  We propose in this paper a new nonlinear mathematical model of an oscillating
water column. The one-dimensional shallow water equations in the presence of
this device are essentially reformulated as two transmission problems: the
first one is associated with a step in front of the device and the second one
is related to the interaction between waves and a fixed partially-immersed
structure. By taking advantage of free surface Bernoulli's equation, we close
the system by deriving a transmission condition that involves a time-dependent
air pressure inside the chamber of the device, instead of a constant
atmospheric pressure as in the previous work \cite{bocchihevergara2021}. We
then show that the second transmission problem can be reduced to a quasilinear
hyperbolic initial boundary value problem with a semilinear boundary condition
determined by an ODE depending on the trace of the solution to the PDE at the
boundary. Local well-posedness for general problems of this type is established
via an iterative scheme by using linear estimates for the PDE and nonlinear
estimates for the ODE. Finally, the well-posedness of the transmission problem
related to the wave-structure interaction in the oscillating water column is
obtained as an application of the general theory.
