  In this paper, we want to understand the Proudman resonance. It is a resonant
respond in shallow waters of a water body on a traveling atmospheric
disturbance when the speed of the disturbance is close to the typical water
wave velocity. We show here that the same kind of resonance exists for
landslide tsunamis and we propose a mathematical approach to investigate these
phenomena based on the derivation, justification and analysis of relevant
asymptotic models. This approach allows us to investigate more complex
phenomena that are not dealt with in the physics literature such as the
influence of a variable bottom or the generalization of the Proudman resonance
in deeper waters. First, we prove a local well-posedness of the water waves
equations with a moving bottom and a non constant pressure at the surface
taking into account the dependence of small physical parameters and we show
that these equations are a Hamiltonian system (which extend the result of
Zakharov [33]). Then, we justify some linear asymptotic models in order to
study the Proudman resonance and submarine landslide tsunamis; we study the
linear water waves equations and dispersion estimates allow us to investigate
the amplitude of the sea level. To complete these asymptotic models, we add
some numerical simulations.
