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

Abstract

This paper explores the acoustics of porous media characterized by fractal, or self-similar, structures. Employing a fractal approach, we use differential operators in non-integer dimensional spaces to address the fundamental equations of acoustics in such media. The primary aim is to examine the transmission of ultrasonic waves within a fractal porous medium. Our findings reveal that the fractal dimension significantly influences wave transmission. In fractal porous materials, waves travel along more complex and intricate paths, resulting in increased tortuosity and attenuation. We introduce the concept of an effective path length leff , which is dependent on the fractal dimension Dx , to describe the actual trajectory of wave propagation. Additionally, we define an effective tortuosity αDx , directly proportional to l e f f 2 , to quantify the additional tortuosity brought about by the fractal structure. The insights gained from this study are crucial, as they enhance our understanding of wave behavior in self-similar porous media, which are prevalent in various natural settings and have multiple practical applications, including sound insulation and the design of acoustic materials. Furthermore, understanding the impact of fractal dimensions on wave behavior is vital for developing more efficient acoustic solutions. This research also sets the stage for further theoretical and experimental work on applying fractal geometry to analyze wave propagation in porous structures.