Scattering waves off resonant structures, with the waves coupling into and out of the structure at a finite number of locations (`ports'), is an extremely common problem both in theory and in real-world applications. In practice, solving for the scattering properties of a particular complex structure is extremely difficult and, in real-world applications, often impractical. In particular, if the wavelength of the incident wave is short compared to the structure size, and the dynamics of the ray trajectories within the scattering region are chaotic, the scattering properties of the cavity will be extremely sensitive to small perturbations. Thus, mathematical models have been developed which attempt to determine the statistical, rather than specific, properties of such systems. One such model is the Random Coupling Model.

The Random Coupling Model was developed primarily in the frequency domain. In the first part of this dissertation, we explore the implications of the Random Coupling Model in the time domain, with emphasis on the time-domain behavior of the power radiated from a single-port lossless cavity after the cavity has been excited by a short initial external pulse. In particular, we find that for times much larger than the cavity's Heisenberg time (the inverse of the average spacing between cavity resonant frequencies), the power from a single cavity decays as a power law in time, following the decay rate of the ensemble average, but eventually transitions into an exponential decay as a single mode in the cavity dominates the decay. We find that this transition from power-law to exponential decay depends only on the shape of the incident pulse and a normalized time.

In the second part of this dissertation, we extend the Random Coupling Model to include a broader range of situations. Previously, the Random Coupling Model applied only to ensembles of scattering data obtained over a sufficiently large spread in frequency or sufficiently different ensemble of configurations. We find that by using the Poisson Kernel, it is possible to obtain meaningful results applicable to situations which vary much less radically in configuration and frequency. We find that it is possible to obtain universal statistics by redefining the radiation impedance parameter of the previously developed Random Coupling Model to include the average effects of certain classical trajectories within the resonant structure. We test these results numerically and find good agreement between theory and simulation.