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

Abstract

Acoustic black hole (ABH), a thin wedge type structure with its thickness is tailored according to the power-law of power (m) greater than or equal to two, has received much attention from the researchers due to its potential as a light and effective absorber of flexural waves propagating in beams or plates. In this paper, the Euler-Bernoulli equation for the ABH of m>2 is reformulated into the form of a generalized hypergeometric differential equation. The exact solution is then derived in terms of generalized hypergeometric functions (pFq) where p=0 and q=3 by classifying the power m into four cases. The derived solution is in linearly independent form without singularities 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 the ABH are calculated to show the applicability of the present solution. This paper aims at establishing a mathematical and theoretical foundation for the study of the ABHs.