Optimal Profile Design for Acoustic Black Holes using Timoshenko beam Theory

TL;DR

This paper develops an optimal profile design for acoustic black holes using Timoshenko beam theory, providing more accurate high-frequency models and comparing results with traditional Euler-Bernoulli theory.

Contribution

It introduces a Timoshenko-based approach for designing acoustic black holes, improving high-frequency accuracy over Euler-Bernoulli models.

Findings

Optimal profiles align at low frequencies.

Profiles differ significantly at higher frequencies.

Wave number dependence on height ratio affects design.

Abstract

We revisit the problem of constructing one-dimensional acoustic black holes. Instead of considering the Euler-Bernoulli beam theory, we use Timoshenko's approach instead, which is known to be more realistic at higher frequencies. Our goal is to minimize the reflection coefficient under a constraint imposed on the normalized wave number variation. We use the calculus of variations in order to derive the corresponding Euler-Lagrange equation analytically and then use numerical methods to solve this equation in order to find the optimal height profile for different frequencies. We then compare these profiles to the corresponding ones previously found using the Euler-Bernoulli beam theory and see that in the lower range of the dimensionless frequency $\Omega$ (defined using the largest height of the plate), the optimal profiles almost coincide, as expected. For higher such frequencies, even for values where Euler-Bernoulli theory should still be marginally valid, the profiles predicted using Euler-Bernoulli differ substantially from the correct ones predicted by Timoshenko theory. One explanation for this phenomenon is that unlike in the constant height case, in our setting the wave numbers also depend on the ratio between the smallest and the largest heights.