Wave pattern formation in a fluid annulus with a radially-vibrating inner cylinder

The phenomenon of pattern formation of free-surface waves of a fluid confined in an annulus the inner wall of which vibrates radially, is investigated both theoretically and experimentally. Although the waves are excited by harmonic axisymmetric deformations of the inner shell, depending on the vibration frequency both axisymmetric and non-symmetric wave patterns may arise. Experimental observations have revealed that waves are excited in two different resonance regimes. The first type corresponds to forced resonance, in which axisymmetric patterns are realized with eigenfrequencies equal to the frequency of excitation. The second kind is parametric resonance, in which case the waves are ‘transverse’, with their crests and troughs aligned perpendicular to the vibrating wall. These so-called cross-waves have frequencies equal to half of that of the wavemaker. Both kinds of resonance were investigated theoretically using Lamé's method of superposition. It was shown experimentally that the pure forced resonant standing waves are not realized when the amplitude of excitation is beyond the threshold of parametric resonance for non-symmetric waves. The experimental observations agree very well with the theoretical results.