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In this dissertation we discuss gravitational waves (GWs) and their neutron star (NS) sources. We begin with a general discussion of the motivation for searching for GWs and the indirect experimental evidence of their existence. Then we discuss the various mechanisms through which NS can emit GWs, paying special attention the r-mode oscillations. Finally we end with discussion of GW detection. In Chapter 2 we describe research into the frequencies of r-mode oscillations. Knowing these frequencies can be useful for guiding and interpreting gravitational wave and electromagnetic observations. The frequencies of slowly rotating, barotropic, and non-magnetic Newtonian stars are well known, but subject to various corrections. After making simple estimates of the relative strengths of these corrections we conclude that relativistic corrections are the most important. For this reason we extend the formalism of K. H. Lockitch, J. L. Friedman, and N. Andersson [Phys. Rev. D 68, 124010 (2003)], who consider relativistic polytropes, to the case of realistic equations of state. This formulation results in perturbation equations which are solved using a spectral method. We find that for realistic equations of state the r-mode frequency ranges from 1.39–1.57 times the spin frequency of the star when the relativistic compactness parameter (M/R) is varied over the astrophysically motivated interval 0.110–0.310. Following a successful r-mode detection our results can help constrain the high density equation of state. In Chapter 3 we present a technical introduction to the data analysis tools used in GW searches. Starting from the plane-wave solutions derived in Chapter 1 we develop the F-statistic used in the matched filtering technique. This technique relies on coherently integrating the GW detector’s data stream with a theoretically modeled wave signal. The statistic is used to test the null hypothesis that the data contains no signal. In this chapter we also discuss how to generate the parameter space of a GW search so as to cover the largest physical range of parameters, while keeping the search computationally feasible. Finally we discuss the timedomain solar system barycentered resampling algorithm as a way to improve to the computational cost of the analysis. In Chapter 4 we discuss a search for GWs from two supernova remnants, G65.7 and G330.2. The searches were conducted using data from the 6th science run of the LIGO detectors. Since the searches were modeled on the Cassiopeia A search paper, Abadie et. al. [Astrophys. J. 722,1504–1513, 2010], we also used the frequency and the first and second derivatives of the frequency as the parameter space of the search. There are two main differences from the previous search. The first is the use of the resampling algorithm, which sped up the calculation of the F-statistic by a factor of 3 and thus allowed for longer stretches of data to be coherently integrated. Being able to integrate more data meant that we could beat the indirect limit on GWs expected from these sources. We…