  The Miles theory of wave amplification by wind is extended to the case of
finite depth h and a shear flow with (constant) vorticity {\Omega}. Vorticity
is characterised through the non-dimensional parameter {\nu} = {\Omega} U_1 /g,
where g the gravitational acceleration, U_1 a characteristic wind velocity and
k the wavenumber. The notion of 'wave age' is generalised to account for the
effect of vorticity. Several widely used growth rates are derived analytically
from the dispersion relation of the wind/water interface, and their dependence
on both water depth and vorticity is derived and discussed. Vorticity is seen
to shift the maximum wave age, similar to what was previously known to be the
effect of water depth. At the same time, a novel effect arises and the growth
coefficients, at identical wave age and depth, are shown to experience a net
increase or decrease according to the shear gradient in the water flow.
