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In this paper, we are interested in the transition between regimes hereeither visco-thermal or multiple scattering effects dominate for thepropagation of acoustic waves through a 2D regular square array of rigidcylinders embedded in air. An extension of the numerical method usingSchl\"omilch series is performed in order to account for visco-thermal losses.Comparison withexperimental data and results from classical homogenizationtheory allows to study the transition between a low frequency limit (whereviscous and thermal effects dominate) and a high frequency regime (wheremultiple scattering effects become predominant). For this particular geometry,a large frequency domain where visco-thermal and multiple scattering effectscoexist is found.

Multiple Scattering and Visco-Thermal Effects on 2D Phononic Crystal