Ultrasonic Propagation in Fractal Porous Material Having Rigid Frame

Abstract

This paper discusses the impact of the fractal structure of porous materials on wave behavior. In addition to commonly used parameters such as porosity, tortuosity, viscous and thermal characteristic lengths, a fractal dimension (α) can be introduced to represent the selfsimilarity of the material. The Helmholtz equation for wave propagation in a porous medium can then be modified to depend on non-integer dimensions. The fractal structure affects the wave's speed, attenuation, and phase shifts, with supersonic wave speeds possible for for certain values of the fractal dimension. The equivalent thickness of the material also varies with the fractal dimension becoming larger for low values of the fractal dimension. This provides us with valuable insights into the potential for creating novel metamaterials with exceptional acoustic properties by adjusting the fractal dimension value, enabling the attainment of supersonic velocities and substantial equivalent thicknesses. Understanding the effect of the fractal dimension on wave behavior has important implications for developing effective acoustic materials and can inform research in noise reduction, metamaterials, medicine, seismology, geophysics, and petroleum engineering.

Non-Integer Dimensional Analysis of Ultrasonic Wave Propagation in Fractal Porous Media

“…This advancement in generalizing differential vector operators to non-integer dimensions supports the use of continuous models for fractal media in NIDS [13,14]. The NIDS calculus developed from this allows for the description of both isotropic and anisotropic fractal media, and has been instrumental in advancing the study of fractal hydrodynamics [15,16,17,18,19], fractal electrodynamics, the elasticity of fractal materials, and acoustics of fractal porous media [20,21,22]. This study aims to apply the NIDS operators referenced in prior research to model ultrasonic wave propagation through a fractal porous medium.…”