  We propose in this paper a new nonlinear mathematical model of an oscillating
water column (OWC). The one-dimensional shallow water equations in the presence
of this device is reformulated as a transmission problem related to the
interaction between waves and a fixed partially-immersed structure. By imposing
the conservation of the total fluid-OWC energy in the non-damped scenario, we
are able to derive a transmission condition that involves a time-dependent air
pressure inside the chamber of the device, instead of a constant atmospheric
pressure as in \cite{bocchihevergara2021}. We then show that the transmission
problem can be reduced to a quasilinear hyperbolic initial boundary value
problem with a semi-linear 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.
