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

Miles' mechanism for generating surface water waves by wind, in finite water depth and subject to constant vorticity flow