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In this study, we start with the Landau-Lifshitz-Gilbert equation to describe the spin waves in magnetic nanowires using the linearization approach. In the long wavelength regime, the spin-wave propagation and attenuation along a uniform magnetic nanowire are first investigated. Then, the study is extended to magnetic nanowires composed of multiple segments. Spin waves are found to be reflected and transmitted at the interfaces. In particular, resonance behavior of the spin-wave transmission can be observed in a three-piece magnetic nanowire structure. The resonance is then characterized in terms of resonance wavelength and linewidth and analyzed based on the interface coupling and the structural and magnetic properties of the nanowires.