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.