A Polar Form Rayleigh Resonator with Future Applications as a Rate Gyroscopic

It is shown that a circular acoustic cavity bounded by a set of circular reflector grooves and formed from elastically isotropic material, can support degenerate resonant behaviour for circular wave-fronted Rayleigh waves. The Rayleigh wave response to a time-harmonic, circular line-load possessing both in-plane and out of plane components, located within a circular acoustic cavity is determined. The forced response is expressed in terms of the family of Bessel functions whose order is determined by the cyclic order of the circular line-load. A multiple scales perturbation method is applied to the homogenised equations of motion and boundary conditions. It is shown that significant accumulative reflection of the circular wave-fronted Rayleigh waves from the reflective array can occur and that the axisymmetric design satisfies the conditions for degenerate resonance. Thus, the resonator has potential application as a Coriolis gyroscope. The perturbation/homogenisation method of analysis shows that the zeroth order solution in the array is affected at first order by periodic irregularities at approximately half the wavelength of the incident Rayleigh wave and that the form of the surface is similar in this respect to that of the conventional 1-D Rayleigh resonator. The Q-factor of the resonator is calculated and is expressed in terms of the wavenumber defining the reflective array. It is shown that the Q-factor depends exponentially on the radial length of the reflective array. For the geometry considered a Q-factor of 21000 is shown to be possible.