The method of geodesic deviations has been applied to derive accurate analytic approximations to geodesics in Schwarzschild spacetime. The results are used to construct analytic expressions for the source terms in the ReggeWheeler and ZerilliMoncrief equations, which describe the propagation of gravitational waves emitted by a compact massive object moving in the Schwarzschild background spacetime. The wave equations are solved numerically to provide the asymptotic form of the wave at large distances for a series of non-circular bound orbits with periastron distances up to the ISCO radius, and the power emitted in gravitational waves by the extreme mass-ratio binary system is computed. The results compare well with those of purely numerical approaches.
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