  In order to study the growth of wind waves in finite depth we extend Miles'
theory to the finite depth domain. A depth-dependent wave growth rate is
derived from the dispersion relation of the wind/water interface. A suitable
dimensionless finite depth wave age parameter allows us to plot a family of
wave growth curves, each family member characterized by the water depth. Two
major results are that for small wave age, the wave growth rates are comparable
to those of deep water and for large wave age, a finite-depth wave-age-limited
growth is reached, with wave growth rates going to zero. The corresponding
limiting wave length and limiting phase speed are explicitely calculated in the
shallow and in the deep water cases. A qualitative agreement with well-known
empirical results is established and shows the robust consistency of the linear
theoretical approach.
