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In this paper,we derive discrete transparent boundary conditions for a class of linearized Boussinesq equations.These conditions happen to be non-local in time and we test numerically their accuracy with a Crank-Nicolson time-discretization on a staggered grid.We use the derived transparent boundary conditions as interface conditions in a domain decomposition method,where they become local in time.We analyze numerically their efficiency thanks to comparisons made with other interface conditions.