Amplification of the vibrational motion of bodies in a stratified liquid

The motion in a liquid of a body having a vibrational degree of freedom may be accompanied by instability, namely, an excitation of vibrations through the energy of translational motion [1]. The possibility of such an instability is basically due to the emission of a vibrational component in the region of the ship angle _s, which satisfies the condition cos _s = U_ph/U, where U is the velocity of the translational motion and U_ph is the phase velocity of the wave at the given frequency. The motion of a small sphere (on an elastic spring parallel to the interface of two liquids) was examined in [1], and the importance of taking the viscosity and nonlinear effects into account was noted. Allowance for these factors is necessary in order to draw any realistic conclusions about the thresholds and nature of the aforementioned instabilitv. This question is examined in the present article in the context of a two-dimensional model of the motion of a circular cylinder perpendicular to its generator and parallel to the interface of two liquids of different densities (Fig. 1) under the action of a given force applied to the cylinder through a two-dimensional elastic spring. Such a model provides a greater understanding of the problem and enables the aforementioned questions to be examined analytically. In the present case the moving body is equivalent to a dipole source of the typeq=-Q (U+U cos t) z+H (x-Ut-a sin t) where U_v = a, Q = 2R², so that QU is the dipole moment corresponding to uniform motion of the cylinder. This expression is true for waves with wavelengths much greater than R.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 4, pp. 87–93, July–August, 1984.In conclusion, I wish to thank L. A. Ostrovskii for his valuable comments and discussion of the results.