Optimization of an acoustic horn with respect to efficiency and directivity

We consider the problem of designing an acoustic horn in order to efficiently transmit the incoming wave energy and favorably distribute the energy in the far field. A finite element solution of the Helmholtz equation, in planar or cylindrical symmetry, models the wave propagation. The transmission efficiency is monitored by measuring the back reflections into the feeding waveguide, and the far-field directivity pattern is computed using an integral expression known from scattering theory. The design problem is formulated as a non-linear least-squares problem, which is solved using a gradient-based algorithm, where the gradients are provided by solutions of the associated adjoint equations. The results demonstrate that this approach can generate horns with almost perfect transmission in a wide frequency band. Due to the improved transmission properties at the lower-frequency region, the optimization with respect to efficiency also generates improved far-field directivity patterns, that is, patterns that vary less with frequency. It is possible to obtain even more uniform directivity patterns by explicitly including directivity requirements in the optimization. However, those improvements in directivity seem to be associated with a substantial loss of efficiency. Copyright © 2007 John Wiley & Sons, Ltd.