This paper develops models using fractional calculus for the reflection of sound waves in acoustic ducts with acoustic black hole (ABH) terminations. While previous research has illustrated the desirable behavior of acoustic black holes for passive vibration and noise control, the integration of ABH elements in larger components and periodic assemblies results in computationally intensive models. In order to capture the power law geometric variation required to achieve the ABH effect, most computational models require a very fine spatial discretization, leading to larger computational costs. To combat this, we develop one-dimensional fractional order models whose effective properties are capable of emulating the absorbing behavior of an ABH termination in an acoustic duct. Two modeling approaches are studied. The first model is based on a fractional order boundary condition while the second model homogenizes the ABH termination using a fractional order acoustic domain. Both methodologies incorporate fractional differential equations with frequency-dependent fractional orders.
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