Calculations of normal modes of internal waves on a weakly sheared flow

In the context of a first-order regular perturbation theory, an approximate solution of the Taylor-Goldstein spectral problem is found as a series in terms of internal-wave mode functions of a motionless medium. The ratio of the characteristic shear of flow to the phase velocity of a mode is used as a small parameter. Formulas are presented for the coefficients of the series and for the first-order correction to the phase velocity. On the basis of these results, the second-order correction to the phase velocity is found. Simple examples illustrating the formulas derived are presented.