Observations on a spatial-resonance phenomenon

Abstract

The phenomenon was observed during experiments in which a beaker containing water was vibrated in one of its bell modes (the inextensional flexural vibrations of the wall). For certain combinations of driving force and frequency, a standing water wave of large amplitude was generated whose peripheral wavenumber might be either zero (i.e. the wave was radially symmetric) or twice that of the bell mode. This relationship between the wavenumbers of the bell mode and water wave, and the fact that the driving frequency was many times that of the water wave, indicated that this was an instance of a general mechanism that has been studied theoretically by Mahony & Smith (1972). For a model situation, allowing for dissipative effects and nonlinear coupling between nearly resonant oscillations at greatly differing frequencies, they derived a relation-ship between the driving force and frequency representing conditions of neutrals tability (i.e. such that the rate of energy transfer from the high frequency to the low frequency oscillations is zero). The aim of the experimental observations reported here was to check this relationship and other predictions of their theory.

Nonlinear low-frequency gravity waves in a water-filled cylindrical vessel subjected to high-frequency excitations

“…The developed model therefore gives quantitative prediction of the critical force-frequency curve other than simple qualitative predictions; -this study demonstrated that parametric nonlinearity exists in the system, which extends the instability region to excitation frequencies equal or below the natural frequency with sufficient large excitation amplitudes, as verified by experimental observations. However, previous researches by Huntley [4] and Mahony & Smith [5] predicted that excitation force should be infinite when excitation frequency is equal to or below the natural frequency; and -this paper successfully predicted the transition between different gravity waves, which agrees with experimental observations.…”

“…The mechanism of this type of subharmonic resonance in nonlinear dynamical systems has been well investigated and modelled (for example, the book by Guckenheimer & Holmes [3]). Therefore, we do not intend to discuss this type of phenomena in this short paper but we try to address the mechanism of the phenomenon described by Huntley [4], who discovered another type of large amplitude gravity waves of lower frequencies, produced in a half water-filled cylindrical beaker subjected to a more high-frequency horizontal excitation. When a water-filled beaker is subjected to an excitation of frequency near to one of natural bell modes of the structure, large amplitude standing water waves may be generated.…”

Nonlinear low frequency water waves in a cylindrical shell subjected to high frequency excitations – Part II: Theoretical analysis

“…They studied a simple 2-dimensional (2-D) rectangular enclosure half-filled with water and then driven as an acoustic resonance tube. Considering nonlinear aerial and water wave interactions, they developed an equation governing the critic acoustic pressure amplitude with respect to excitation frequency when this phenomenon occurs, which was validated by experimental results of Huntley [1] and Franklin et al [5]. Huntley [6] extends Mahony and Smith's model to real three-dimensional (3-D) cases.…”

“…Large amplitude low frequency gravity water waves generated by a high frequency excitation at a water-filled beaker is firstly discovered by Huntley [1]. Wang et al observed the same phenomenon when doing an experimental research on Dragon Washbasin [2].…”

Experimental studies of surface waves inside a cylindrical container

“…Miles & Henderson (1990) revisited the phenomenon in detail. Sidewall forcing induced free-surface oscillations were experimentally observed in a tank with a free fluid surface and described as a spatial-resonance phenomenon (Huntley 1972;Mahony & Smith 1972). Nonlinear wave interactions can also occur via non-resonant mode interaction between the short and long waves.…”