Acoustic black hole profile optimization

The effectiveness of a one-dimensional acoustic black hole has often been characterized by its reflection coefficient R and its so-called normalized wave number variation, both of which depend on the height profile of the beam. The underlying assumptions of existence of acoustic black holes require the normalized wave number variation to be small. In this work, we consider an acoustic black hole with fixed parameters of geometry (i.e. length, maximal and minimal height) and pose the variational problem of finding a height profile which minimizes R under the additional constraint of keeping L²ⁿ norm of the normalized wave number variation small, since for large n this norm approximates a point-wise estimate. Using the method of Lagrange multipliers, we solve this variational problem and prove that the optimal profile is one of three different types. These profile types can be seen as direct generalizations of the well-researched profile h(x) = ex². In the limiting case n ->∞ we obtain closed form expressions for the optimal profiles as well as the corresponding reflection coefficient and the normalized wave number variation. Furthermore, we compare numerically the performance of the optimal profiles and profiles previously considered in the literature. In this comparison, we demonstrate that for comparable reflection coefficients the normalized wave number variation of the optimal profile is far superior to the other profiles.