Global Journal of Researches in Engineering, I: Numerical Methods, Volume 23 Issue 1

buildings, aircraft, automobile industries, and civil engineering. II. M odel Acoustic propagation in porous materials is a complex phenomenon that involves the interaction of sound waves with fluid and solid components of the porous medium. When considering air-saturated porous materials with immobile solid skeletons, wave propagation is confined to the fluid, and this behavior is typically modeled using the equivalent fluid model [5,6], which is a particular case of Biot theory [13-15]. The two frequency response factors, the dynamic tortuosity of the medium α ( ω ) and the dynamic compressibility of air in the porous material β ( ω ), are used to account for structure-fluid interactions. The dynamic tortuosity is provided by Johnson et al [5,6], while the dynamic compressibility is given by Allard [7]. In the frequency domain, these factors are multiplied by the density and compressibility of the fluid. At extremely low and high frequencies, the equations governing the acoustic behavior of the fluid simplify and the parameters involved are different. In the high-frequency range [7], this simplification occurs when the viscous and thermal skin thicknesses ( ) = � 2 0 and ′ ( ) = � 2 0 are smaller than the pore radius r. (Here, the density of the saturating fluid is represented by ρ 0, the viscosity by η , the pulse frequency by ω , and the Prandtl number by Pr). In the low-frequency range of the ultrasonic domain, the dynamic tortuosity and compressibility are given by [12]: ( ) = ∞ �1 + ( ) Λ � 2 � 1 2 + � ( ) Λ � 2 � 2 � + ⋯� (1) ( ) = 1 + ( − 1) � ′ ( ) Λ ′ � 2 � 1/2 + ( ′ − 1) � ′ ( ) Λ ′ � 2 � 2 � + ⋯ � (2) where, = √−1 and γ is the adiabatic constant. The relevant physical parameters of the models are the high-frequency limit of the tortuosity α ∞ , the viscous and thermal characteristic lengths Λ and Λ ', respectively, and the dimensionless parameter ξ introduced by Sadouki [12], which is a shape factor related to the correction of the viscous skin depth of the air layer near the tube surface where the velocity distribution is significantly perturbed by the viscous forces generated by the stationary frame in the low- frequency ultrasonic regime. ξ ' is the associated thermal counterpart. Consider a homogeneous porous material that occupies the region 0≤ x ≤ L. A sound pulse normally strikes the medium, generating an acoustic pressure field p(x,t) and an acoustic velocity field v(x, ω ) within the material (Fig. 1). These fields satisfy the Euler equation and the constitutive equation along the x-axis: 0 ( ) ( , ) = ( , ) , ( ) ( , ) = ( , ) , (3) Here, K a is the compressibility modulus of the fluid. Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 34 Year 2023 © 2023 Global Journals ( ) I Investigating the Effects of Physical Parameters on First and Second Reflected Waves in Air-Saturated Porous Media under Low-Frequency Ultrasound Excitation