Using high temporal and high spatial resolution observations taken by the Atmospheric Imaging Assembly on board the Solar Dynamics Observatory, we present a detailed observational analysis of a high-quality quasi-periodic fast-propagating (QFP) magnetosonic wave that was associated with the eruption of a magnetic flux rope and a GOES C5.0 flare. For the first time, we find that the QFP wave lasted for the entire flare lifetime rather than only during the rising phase of the accompanying flare, as reported in previous studies. In addition, the propagation of the different parts of the wave train showed different kinematics and morphologies. For the southern (northern) part, the speed, duration, and intensity variation are about 875 +/- 29 (1485 +/- 233) km s(-1), 45 (60) minutes, and 4% (2%), and their pronounced periods are 106 +/- 12 and 160 +/- 18 (75 +/- 10 and 120 +/- 16) s, respectively. It is interesting that the northern part of the wave train showed an obvious refraction effect when it passed through a region of strong magnetic field. The result of a periodicity analysis indicates that all of the periods of the QFP wave can be found in the period spectrum of the accompanying flare, suggesting their common physical origin. We propose that the quasi-periodic nonlinear magnetohydrodynamics process in the magnetic reconnection that produces the accompanying flare should be important in exciting a QFP wave, and the different magnetic distributions along different paths can account for the different speeds and morphology evolution of the wave fronts.