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Liquid surface waves in parabolic tanks
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15.In the present paper, we studied the stationary surface waves modes bounded by two conformal parabolas with fixed curvatures. The standard time-harmonic wave problem with a given frequency can be analyzed by following the evolution of the velocity potential. The vector potential can be written as a linear combination of the eigenfunctions of the particular involved geometry (for a detailed description see Ref. 24). In our case the complete set of functions taking into account the symmetry properties of the confocal parabolas are well described by Eqs. (11)–(14) or in the more symmetric case by Eqs. (20) and (21).
16.This procedure had been used in Ref. 11 to measure the wave patterns of water surface and to study the chaos phenomenon in a stadium-shaped cylindrical walls.
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19.In the calculation we used and the parameters given in Ref. 11.
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