By obtaining solutions expressed as power series, eigenvalues and eigenfunctions are determined to high accuracy. In the case I, it is shown that normal modes of gravity waves in a channel are modified by the basic shear flow to become unstable in some discrete ranges of the wavenumber, if Froude number, Fr, is greater than 2. In the case II, it is shown that two types of unstable waves are found for Fr>1: One is similar to the unstable waves in the case I but modified by the presence of the rest fluid. The wave is trapped near the boundary. The other is a wave destabilized by the shear too, but radiates its energy to the unbounded rest fluid. The unstable regions of the waves of this type are continuous in wave number space. The structure of all unstable waves found in this paper are similar to that of gravity waves.

It is shown that the unstable waves extract their energy not from the "ordinary" mean kinetic energy, but from an additional term of the mean kinetic energy arising from the correlation between perturbation zonal velocity and perturbation depth, and thus reduce the "depth-weighted" mean kinetic energy. Variation of the basic flow with time and some possibilities of redistribution of momentum by these unstable gravity waves are discussed.

Relation between our results and Blumen et al. (1975) is also commented.