Nonpropagating excitations have been studied in 1-D and in 3-D, but not so much in 2-D. So, this thesis concerns 2-D nonpropagating excitations. In particular, we construct nonpropagating excitations confined to an annulus under a time-independent scheme, and then we construct nonpropagating excitations confined to a sector of an annulus under a time-dependent scheme. In the time-dependent scheme, we have our nonpropagating excitation orbit the center of our annulus, demonstrating for the first time a nonpropagating excitation with significant translational motion.In this thesis, we also reproduce the results of a 2009 paper involving 1-D nonpropagating excitations, and we extend the theory in this 2009 paper concerning 1-D nonpropagating excitations to 2-D nonpropagating excitations.