In this work we present the mathematical model and simulations of a particular wave energy converter, the so-called oscillating water column. In this device, waves governed by the one-dimensional nonlinear shallow water equations arrive from offshore, encounter a step in the bottom and then arrive into a chamber to change the volume of the air to activate the turbine. The system is reformulated as two transmission problems: one is related to the wave motion over the stepped topography and the other one is related to the wave-structure interaction at the entrance of the chamber. We finally use the characteristic equations of Riemann invariants to obtain the discretized transmission conditions and we implement the Lax-Friedrichs scheme to get numerical solutions.

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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 timedependent air pressure inside the chamber of the device, instead of a constant atmospheric pressure as in the previous work [7]. 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.