QRD Diffuser

An acoustic diffuser using wells of mathematically calculated depths based on the quadratic residue sequence.

A Quadratic Residue Diffuser (QRD) is a reflection phase grating that scatters sound waves across a wide frequency range. Developed by physicist Manfred Schroeder in the 1970s, QRDs use number theory to create an optimal pattern of well depths.

The key principle: each well causes a different phase shift in the reflected sound depending on its depth. When these scattered waves recombine, they interfere constructively at many different angles, spreading the acoustic energy evenly rather than reflecting it like a mirror.

Well depths follow the quadratic residue sequence: r_k = k² mod N, where N is a prime number (typically 7, 11, 13, or 17) and k is the well position. This mathematical basis ensures flat, uniform scattering across the design bandwidth.

QRDs are used in recording studios, control rooms, concert halls, and home theaters to add spaciousness and reduce flutter echo while maintaining acoustic energy in the room.

Formula

d_k = (r_k × λ) / (2N)

d_k = Well Depth (meters)
r_k = Quadratic Residue (k² mod N)
λ = Wavelength (c / f₀)
N = Prime Number (7, 11, 13...)

Practical Example

Design a 7-well QRD for 500 Hz

λ = 343/500 = 0.686m. Sequence: 0,1,4,2,2,4,1. Max depth: (6 × 0.686)/(2 × 7) = 29.4 cm

The seven wells would be: 0, 4.9, 19.6, 9.8, 9.8, 19.6, 4.9 cm deep.

Standards: ISO 17497-2

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