). Chaplygin, S. A. (1910), “About pressure of a parallel-plane flow on blocking bodies”, Moscow mathematical collection, ¹ 28. (in Russian)
(49). Longue –Higgins, M. S., Fox, M. J. H.(1977), Theory of the almost -highest wave. The inner solution”, Journal of Fluid Mechanics, vol.80, ð.721-742.
(50). Lighthill, J.(1978), “Waves in fluids”, Cambridge University Press, London - New York - Melbourne.
(51).Yuen, H.C., Lake, B.M. (1982), “Non-linear dynamics of deep water gravity waves”, Advanced and applied mechanics, Vol.22. Edited by Chia –Shun Yih, Academic press, New York-London-Paris- San Diego-San Francisko – Sao Paulo - Sydney-Tokio- Toronto, p.67-229
(52). Amromin, E.L., Basin, M.A., BushkovskyV.A.(1990), “Two solutions of a spatial problem on limiting waves on a surface of a heavy liquid”, Applied mathematics and mechanics, vol.54. N1. (in Russian)
(53). Maklakov, D. V. (1997),”Non-linear problems of hydrodynamics of potential flows with unknown boundaries”. M.:”Janus-K”, 280p.(in Russian)
(54). Basin, M. A. (1984), “About change of vector of a summar vorticity of continuum at motion in it of a solid body and at motion it inside a solid vessel.”, “Proceedings of AN of USSR”, Mechanics of a liquid and gas, ¹ 1(in Russian)
(55). Basin, M. A. (1984), “Change of the moments of a vorticity field of a liquid at motion in it of a solid body”, Perfecting of running, seaworthy and maneuvering capabilities of vessels. Materials on exchange of experience, ¹ 400, Leningrad, "Sudoctrojenie", p.49-54. (in Russian).
(56). Basin, M. A. (1990), ”Âasic equations of vortex fluid motion. Vortex-wave resonance”, IUTAM Symposium on Separated Flows and Jets, Novosibirsk: USSR p.39-41; Springer - Verlag. Berlin -Heidelberg , (V. V. Kozlov, A.V. Dulov editors), p.113-116.
(57). Basin, M. A.(1993), Basic equations of vortex fluid motion. Selected Papers,. Vol.1,. Applied Hydrodynamics, St-Petersburg. State Marine Technical University, p. 23-34.
(58).Basin, M. A., Kornev, N.V., Zacharov, A.B.(1993), “Approximation of three-dimensional vortex fields”. Transactions of Central research institute of a maritime fleet, SPb, .p.184-196. (in Russian).
(59). Basin, M. A., Kornev, N. V. (1994), “Field of vortitity approximation in unbounded media”, “JTF of RAN”, November, SPb, p. 179-185. (on Russian).
(60). Basin, M. A., Kornev, N. V. (1994), “New Computational Method of Vortex Dynamics”, Report on the Euromech Colloquium 315,. Nurnberg – Erlangen, Germany,. March.
(61). Basin, M.A., Kornev, N.V.(1994), ”Approximation of Vorticity Field in an Unbounded Volume”. Tech. Phys. 39 (11) , American Institute of Physics. p.1184-1187.
(62). Basin, M.A., Kornev, N.V.(1995), “Vortex Methods in Hydrodynamics”, International Symposium on Ship Hydrodynamics Devoted to 85-th Anniversary of Birthday of Abram M. Basin (ISSH), .St-Petersburg, May, p.431-450
(63). Basin, M.A., Kornev, N.V.(1998), “Incorporation of the viscosity in the vortex mrthod” ZAMM,78, Issue5, p 335-344. (in German)
(64). Egorov, I. T., Sadovnikov, Ju. M., Isaev, I. I., Basin, M. A. (1971), “Artifical cavitation”, Leningrad : “Sudostrojenie”, 284 p.(in Russian).
(65). Egorov, I. T., Basin, M. A., Sadovnikov, Ju. M., Shallar, A. V. (1972),. “A technique of experimental research of non-steady hydrodynamic parameters of cavitating hydrofoils”, Experimental hydrodynamics of a vessel. Materials on exchange of experience, Issue190, .Leningrad: "Sudostrojenie", p.94-100. (in Russian)
(66). Basin, M. A., Shaposhnikov, I. G., Zilist, L. P.(1994), “Problems, methods and results in hydrofoil cavitation”, Proceedings of the Second International Symposium on Cavitation. April ,Tokyo, Japan, p.99-105.
(67). Basin, M. A., Borisov, R. V., Greengoltz, A. I., Guseev, A. S. Kagan I. S. (1986),“Theoretical and experimental determination of the forces of viscous nature by the vibration of the bodies at the fluid”, Proceedings of XV Ubileum Seminar on the Ship Hydrodynamics, Varna: Bulgaria.
(68) Basin M. A(1999). "Vortex-Wave Resonance" . Proceedings of the First International Conference on Vortex Methods, November 4-5,1999, Kobe, Japan.Pp.303-310.
(69) Basin M.A. (2002) "Computers.Vortexes.Resonanses. Wave Theory of Structures and Systems Interaction. Part 2".SPb.:"Norma"..144p.(in Russian)
(70) Basin M.A. (2002) "Information -wave Theory of Structures and Systems". Physics of Consciousness and Life, Cjsmology and Astrophysics ¹¹1-3.(in Russian) .