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The low lying characteristic (quasinormal) modes of black holes can be evaluated with the help of the semi-analytic method based on a modified WKB approach. We extend the 3rd order WKB formula of Iyer and Will to the 6th order beyond the eikonal approximation and attain thereby an accurate formula for computing QN modes. With the help of the obtained formula we find quasinormal modes corresponding to decay of scalar, dirac, electromagnetic, gravitino, and graviton fields in the Schwarzschild background. For Schwarzschild BH the 6th order WKB lower overtone modes, corresponding to scalar, elctromagnetic and gravitational perturbations, coincide with numerical results already for l 1, while the 3rd order formula does the same at l 4.
R. A. Konoplya (2004) studied this question.