Exact solution of Euler-Bernoulli equation for acoustic black holes via generalized hypergeometric differential equation

Abstract

Acoustic black hole (ABH), a thin wedge-shaped structure tapered according to the power-law of power (m) greater than or equal to two, has received much attention to the researchers in wave dynamics due to its potential as a light and effective absorber of elastic waves propagating in beams or plates. In this paper, the Euler-Bernoulli equation for the ABH is reformulated into the form of a generalized hypergeometric differential equation for m>2. Then, the exact solution is derived in terms of generalized hypergeometric functions (F-p(q)) where p = 0 and q = 3 by dividing the power m into four cases. The derived exact solution is in linearly independent and regular form for arbitrary m. In addition, by using the exact solution, the displacement field of a uniform beam with an ABH and the reflection coefficient from an ABH are calculated to demonstrate the applicability of the present exact solution. This paper aims at providing a mathematical and theoretical basis for the study of the ABHs. (C) 2019 Elsevier Ltd. All rights reserved.