Optimal profile design for acoustic black holes using Timoshenko beam theory

We revisit the problem of constructing one-dimensional acoustic black holes. Instead of considering the Euler-Bernoulli beam theory, we use Timoshenko's approach, 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 wavenumber variation. We use the calculus of variations to derive the corresponding Euler-Lagrange equation analytically and then use numerical methods to solve this equation 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 Ω (defined using the largest height of the plate), the optimal profiles almost coincide, as expected.