A reflection grating that nullifies the specular reflection: A cone of silence

This paper describes the geometry of an acoustical diffusor that nullifies the specular reflection of plane sound waves whose frequencies include a design frequency and all its harmonics up to a certain limit. The diffusor is a specially designed acoustical phase grating whose phase cancellation properties are governed by concepts from pure number theory going back to the great mathematician Karl Friedrich Gauss in the early 1800s and in the late 1920s to applications of the theory by Manfred R. Schroeder of Götingen and Bell Laboratories. Presented here is an exposition of Schroeder’s ground-breaking quadratic residue diffusor, using sufficient mathematical rigor to prove its astounding uniform reflection property, originally conceived for improved concert hall acoustics. It then becomes clear how to modify the quadratic residue sequences to what is called modified primitive root sequences to achieve the nullification of the specular reflection. Since the method, for any angle of incidence, results in cancellation in the specular direction and throughout an angular region on each side of the specular direction, the cancellations occur in a cone-shaped region. In particular, for normal incidence the sender of a signal sits in the cone—a cone of silence.