In this paper, a model for sound propagation in double porosity materials with elastic frame is proposed. The main originality of this paper is to develop a model which is suitable for sound absorbing materials. The original idea is to replace the fluid-phase related properties of a single porosity material in Biot transverse isotropic equations with those of a double porosity medium. The model is then developed for 1D problem and used to derive the surface impedance and the sound absorption coefficient of double porosity samples. To quantify the influence of the porous sample dimensions, a numerical procedure based on a finite element simulations is proposed. It is applied to perforated porous samples (a polymer foam and one fibrous material with two different size of perforation) subject to a plane wave at normal incidence. Numerical results are compared to impedance tube measurements.