Geometric design and acoustic performance of dihedral focusing retroreflectors based on confocal conics

• Confocal conics intersecting at a right-angle form focusing retroreflectors. • Geometric solutions can include circle, ellipse, parabola, hyperbola and/or line. • Retroreflection is characterised by an auto-caustic for which formulas are given. • Acoustic wave-based simulations and measurements confirm applicability to sound. This study investigates dihedral retroreflectors, and how they can be modified for increased retroreflection in their vicinity by warping one or both of the dihedron’s surfaces thereby creating a focusing retroreflector. Previous research on this type of focusing acoustic retroreflector was restricted to identical confocal parabolas, and did not theorise the pattern of retroreflection beyond the focus. The present study shows that a wide range of solutions exists, for which retroreflection is particularly strong over a distinctive curve referred to as an auto-caustic, the shape of which can be calculated from the dihedron’s geometric parameters. Five cases were evaluated using raytracing, finite-difference time-domain acoustic simulation, and physical acoustic measurement. The cases include combinations of a confocal ellipse and hyperbola (convex and concave hyperbola), a confocal pair of parabolas, an ellipse and line, and a circle and line. Results highlight the importance of the auto-caustic curve, over which retroreflection is strong and similar to that at the geometric focus. For the cases evaluated (with 1.22 m dihedron face lengths and a focal length of 2 m), focusing retroreflection is particularly evident in the 2 kHz and 4 kHz octave bands.