Through the use of both experiments and theoretical modelling, this thesis examines the weakly non-linear dynamics of an internal wave beam becoming unstable to Triadic Resonance Instability (TRI). To date, most theoretical work examines the instability in the context of monochromatic plane waves. In the ocean, however, waves seldom take this form, rather they manifest as beams that span the wavenumber spectra. With an aim of developing our understanding of the role of TRI in oceanic settings, this thesis focuses on how the instability evolves in finite-width internal wave beams. Experiments have been conducted using a new generation of wave maker, featuring a flexible horizontal bottom boundary driven by an array of independently controlled actuators. Using this wavemaker, finite-width internal wave beams of varying amplitude were generated in a linear stratification. Novel experimental results show that when one of these beams becomes unstable via TRI, the approach to a saturated equilibrium state for the triadic waves is not monotonic, rather their amplitudes continue to oscillate without reaching a steady equilibrium. Further diagnostics reveal how the frequencies of the two secondary waves involved in the resonant triad also modulate over time. This behaviour is shown to be a result of the finite spatial extent of the primary beam, which causes cyclic growth and decay of different triadic perturbations. As previously published literature does not consider the triadic energy exchange to be a function of space, it is unable to predict this oscillatory behaviour. Theoretical modelling is developed to capture the essence of the experimental setup. The model uses numerical solutions of a weakly non-linear system in a two-dimensional framework. A detailed study looks at how different wavenumbers and frequencies of the secondary waves affect the development of TRI in a finite-width beam. The results show how the orientation of the secondary waves has a strong influence on the evolution of the instability, as this determines the duration over which the triadic energy exchange can occur. By including multiple possible resonant waves in the system, the results also capture both the amplitude and frequency oscillations exhibited experimentally. This model not only recapitulates the experimental findings, but provides a tool for the community to dissect the underlying dynamics of the instability. Both the experimental and theoretical findings presented in this thesis reveal novel insights into how TRI evolves in finite-width internal wave beams. This work thus provides a key to understanding how this instability mechanism may manifest in oceanic scenarios and its potential role in global ocean circulation.