Transient ultrasonic wave propagation in porous material of non-integer space dimension

Acoustic propagation in a self-similar porous medium having a rigid frame is studied. A fractional propagation equation in a porous material of non-integer dimension is established using the variational method (Stillinger-Palmer-Stavrinou formalism). The wave equation is solved analytically in the time domain using the Laplace transform method. The analytical solution of the propagation equation shows the existence of a supersonic wave whose front wave velocity depends on the non-integer dimension and the tortuosity of the self-similar porous material. Numerical simulations of the amplitude of the ultrasonic wave inside the material show the sensitivity of the main important parameters describing the propagation (non-integer dimension, tortuosity, viscous and thermal characteristic lengths). The non-integer dimension seems to be the only parameter which acts on both the amplitude and the velocity of the acoustic wave.