  This work is an analytical investigation of the evolution of surface water
waves in Miles and Jeffreys theories of wind wave interaction in water of
finite depth. The present review is divided into two major parts. The first
corresponds to the surface water waves in a linear regime and its nonlinear
extensions. In this part, Miles theory of wave amplification by wind is
extended to the case of finite depth. The dispersion relation provides a wave
growth rate depending on depth. Our theoretical results are in good agreement
with the data from the Australian Shallow Water Experiment and the data from
the Lake George experiment. In the second part of this study, Jeffreys theory
of wave amplification by wind is extended to the case of finite depth, where
the Serre-Green-Naghdi is derived. We find the solitary wave solution of the
system, with an increasing amplitude under the action of the wind. This
continuous increase in amplitude leads to the soliton breaking and blow-up of
the surface wave in finite time. The theoretical blow-up time is calculated
based on actual experimental data. By applying an appropriate perturbation
method, the SGN equation yields Korteweg de Vries Burger equation (KdVB). We
show that the continuous transfer of energy from wind to water results in the
growth of the KdVB soliton amplitude, velocity, acceleration, and energy over
time while its effective wavelength decreases. This phenomenon differs from the
classical results of Jeffreys approach due to finite depth. Again, blow-up and
breaking occur in finite time. These times are calculated and expressed for
soliton- and wind-appropriate parameters and values. These values are
measurable in usual experimental facilities. The kinematics of the breaking is
studied, and a detailed analysis of the breaking time is conducted using
various criteria. Finally, some integrability perspectives are presented.
