Transient evolution of eigenmodes in dynamic cavities and time-varying media

In this paper, we investigate the transient evolution of the natural modes of dynamic cavities and time-varying media. The mode amplitude is modelled as a damped harmonic oscillator with time-varying coefficients, i.e., a parametric oscillator. An approximate closed-form solution is found in terms of the modified Airy function method. The solution for the Doppler shifted mode spectrum is specialized to a mode-stirred cavity. The time dependence of the coefficients is being related to the physical dimensions and the speed of a rotating perturbation (stirrer) in a deterministic way. Because of the stochastic nature of mode-stirred cavities, the effect of random Doppler shifts are also investigated, leading to a Fokker-Planck equation whose diffusion coefficient shows quadratic dependence on the mode amplitude. The analytical results obtained from this analysis are useful for comparison with other solution techniques, particularly those involving Green functions, and is of interest in continuous-time mode-stirred reverberation chambers, as well as in other fields of physics involving dynamic cavities and random media.