XXXVII international conference on plasma physics and CF, February 8 – 12, 2010, Zvenigorod.

SHALLOW-WATER WAVES DUE TO CIRCULAR ‘SUPERSONIC’ MOTION OF A DISTURBANCE Chernikov G.P., Snezhkin E.N. RRC ‘Kurchatov Institute’, Moscow, RUSSIA, e-mail: [email protected]

Initially wave generation due to curvilinear supersonic motion of a disturbance came to the attention in practical and theoretical aeronautics because of non-trivial behaviour of the sonic boom fronts from smoothly-turning jet planes. Having passed some distance to the centre of curvature of the trajectory, the ‘inner’ one of the two fronts suffered a sharp kink, which looks like a kind of reflection, and thence went backwards, from the centre [1, 2]. Later these aeronautical data were not confirmed in a more general theoretical study of dispersive media based on Lighthill's kinematic wave method and combined with hydrodynamic experiments on soap films [3, 4]. The confirmation has only been obtained in our laboratory simulations reported here: the shallow-water waves from a small body (artificial disturbance), which is dipped into a thin liquid layer and revolves there in a circular orbit, do behave exactly like the shock fronts from the supersonic planes. The locus of the waves observed (Huygens-Mach fronts) turns out to be excellently described both qualitatively and quantitatively by an equation of a circle involute with the radius of the evolute circle equal to the linear velocity of the (‘sonic’) waves divided by the angular velocity of the disturbance. (Similar descriptions are of frequent occurrence in theoretical studies of radiation from superluminal motion of charges [5, 6]; see also examples of relating works in chemistry [7], mathematics [8] and electrodynamics [9].) The phenomenon under study can play an important role in dynamics of non-straight supersonic flows. If such a disturbance happens on a streamline in a boundary or shear zone of the flow, the wave from it, returning back to the same streamline, can trigger generation of noise or even destabilization of the flow due to the mechanism of wave over- reflection [10, 11]. Processes of the type are shown to be key elements in developing the over-reflection instability of differentially rotating compressible media [12], which is significant for understanding dynamics of astrophysical discs. References [1]. Batdorf S.B., Aeronaut. J. RAeS 76, No 741, 541 (1972). [2]. Meyer F., AIAA J. 11, No. 3, 408 (1973). [3]. Alekseenko S.V., Cherep A.A., Dokl. Ross. Akad. Nauk 327 (3), 306 (1992). [4]. Alekseenko S.V., Cherep A.A., Acta Mech. 105, 143 (1994). [5]. Ardavan H., Proc. R. Soc. London A 424, No. 1866, 113 (1989). [6]. Bolotovskii B.M., Bykov V.P., Sov. Phys. Usp. 33, No. 6, 477 (1990). [7]. Winfree A.T., Science 175, No. 4022, 634 (1972). [8]. Arnold V.I., Huygens and Barrow, Newton and Hooke: pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals. Birkhaüser Verlag, Basel, 1990. [9]. Hannay J.H., Jeffrey M.R., Proc. R. Soc. A 461 3599 (2005). [10]. Ribner H.S., J. Acoust. Soc. Am. 29, 435 (1957). [11]. Miles J.W., J. Acoust. Soc. Am. 29, 226 (1957). [12]. Fridman A.M., Snezhkin E.N., Chernikov G.P., Rylov A.Yu., Titishov K.B., Torgashin Yu.M., Phys. Lett. A 372, 4822 (2008).

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