Planar-flow melt spinning (PFMS) is a single-stage rapid manufacturing technique for producing thin metal sheets or ribbons. During the processing molten metal flows through a nozzle onto a moving substrate where a puddle is formed. This study focuses on the time dependent behavior of the process and looks at the puddle dynamics during the cast. Understanding how the issues of heat transfer, fluid flow and contacting dynamics influence the quality of the cast ribbon are the primary focus of the study. The commercial acceptance of PFMS requires ribbons to be cast with good quality (e.g. uniform thickness). There are a large range scales that are relevant to the ribbon product. Thickness variations occur on the macro-scale over the length of the cast (50 m). There is a steady decrease in ribbon thickness over the length of the cast. There is also a periodic variation on the length scale of the wheel circumference (2-3 m). Steady mass, momentum and energy balances are used to understand these long lengthscale thickness variations. There are also thickness variations that occur on smaller scales, cm to mm. A periodic thickness variation across the width of the ribbon is a common defect observed in our casting, referred to as the cross-wave defect. The molten metal puddle is subject to capillary vibrations. We find that these oscillations correspond to the cross-wave defect. Motivated by the physics of the cross-wave defect, a more generalized problem of the vibrations of coupled capillary surface is identified. The motions of the puddle are similar to the classical problem of the vibration modes of a 'plucked' sphere. In considering the possible modes of vibration for a droplet confined between two plates, we find a novel vibration mode where the center-of-mass oscillates. This model is extended to account for a droplet confined within a tube, with two sections of the droplet extending from each end. Restricting to quasi-steady interface shapes, we find a translational vibration mode which can occur with lower frequency than the traditional Rayleigh modes. The vibration frequencies are compared to experimental values. (Abstract).