Faraday excitation, hysteresis and wave instability in a narrow rectangular wave tank

Abstract

Experiments are described on surface gravity-capillary waves in a 70 cm × 2.8 cm rectangular tank undergoing small vertical oscillations. The large aspect ratio of the tank permits study of several neighbouring two-dimensional wave modes; their onset, hysteresis and instability. Two water depths, close to one and two centimetres, are investigated in detail. Thresholds are determined for wave onset from a flat surface and the hysteretic return to a flat surface from finite-amplitude waves. Also, conditions at which a pure standing wave becomes unstable to other modes are established. Our finding of hysteresis below the minimum forcing for linear wave-onset suggests that nonlinear forcing and nonlinear damping are both significant, as recently suggested by Miles (J. Fluid Mechanics 248 (1993) 67l.) A theoretical model, incorporating these effects, is analysed and the corresponding hysteresis boundaries established. These are broadly consistent with our experimental results.

Standing wave description of nearly conservative, parametrically driven waves in extended systems

“…The coefficient of the quintic term β 2 > 0, as explained just after Eq. (19), which means that all solutions of (4) and (6) are bounded. The simultaneous presence of cubic and quintic terms in this equation does not require to be near a codimension-two point of the parameter space, but is a consequence of the smallness of dissipation (which of course can be seen as a codimension-two point of the parameter space of a fully dissipative system).…”

“…(18) to obtain Eq. (4), where β 0 , β 1 , β 2 , β 4 , and µ c are given by (19), (20), and (27). The coefficient of the new cubic term is…”

“…Finally, using (19), (20), (41), and (49), we readily obtain the following expressions for the remaining coefficients in Eq. (39)…”

“…Also note that β 3 and β 4 change sign at S = 1/3, while β 2 is always positive, as we already pointed out just after Eq. (19). Unfortunately, the amplitude Eq.…”

Amplitude measurements of Faraday waves

“…Several other authors report amplitude measurements in small aspect ratio experiments at driving frequencies between 5 and 10Hz [30][31][32]. In this situation a considerable mismatch between frequency and wavelength may occur.…”