Radiation-pressure induced frequency shift of liquid capillary bridge transverse modes: A string approximation

Abstract

Liquid capillary bridges in air have been stabilized by placing them at the velocity anti-node of an ultrasonic standing wave. This radiation-pressure induced stabilization depends on an acoustic parameter q that is proportional to the square of the ultrasonic pressure amplitude [M. J. Marr-Lyon et al., Phys. Rev. Lett. 86, 2293–2296 (2001)]. An approximation developed here suggests that an independent measure of q could be obtained by measuring the increase in the natural frequency of the lowest transverse bridge mode. That mode is known as the (1,1) mode. Even for an inviscid bridge, the exact (1,1) mode description is complicated. This complication is avoided by showing that the natural frequency for long round bridges is approximately that of a string having an effective tension CT/2 where C is the bridge circumference and T the surface tension. The additional restoring force (associated with the acoustic radiation pressure) for a bridge at a velocity anti-node is modeled as a uniform elastic support. When q is small, the frequency shift is predicted to be proportional to q. The calculation should also apply to the use of vibrating strings to measure acoustic standing wave amplitudes. [Work supported by NASA.]