A rigorous approach to optimal profile design for acoustic black holes

Calculus of variations is used to determine a profile shape for an acoustic black hole without a layer of viscoelastic dampening material with fixed parameters of geometry (i.e., length, maximal and minimal thickness), which minimizes the reflection coefficient, without violating the underlying assumptions of existence for acoustic black holes. The additional constraint imposed by keeping the normalized wave number variation (NWV) small everywhere in the acoustic black hole is handled by the use of Lagrange multipliers. From this method, closed-form expressions for the optimal profile, its reflection coefficient, and the NWV are derived. Additionally, it is shown that in the special case where only the NWV (and not the reflection coefficient) is considered, the optimal profile reduces to the well-known thickness profile for acoustic black holes, h(x)=ϵx². We give a numerical example of the difference between an acoustic black hole with optimal profile and classical profile, h(x)=ϵx^m, m > 2. For close to identical reflection coefficients, the optimal profile vastly outperforms the classical profile in terms of having low NWV at a large range of frequencies.