Triple-wave ensembles in a thin cylindrical shell — страница 6
be unstable one provided the following Lighthill criterion (23) is satisfied. Envelope solitons The experiments described in the paper  arise from an effort to uncover wave systems in solids which exhibit soliton behavior. The thin open-ended nickel cylindrical shell, having the dimensions cm, cm and cm, was made by an electroplating process. An acoustic beam generated by a horn driver was aimed at the shell. The elastic waves generated were flexural waves which propagated in the axial, , and circumferential, , direction. Let and , respectively, be the eigen numbers of the mode. The modes in which is always one and ranges from 6 to 32 were investigated. The only modes which we failed to excite (for unknown reasons) were = 9,10,19. A flexural wave pulse was generated by blasting the shell with an acoustic wave train typically 15 waves long. At any given frequency the displacement would be given by a standing wave component and a traveling wave component. If the pickup transducer is placed at a node in the standing wave its response will be limited to the traveling wave whose amplitude is constant as it propagates. The wave pulse at frequency of 1120 Hz was generated. The measured speed of the clockwise pulse was 23 m/s and that of the counter-clockwise pulse was 26 m/s, which are consistent with the value calculated from the dispersion curve (6) within ten percents. The experimentally observed bending wavetrains were best fitting plots of the theoretical hyperbolic functions, which characterizes the envelope solitons. The drop in amplitude, in 105/69 times, was believed due to attenuation of the wave. The shape was independent of the initial shape of the input pulse envelope. The agreement between the experimental data and the theoretical curve is excellent. Figure 7 displays the dependence of the nonlinearity coefficient and eigen frequencies versus the wave number of the cylindrical shell with the same geometrical dimensions as in the work . Evidently, the envelope solitons in the shell should arise accordingly to the Lighthill criterion (23) in the range of wave numbers =6,7,..,32, as . REFERENCES Bretherton FP (1964), Resonant interactions between waves, J. Fluid Mech., 20, 457-472. Bloembergen K. (1965), Nonlinear optics, New York-Amsterdam. Ablowitz MJ, H Segur (1981), Solitons and the Inverse Scattering Transform, SIAM, Philadelphia. Kubenko VD, Kovalchuk PS, Krasnopolskaya TS (1984), Nonlinear interaction of flexible modes of oscillation in cylindrical shells, Kiev: Naukova dumka publisher (in Russian). Ginsberg JM (1974), Dynamic stability of transverse waves in circular cylindrical shell, Trans. ASME J. Appl. Mech., 41(1), 77-82. Bagdoev AG, Movsisyan LA (1980), Equations of modulation in nonlinear dispersive media and their application to waves in thin bodies, .Izv. AN Arm.SSR, 3, 29-40 (in Russian). Kovriguine DA, Potapov AI (1998), Nonlinear oscillations in a thin ring - I(II), Acta Mechanica, 126, 189-212.