Modulations of Driven Nonlinear Surface Waves on Water and Liquid Helium-4

摘要

When a trough of water or liquid helium-4 is oscillated vertically at frequency f, waves can form on the surface through the process of parametric resonance at half the frequency of the drive, at f/2. The principle measurement is the liquid height at the end of the trough as a function of time. Nonlinearities determine the response amplitude and give rise to hysteresis. At larger drive amplitudes, the signal at f/2 becomes modulated at frequencies much lower than that of the drive in a variety of ways. One such modulation is smooth with a frequency near one of the low modes of the trough even though it is one of the higher modes that is driven. The surface amplitude is first larger on one end of the trough then larger on the other, oscillating back and forth at the modulation frequency which is of the order of f/20. Several of these frequencies can be present at once; they may be incommensurate (irrationally related) with each other or the drive or they may phase lock. The cause of these modulations are three wave resonant triads formed by a low mode, an intermediate mode, and the driven mode of the trough. Another modulation is characterized by short, sudden drops in amplitudes. These sudden drops are localized in space and propagate back and forth in the trough at nearly the gravity wave velocity. Several drops may be present at once. A third modulation is characterized by extremely deep frequencies--as low as f/2000. It seems to be caused by two adjacent modes, nonlinearly coupled together, slowly exchanging energy back and forth. At still higher drive amplitudes the surface response is chaotic. (Theses)