A quadratic residue diffuser (QRD) designed for 250 Hz needs wells about 56 cm deep, while an optimized acoustic metadiffuser only 3 cm thick covers 250 Hz to 2 kHz (Jiménez, Cox, Romero-García and Groby, Scientific Reports 2017). On a measured panel, a plywood QRD reached a diffusion coefficient of 0.63 at 1 kHz and 2 kHz, against roughly 0.10 for the flat reflector alone (Tronchin et al., Applied Sciences 2021). The free-field test behind those numbers, ISO 17497-2, puts the source 10 m from the sample and the receivers on a 5 m arc, a setup that has more in common with an anechoic chamber than with a living room. In the Lviv Opera, a 998-seat house of 5,500 m³, a Schroeder diffuser installed in 2012 diffused mainly above about 1 kHz (Kamisiński, Archives of Acoustics 2012). We aggregated data from Schroeder’s original JASA papers, ISO 17497 listings at BSI and DIN Media, the University of Parma, MDPI Acoustics and Applied Sciences studies, Scientific Reports, Physical Review X, RPG Acoustical Systems data sheets and EBU Tech 3276.
TL;DR
- QRD well depth follows d_n = s_n x λ0 / 2N, with N an odd prime (Stanford CCRMA, QRD waveguide mesh paper)
- A 500 Hz QRD with 9.4 cm wells is usable only from 375 Hz to 1,824.8 Hz (Khrystoslavenko and Grubliauskas, VILNIUS TECH 2019)
- An N = 7 QRD for 500 Hz to 5 kHz needs a deepest well of 0.2 m (Pardo-Quiles et al., Acoustics 2024)
- ISO 17497-2, published in May 2012, runs 15 pages and measures the directional diffusion coefficient in a free field (DIN Media, ISO 17497-2 listing)
- A flat MDF panel scored a scattering coefficient of 0.117 but a diffusion coefficient of 0.35 at 1 kHz (Farina, University of Parma 2000)
- Plywood QRD diffusion coefficient: 0.63 at 1 kHz and 2 kHz vs about 0.10 for a flat panel (Tronchin et al., Applied Sciences 2021)
- A QRD-based acrylic panel climbs from d = 0.10 at 400 Hz to 0.62 at 2 kHz (RPG Diviewsor 79 data sheet)
- A 250 Hz QRD needs about 56 cm of depth; a 3 cm metadiffuser covers 250 Hz to 2 kHz (Jiménez et al., Scientific Reports 2017)
- Conventional Schroeder diffusers are about one wavelength thick; metasurface versions are about an order of magnitude thinner (Zhu et al., Physical Review X 2017)
- The Lviv Opera diffuser raised GL at 4 kHz by about 0.7 dB across the stalls (Kamisiński, Archives of Acoustics 2012)
- One to three diffusers in a 25.2 m³ room raised C80 by 3.3 to 12.4 dB at 1 and 2 kHz (Azad et al., Acoustics 2019)
1. Design Frequency, Well Depth and Well Width
Schroeder first proposed number-theory surfaces in 1975, using maximum-length sequences (Schroeder, JASA 57, 149-150, 1975), and a 1979 extension switched to quadratic residue sequences to widen the bandwidth. The appeal is that the bandwidth is set by two dimensions you can read off a drawing: well depth fixes the bottom of the useful range, and well width fixes the top. The deepest well is what costs space, because depth is a fraction of the design wavelength.
The design study from VILNIUS TECH shows how narrow that window can be. A 500 Hz QRD with 9.4 cm wells is useful only from 375 Hz to 1,824.8 Hz, a ratio of 4.87, or about 2.3 octaves (1,824.8 / 375 = 4.87; log2 of 4.87 = 2.28). Keeping the panel 1 m wide and raising the prime N narrows the wells, which pushes the upper limit up but also drags the design frequency up with it (Khrystoslavenko and Grubliauskas 2019). The depth formula itself is laid out in the Stanford CCRMA QRD paper.
| Metric | Value | Source |
|---|---|---|
| Well-depth rule | d_n = s_n x λ0 / 2N, N an odd prime | Stanford CCRMA, 2-D waveguide mesh QRD paper (ICMC 2004) |
| N = 7 quadratic residue sequence | 0, 1, 4, 2, 2, 4, 1 | Pardo-Quiles, Rodríguez and Rodríguez-Rodríguez, Acoustics 2024 |
| N = 7 QRD for 500 Hz to 5 kHz | λ0 = 0.682 m, deepest well 0.2 m | Pardo-Quiles et al., Acoustics 2024 |
| 500 Hz design, 9.4 cm wells | Usable from 375 Hz to 1,824.8 Hz | Khrystoslavenko and Grubliauskas, VILNIUS TECH 2019 |
| 1 m wide panel, N = 7 vs N = 23 | Design 330 Hz (upper limit 1,204.4 Hz) vs 1,100 Hz (upper limit 4,014.6 Hz, 4.3 cm wells) | Khrystoslavenko and Grubliauskas, VILNIUS TECH 2019 |
| Well-width rule | w = λmin / 2 | Suyatno and Lianto, Journal of Physics: Conference Series 2021 |
| Onset of sound redirection | About 0.5 x f0 | Kamisiński, Archives of Acoustics 2012 |
Context note: published rules of thumb for the low end disagree. The VILNIUS TECH tables put the lower limit at 0.75 x f0 (375 / 500 = 0.75 and 247.5 / 330 = 0.75), while Kamisiński states that the diffuser starts redirecting sound at 0.5 x f0. Treat any “useful from” figure as design-method specific.
2. Measuring Diffusion: ISO 17497 and the AES Coefficient
There are two standardized numbers for a diffusing surface, and they answer different questions. ISO 17497-1 measures a random-incidence scattering coefficient in a reverberation room: how much energy leaves the specular direction. ISO 17497-2 measures a directional diffusion coefficient in a free field: how evenly the reflected energy spreads across a polar response. BSI states plainly that Part 1 is not intended to characterize the spatial uniformity of scattering (BSI, BS ISO 17497-1:2004+A1:2014).
The gap between the two is not academic. A flat MDF panel measured at 1 kHz had a scattering coefficient of 0.117 but a diffusion coefficient of 0.35, and the author concluded that the scattering coefficient “is generally a quantity very different from the diffusion coefficient” (Farina, University of Parma 2000). On the free-field side, Hargreaves, Cox, Lam and D’Antonio compared free-field definitions and put forward an autocorrelation-based coefficient as the best one (JASA 108, 1710-1720, 2000). The ISO Part 2 listing is at DIN Media.
| Metric | Value | Source |
|---|---|---|
| ISO 17497-1 publication and amendment | Published 22 July 2004, amended in 2014; status current, under review | BSI, BS ISO 17497-1:2004+A1:2014 |
| ISO 17497-2 publication | May 2012, 15 pages, listed as current | DIN Media, ISO 17497-2:2012 listing |
| ISO 17497-2 free-field geometry | Source 10 m from sample, receivers on a 5 m radius, max 5° spacing, at least 80% of receivers outside the specular zone | Tronchin, Farina, Bevilacqua, Merli and Fiumana, Applied Sciences 2021 |
| Recommended free-field diffusion measure | Autocorrelation-based coefficient | Hargreaves, Cox, Lam and D’Antonio, JASA 2000 |
| Flat MDF panel, 1 kHz | Scattering 0.117, diffusion 0.35 | Farina, University of Parma 2000 |
| Hemicylindrical panel rotated 90°, 1 kHz | Scattering 0.20, diffusion 0.52 | Farina, University of Parma 2000 |
| Microphone array in Farina’s simulation model | 37 microphones at 5° spacing, source 10 m, receivers 5 m | Farina, University of Parma 2000 |
Context note: the AES information document on surface scattering uniformity (designated AES-4id-2001) could not be retrieved from aes.org during this research, so its content is described here only through the Hargreaves et al. paper and Tronchin et al., who note that the AES “proposed a measuring technique of the diffusion uniformity coefficient” and that both ISO parts measure from 100 Hz to 5 kHz in third-octave bands.
3. Measured Performance: QRDs vs Flat Panels
Even a well-built QRD does not get close to a diffusion coefficient of 1. The best plywood QRD in the 2021 Parma tests reached d = 0.63 at 1 kHz and 2 kHz, about six times the roughly 0.10 of the flat reflector alone (0.63 / 0.10 = 6.3). Outside its design band the same panel fell to 0.23 at 250 Hz, 37% of that peak (0.23 / 0.63), and orientation mattered as much as construction (Tronchin et al., Applied Sciences 2021).
A commercial QRD-based panel shows the same shape: weak below its design range and uneven inside it. RPG’s acrylic Diviewsor 79 goes from 0.10 at 400 Hz to 0.48 at 500 Hz, dips to 0.30 at 1.25 kHz, and ends at 0.62 at 2 kHz (RPG Acoustical Systems, Diviewsor 79). A dip like the one at 1.25 kHz is why a single averaged rating can mislead for this product class.
| Metric | Value | Source |
|---|---|---|
| Plywood QRD, vertical, no backing, 1 kHz and 2 kHz | d = 0.63 | Tronchin et al., Applied Sciences 2021 |
| Same QRD at 250 Hz / 8 kHz | d ≈ 0.23 / 0.22 | Tronchin et al., Applied Sciences 2021 |
| Flat reflecting panel alone | d ≈ 0.10 from 1 kHz upward | Tronchin et al., Applied Sciences 2021 |
| QRD-based acrylic panel, 400 Hz / 500 Hz / 2 kHz | d = 0.10 / 0.48 / 0.62 | RPG Acoustical Systems, Diviewsor 79 data sheet |
| Same panel, 1.25 kHz dip | d = 0.30 | RPG Acoustical Systems, Diviewsor 79 data sheet |
| Galav2 cavity diffuser vs flat MDF, 1 kHz | d = 0.86 vs 0.35 | Farina, University of Parma 2000 |
| Predicted QRDs, 500 Hz designs | Normalized d peak 0.61 at 800 Hz (N = 11); scattering up to 0.99 (N = 23) | Khrystoslavenko and Grubliauskas, VILNIUS TECH 2019 |
Outlier note: the measuring room changes the answer. The Parma team also tested in a semi-anechoic chamber, where extraneous wall reflections made values “consistently high” and the results were judged invalid. The valid runs used a 16 x 20 x 7 m reverberant laboratory with time windowing, 25 microphones at 7.5° spacing and a 900 x 600 mm panel built on primes 17 and 13 for a 250 Hz to 2 kHz target.
4. Depth, Weight and Absorption Trade-offs
Every octave of extra bass diffusion doubles the depth a classic QRD needs, since well depth scales with the design wavelength, which is why bass-range Schroeder diffusers eat so much room depth. A QRD designed for 250 Hz needs about 56 cm of depth, and an N = 5 QRD for 500 Hz is already 27.4 cm thick; folding the wells only gets that to about half (Jiménez, Cox, Romero-García and Groby, Scientific Reports 2017). That research line, slow-sound slits loaded with Helmholtz resonators, belongs to the wider acoustic metamaterials field.
The metadiffuser results suggest the depth rule is a property of wells, not of diffusion. The same paper reports panels between 1/46 and 1/20 of the design wavelength, and a separate team described conventional Schroeder diffusers as “on the order of one wavelength” thick before building a metasurface version about an order of magnitude thinner (Zhu, Fan, Liang, Cheng and Jing, Physical Review X 2017). Wells also absorb, which matters when a diffuser replaces a reflective finish.
| Metric | Value | Source |
|---|---|---|
| Conventional QRD for 250 Hz | About 56 cm deep | Jiménez et al., Scientific Reports 2017 |
| N = 5 QRD for 500 Hz | 27.4 cm thick | Jiménez et al., Scientific Reports 2017 |
| Optimized broadband metadiffuser | 3 cm thick, 250 Hz to 2 kHz, mean normalized d about 0.65, peaks of 0.9 | Jiménez et al., Scientific Reports 2017 |
| Metasurface Schroeder diffuser | About an order of magnitude thinner than the about-one-wavelength conventional design | Zhu et al., Physical Review X 2017 |
| Membrane-resonator QRD | About 1 order of magnitude thinner than well-based diffusers | Zhao et al., Applied Physics Express 2021 |
| Commercial QRD-based panel size and mass | 23-5/8 in square, 9-1/8 in deep, about 10 lb/ft² | RPG Acoustical Systems, Diviewsor 79 data sheet |
| Same panel, absorption ratings | NRC 0.30, SAA 0.26 | RPG Acoustical Systems, Diviewsor 79 data sheet |
| Resonator-hybrid skyline QRD, 3D-printed | Peak absorption coefficient 0.9 near 2,000 Hz (impedance tube) | Suyatno and Lianto, Journal of Physics: Conference Series 2021 |
Context note: a 2020 follow-up from the metadiffuser group describes 3 cm panels as 20 times thinner than traditional designs (Jiménez et al., Tecniacústica 2020). On absorption by groups of QRDs, Yang, Shen and Wang found that it varies noticeably with installation shape, especially below the design frequency, with two absorption peaks (JASA 2006). The membrane-resonator figure comes from the paper’s abstract only (Zhao et al. 2021).
5. QRDs in Concert Halls, Studios, Classrooms and Listening Rooms
The concert hall case for diffusion came first. Schroeder’s 1979 paper argued that modern low-ceiling halls need special ceiling shapes and surface structures to keep early sound out of the median plane and send it laterally (Schroeder, JASA 65, 958-963, 1979). Retrofits followed: by 1986 QRDs had been added to a 750-seat University of Idaho auditorium originally designed in 1935, with measurements matching computer predictions closely (Graffy and Paoletti, JASA 1986). For the hall metrics these treatments aim at, see our concert hall reverberation and C80 statistics.
In small rooms the effect can be outsized. In a 25.2 m³ test room, one to three pyramidal or convex diffusers raised C80 by 3.3 to 12.4 dB at 1 and 2 kHz, mainly by redirecting sound toward the absorbing floor and ceiling (Azad, Meyer, Lokki and Siebein, Acoustics 2019). These were not QRDs, and the authors warn that rules from non-diffuse rooms do not transfer to other cases. The EBU listening-room recommendation asks for diffusion but stops short of specifying diffusers (EBU Tech 3276, 1998); the low-frequency side of the same rooms is covered in our bass trap absorption statistics.
| Metric | Value | Source |
|---|---|---|
| University of Idaho auditorium QRD retrofit | 750 seats, hall originally designed in 1935 | Graffy and Paoletti, JASA 1986 |
| Lviv Opera rear under-balcony Schroeder diffuser (May 2012) | 5,500 m³ auditorium, 998 seats; GL at 4 kHz up about 0.7 dB across the stalls | Kamisiński, Archives of Acoustics 2012 |
| Lviv diffuser effective band | Diffusion above about 1 kHz; more absorption than the old tapestry below 1 kHz | Kamisiński, Archives of Acoustics 2012 |
| 25.2 m³ room, 1 to 3 diffusers, 1 and 2 kHz | C80 +3.3 to +12.4 dB; T20 down 0.19 to 0.80 s | Azad, Meyer, Lokki and Siebein, Acoustics 2019 |
| Classroom ceiling diffusers on about 8% of the ceiling | C50 +0.8 dB average at 1 kHz; +1.5 to +3.2 dB at a rear receiver, 1 to 4 kHz | Arvidsson, Nilsson, Bard Hagberg and Karlsson, Acoustics 2020 |
| EBU reference listening room | Reverberant field diffuse enough to avoid flutter echoes; listening positions at least 1.5 m from side and back walls | EBU Tech 3276, 2nd edition 1998 |
| Auditorium diffuser shapes simulated | Y-shaped best, QRD rectangles second, edge shapes worst | Pardo-Quiles et al., Acoustics 2024 |
Context note: the classroom study (Arvidsson et al., Acoustics 2020) used curved hardboard diffusers, which also acted as resonant absorbers at 125 to 250 Hz, and its 1 dB C50 just noticeable difference (per ISO 3382-1) means the average ceiling gain was below audibility while the rear-seat gain was not. The Lviv data come from Kamisiński’s Archives of Acoustics paper. Most recent available data for EBU listening-room guidance: EBU Tech 3276, 1998.
Summary: Quadratic Residue Diffuser by the Numbers
| Metric | Value | Source |
|---|---|---|
| First number-theory diffuser paper | 1975, JASA 57, 149-150 | Schroeder, JASA 1975 |
| QRD well-depth rule | d_n = s_n x λ0 / 2N, N an odd prime | Stanford CCRMA, QRD waveguide mesh paper |
| 500 Hz QRD with 9.4 cm wells, usable range | 375 Hz to 1,824.8 Hz | Khrystoslavenko and Grubliauskas, VILNIUS TECH 2019 |
| 1 m wide N = 23 QRD upper limit | 4,014.6 Hz with 4.3 cm wells | Khrystoslavenko and Grubliauskas, VILNIUS TECH 2019 |
| N = 7 QRD for 500 Hz to 5 kHz, deepest well | 0.2 m | Pardo-Quiles et al., Acoustics 2024 |
| ISO 17497-2 publication | May 2012, 15 pages | DIN Media, ISO 17497-2 listing |
| ISO 17497-1 publication | July 2004, amended 2014 | BSI, BS ISO 17497-1:2004+A1:2014 |
| Recommended free-field diffusion measure | Autocorrelation-based coefficient | Hargreaves et al., JASA 2000 |
| Flat MDF panel, 1 kHz | Scattering 0.117, diffusion 0.35 | Farina, University of Parma 2000 |
| Plywood QRD (best configuration) vs flat reflector | d = 0.63 vs about 0.10 at 1 kHz and 2 kHz | Tronchin et al., Applied Sciences 2021 |
| QRD-based acrylic panel, 2 kHz | d = 0.62 | RPG Acoustical Systems, Diviewsor 79 data sheet |
| QRD-based acrylic panel absorption | NRC 0.30, SAA 0.26 | RPG Acoustical Systems, Diviewsor 79 data sheet |
| QRD designed for 250 Hz | About 56 cm deep | Jiménez et al., Scientific Reports 2017 |
| Broadband metadiffuser | 3 cm, 250 Hz to 2 kHz, mean d about 0.65 | Jiménez et al., Scientific Reports 2017 |
| Metasurface Schroeder diffuser | About an order of magnitude thinner | Zhu et al., Physical Review X 2017 |
| QRD auditorium retrofit, 1986 | 750 seats | Graffy and Paoletti, JASA 1986 |
| Lviv Opera diffuser, GL at 4 kHz | Up about 0.7 dB | Kamisiński, Archives of Acoustics 2012 |
| Small-room C80 gain from 1 to 3 diffusers | 3.3 to 12.4 dB at 1 and 2 kHz | Azad et al., Acoustics 2019 |
| Classroom ceiling diffusers, rear receiver | C50 +1.5 to +3.2 dB | Arvidsson et al., Acoustics 2020 |
Methodology and Sources
Every figure above was read in the primary paper, standard listing or manufacturer data sheet linked below. Laboratory and simulation results are labeled as such; manufacturer data are labeled as manufacturer data.
- Schroeder, M. R.: Diffuse sound reflection by maximum-length sequences, JASA 57, 149-150 (1975) and Binaural dissimilarity and optimum ceilings for concert halls, JASA 65, 958-963 (1979).
- Stanford CCRMA: 2-D digital waveguide mesh with a quadratic residue diffuser, QRD design section.
- Khrystoslavenko and Grubliauskas, VILNIUS TECH: Theoretical predictions of sound scattering and diffusion coefficients from quadratic residue diffusers (2019).
- Pardo-Quiles, Rodríguez and Rodríguez-Rodríguez: Design of optimal sound absorbers using acoustic diffusers for multipurpose auditoriums, Acoustics 2024.
- Suyatno and Lianto: Skyline quadratic residue diffuser-resonator (SQRD-R), Journal of Physics: Conference Series 1896 (2021).
- BSI: BS ISO 17497-1:2004+A1:2014, random-incidence scattering coefficient in a reverberation room.
- DIN Media: ISO 17497-2:2012, directional diffusion coefficient in a free field.
- Tronchin, Farina, Bevilacqua, Merli and Fiumana: Comparison failure and successful methodologies for diffusion measurements undertaken inside two different testing rooms, Applied Sciences 11(22) 10523 (2021).
- Farina, University of Parma: A new method for measuring the scattering coefficient and the diffusion coefficient of panels (2000).
- Hargreaves, Cox, Lam and D’Antonio: Surface diffusion coefficients for room acoustics: free-field measures, JASA 108, 1710-1720 (2000).
- RPG Acoustical Systems: Diviewsor 79 product page and data sheet (manufacturer data).
- Jiménez, Cox, Romero-García and Groby: Metadiffusers: deep-subwavelength sound diffusers, Scientific Reports 7, 5389 (2017), plus the Tecniacústica 2020 follow-up.
- Zhu, Fan, Liang, Cheng and Jing: Ultrathin acoustic metasurface-based Schroeder diffuser, Physical Review X 7, 021034 (2017).
- Zhao et al.: Ultrathin Schroeder diffuser based on membrane-type resonators, Applied Physics Express (2021) (abstract only).
- Yang, Shen and Wang: On the sound absorption of quadratic residue diffuser groups, JASA 119 (2006).
- Graffy and Paoletti: In situ auditorium measurements of quadratic residue diffusors, JASA 80 (1986).
- Kamisiński, AGH University: Correction of acoustics in historic opera theatres with the use of Schroeder diffuser, Archives of Acoustics 37(3), 349-354 (2012).
- Azad, Meyer, Lokki and Siebein: Effects of adding pyramidal and convex diffusers on room acoustic parameters in a small non-diffuse room, Acoustics 2019.
- Arvidsson, Nilsson, Bard Hagberg and Karlsson: Combination of absorbers and diffusers in a classroom, Acoustics 2020.
- EBU: Tech 3276, Listening conditions for the assessment of sound programme material, 2nd edition 1998.
- Data watch: this is a mature field, so much of the core data is older than 2023; most recent available data for the standards are ISO 17497-1 (2004, amended 2014) and ISO 17497-2 (2012), and BSI lists Part 1 as under review. The AES-4id-2001 document itself was not reachable during research and is described only through papers that cite it. Source conflicts: the low-frequency limit is stated as 0.75 x f0 in the VILNIUS TECH tables but 0.5 x f0 by Kamisiński; the metadiffuser paper gives 56 cm for its 250 Hz reference QRD in the text but a figure caption lists a 32 cm QRD; the VILNIUS TECH paper’s conclusion that more wells increase diffusion conflicts with its own results section. The Farina, Khrystoslavenko and Pardo-Quiles figures are partly or wholly simulations, the RPG figures are manufacturer data, and the Azad and Arvidsson rooms used non-QRD diffusers.
Last updated: October 18, 2026. We update this roundup quarterly, and the next refresh is expected when ISO completes its current review of ISO 17497-1.