An acoustic diffuser is a surface treatment designed to scatter sound energy uniformly in many directions rather than reflecting it as a single mirror-like beam. The Schroeder Quadratic Residue Diffuser (QRD), invented by Manfred Schroeder in 1975, achieves this by arranging a series of wells whose depths follow the quadratic residue sequence: d(n) = λ·(n² mod N) / (2N), where N is a prime number and λ is the design wavelength. The pseudo-random depths introduce phase differences that cause constructive and destructive interference at different angles, spreading energy across the hemisphere above the surface.
Select a preset (Flat Wall, QRD-7, QRD-11, or Hybrid), then adjust frequency, well count, max depth, and absorption to see how the polar scattering pattern (shown in the energy plot) changes shape. The diffusion coefficient d (0 = specular, 1 = perfectly diffuse) updates in real time. Toggle wavefronts and energy map to switch between the ray-tracing and Fraunhofer far-field views.
What is the difference between a diffuser and an absorber?
An absorber converts sound energy into heat, reducing the overall sound level in a room. A diffuser redistributes sound energy spatially without reducing it significantly — it breaks up flutter echo and standing waves while preserving the sense of spaciousness ("envelopment"). Most acoustic treatments use a combination: low-frequency absorbers (bass traps) with mid/high-frequency diffusers to control the room's character without deadening it.
Why must the number of wells N be a prime number?
The quadratic residue sequence {n² mod N} for n = 0, 1, …, N−1 produces N/2 + 1 distinct depth values only when N is prime. A prime N ensures the sequence is maximally non-repeating within one period, giving the diffuser uniform spectral properties across all scattering angles. Non-prime values produce repeated depths that create preferred scattering directions — exactly what a diffuser is meant to avoid.
What is the design frequency of a QRD diffuser?
The design frequency f₀ is the frequency at which the deepest well is exactly half a wavelength: d_max = λ/2 = c/(2f₀). For a 15 cm maximum depth, f₀ = 343/(2×0.15) ≈ 1143 Hz. The diffuser works best between f₀ and roughly 5–6×f₀; below f₀ the well depths are too shallow to introduce meaningful phase differences, so the surface behaves like a flat wall.
The diffusion coefficient d is defined by ISO 17497-2 as d = (∑Iᵢ)² / (N × ∑Iᵢ²) − 1/N, normalised to a [0,1] range. A flat wall with specular reflection gives d ≈ 0; a perfect hemispherical scatter gives d = 1. The simulator computes d using Fraunhofer far-field diffraction, summing the complex amplitudes from each well and computing the normalised scattered energy distribution across 36 sectors.
Flutter echo occurs when sound bounces back and forth between two parallel reflective surfaces (e.g., opposite walls), producing a rapid series of distinct echoes heard as a metallic ringing "zing" after a sharp sound. A QRD diffuser on one or both walls breaks up the specular reflection, scattering energy into many directions so successive bounces no longer align. Recording studios and concert halls routinely use diffusers on rear walls for this reason.
The Schroeder frequency (f_S ≈ 2000 √(RT60/V), where RT60 is reverberation time in seconds and V is room volume in m³) marks the transition between the discrete modal region (where individual room resonances dominate) and the diffuse field region (where modes overlap and the sound field is statistically uniform). Below f_S, bass traps targeting individual room modes are needed; above f_S, diffusers are effective. In a typical home studio (V ≈ 30 m³, RT60 ≈ 0.3 s), f_S ≈ 200 Hz.
A diffuser must be at least one wavelength wide to produce useful diffusion — otherwise it acts as a point scatterer. At 500 Hz, λ = 0.69 m, so a 70 cm panel (the simulator's default) is marginally adequate. At 200 Hz, λ = 1.7 m and you would need a 1.7 m wide panel or a periodic array of panels. This physical size requirement makes low-frequency diffusion very challenging in domestic rooms.
High absorption reduces the amplitude of reflected rays without changing their direction distribution. A partly absorptive diffuser (the "Hybrid" preset uses 40% absorption) scatters sound in the same angular pattern as a pure QRD but with less energy — useful when you want both reduced reverberation and spatial diffusion. Pure absorption without diffusion can make a room sound unnaturally "dead" even when RT60 is within target range.
Yes — QRD diffusers appear on rear walls and balcony soffits of many concert halls and recording studios worldwide, including the Philharmonie de Paris and various Abbey Road live rooms. They are also used in home cinemas to break up rear-wall reflections without the excessive damping that would result from using absorbers. Skyline diffusers (2D QRD arrays) provide scattering in both horizontal and vertical planes.
The number (7 or 11) refers to the prime N used in the quadratic residue sequence, which equals the number of wells per period. QRD-7 has wells with depths proportional to {0,1,4,2,2,4,1} (mod 7); QRD-11 uses 11 wells. More wells per period means finer angular control and effective diffusion over a wider frequency range, but requires a wider panel. The simulator lets you vary N from 3 to 17 to see how the scattering pattern changes.
This simulator models a Schroeder quadratic-residue diffuser (QRD): a ribbed acoustic panel whose well depths follow the sequence d(n) = λ·(n² mod N)/(2N), where N is a prime number of wells and λ is the design wavelength. Because those depths are derived from number theory rather than being random, they scatter incoming sound into a wide fan of directions instead of bouncing it back as a single specular reflection, which is exactly what the ray-tracing and polar energy plot on the right show in real time.
Purple wells of varying depth sit at the bottom of the canvas; incident wavefronts spread out from the source dot, reflected ray fans fan out from each well at slightly different angles, and the polar plot in the top-right sums the phase-shifted reflections into a scattering pattern. The Diffusion coeff stat reports how close that pattern is to a perfect hemisphere (1) versus a single mirror-like beam (0).
Choose a Preset Scene (Flat Wall, QRD-7, QRD-11 or Hybrid), or build your own with the Wells (N′), Max depth and Absorption sliders — note Wells only accepts odd numbers, since N should be prime for a genuine QRD sequence. Frequency, Source position X and Incident angle change the sound source, and the Wavefronts / Energy map buttons toggle the two visual layers independently.
Manfred Schroeder invented the quadratic-residue diffuser in 1975 while working at Bell Labs, applying number-theory sequences originally studied for radar and cryptography to the completely different problem of room acoustics — QRD panels are now standard fittings in recording studios and concert hall rear walls worldwide.
The quadratic-residue formula s(n) = n² mod N only produces the ideal, evenly-distributed set of well depths when N is a prime number, and aside from 2 every prime is odd. Restricting the slider to odd values (3, 5, 7 … 17) keeps every preset a mathematically valid QRD sequence rather than a degraded, partially-repeating one.
It is calculated from the polar energy plot: the simulator sums the scattered energy across 36 angular sectors and computes a normalised statistical spread, so a value near 1 means energy is scattered almost equally in every direction, while a value near 0 means most of the energy is still concentrated in one or two directions, as it would be for the Flat Wall preset.
The design frequency is set by the deepest well acting as a quarter-wavelength resonator, so f = c/(4·d_max) using the speed of sound c = 343 m/s. Increasing Max depth lowers the frequency at which the diffuser scatters most effectively, which is why real low-frequency diffusers have to be physically much thicker than the shallow high-frequency ones.
Wavefronts is a ray-tracing view: it draws expanding arcs from the source and straight reflected rays bouncing off each well at an angle nudged by that well's depth-dependent phase shift. Energy map is a frequency-domain view: it sums the complex reflected wave from every well using Fraunhofer diffraction maths to predict the overall scattering pattern, independent of any single ray's path.
In the simulator, absorption multiplies the energy and length of every reflected ray by the same factor (1 − absorption/100) regardless of angle, which models a diffuser whose wells are lined with sound-absorbing material. The angular scattering pattern itself, driven purely by the well-depth sequence, is left untouched — so a highly absorptive QRD still diffuses evenly, just with less total energy returned to the room.