Dispersive behaviour of bones with higher order gradient poroelasticity

The purpose of the presentation was to give a theoretical framework for which some characteristics of ultra-sound wave propagation through bones can be justified. In particular, positive and negative dispersion with positive and negative attenuation and the presence of P2 waves. For this purpose a 1D model for a saturated solid-fluid mixture has been derived with the inclusion of higher order terms and through a variational procedure. The model is conceived to get a well-posed system of PDEs and BCs. With this model static and dynamic simulations can be obtained. Thus, dynamic solutions have been obtained in terms of plane waves. The dispersion relation of the mixture is achieved. The diagrams of velocity and of the attenuation have been shown. Nevertheless the simplicity of the model, such diagrams are able to reproduce the experimental evidence of ultrasonic positive and negative dispersion with positive or negative attenuation. Finally, the prediction of P1 fast waves and of P2 slow (highly attenuated) waves have been derived.