Mode analysis for a quartz tuning fork coupled to acoustic resonances of fluid in a cylindrical cavity

Quartz tuning forks are precise electromechanical oscillators mass produced in different sizes around one millimeter for the purpose of providing the reference frequency for watches and such. Usually, they are designed to operate at 2¹⁵ = 32768 Hz in vacuum at room temperature. When refrigerated to cryogenic conditions, they may show extremely high Q-values. Immersion of such an oscillator to fluid medium changes its response due to inertial forces and dissipation exerted by the medium. This makes it very useful in studies of pure and mixed helium fluids at low temperatures. When the wavelength of sound in the medium, determined by the frequency of oscillation and the speed of sound, corresponds to typical dimensions in the fluid volume, the oscillator may produce standing acoustic waves, observed as strong anomalies in the oscillator response. This can happen in helium fluids for both first and second sound under various conditions. We study the character of such modes by computational methods for typical fork geometries in a cylindrical volume. Reasonable correspondence with measurements in helium mixtures both below and above 1 K is obtained. This is the regime of vigorous second sound resonances, since the speed of this unusual mode compares nicely with the typical dimensions and frequency of the tuning forks. The nontrivial geometry of the fork in the cylinder makes the problem somewhat challenging for computations.