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The aim of this work is to introduce and analyze a finite element discontinuous Galerkin method on polygonal meshes for the numerical discretization of acoustic waves propagation through poroelastic materials. Wave propagation is modeled by the acoustics equations in the acoustic domain and the low-frequency Biot's equations in the poroelastic one. The coupling is realized by means of (physically consistent) transmission conditions, imposed on the interface between the domains, modeling different pore configurations. For the space discretization, we introduce and analyze a high-order discontinuous Galerkin method on polygonal and polyhedral meshes, which is then coupled with Newmark-$β$ time integration schemes. Stability analysis for both the continuous and semi-discrete problem is presented and error estimates for the energy norm are derived for the semi-discrete one. A wide set of numerical results obtained on test cases with manufactured solutions are presented in order to validate the error analysis. Examples of physical interest are also presented to investigate the capability of the proposed methods in practical scenarios.