QRD diffuser cutoff frequency explained in one line: there is no single cutoff. The design frequency sets a soft low end, clear well width sets a ceiling near c/(2w), and the flat-plate effect at N times the design frequency caps the top. On paper, a 7-well, 500 Hz panel with 5 cm wells covers about 500 Hz to 3.4 kHz.
A fourth limit catches most DIY builds: a repeating array needs a period width (N × w) of at least one wavelength before it sends sound anywhere but specular at normal incidence. That N7 panel’s 35 cm period puts its first grating lobe near 980 Hz. All figures use 343 m/s for the speed of sound.
What sets a QRD’s cutoff frequencies?
Three dimensions set them: the depth of the deepest well, the clear width of each well, and the prime N. Depth fixes the design frequency. Width fixes the plane-wave ceiling. The prime fixes the flat-plate frequency and, through period width, the lowest frequency a repeating array can send off-axis. The binding limit is whichever is most restrictive.
| Limit | Set by | Formula | N7, f₀ = 500 Hz, w = 5 cm |
|---|---|---|---|
| Design frequency (low) | Deepest well and prime | f₀ = c·s_max / (2N·d_max) | 500 Hz, deepest well 19.6 cm |
| First grating lobe (low, repeating array, normal incidence) | Period width N·w | f = c / (N·w) | 980 Hz |
| Plane-wave limit (high) | Clear well width w | f = c / (2w) | 3,430 Hz |
| First flat-plate frequency (high) | Prime N | f = N·f₀ | 3,500 Hz |
None of these is a brick wall; each marks where an assumption of the ideal model stops holding. For the phase theory behind all four, see how QRD scattering works.
Is the design frequency a hard low cutoff?
No. The design frequency f₀ is the frequency whose wavelength sets every well depth, and below it the panel scatters progressively less rather than stopping. The depth formula is dₙ = sₙλ₀/(2N), with sₙ = n² mod N, as written out in the metadiffuser paper by Jiménez, Cox and colleagues. Rearranged for the low end, f₀ = c·s_max/(2N·d_max).
Use the real maximum residue: 4 for N7, 9 for N11, 12 for N13. Plugging in N − 1 overstates the depth an N7 panel needs. Full depth-to-frequency tables are in the N7 to N17 well depth reference.
The reason shows up in each well’s round-trip reflection phase, φₙ = 2π·sₙ·f/(N·f₀). At f₀ the phases follow the residue pattern exactly. At half of f₀ every phase is halved and the equal-energy property is gone. As frequency keeps falling, all phases shrink toward zero, the condition of a flat wall. The panel slides toward flat-plate behaviour rather than hitting a line.
Depth is the price of a low f₀. The Scientific Reports authors name large depth at low design frequencies as a limitation of Schroeder diffusers; their 500 Hz reference QRD used N = 5 and was 27.4 cm thick.
Why calculators disagree on the low cutoff
Some tools quote the frequency at which the deepest well is a quarter wavelength, c/(4·d_max). That is a real acoustic landmark: the same paper identifies the quarter-wavelength resonance, L = c/(4f), as the point of maximum phase shift for a single well. It is not the Schroeder design frequency, and the gap depends on the prime, because c/(4·d_max) works out to f₀ × N/(2·s_max):
| Prime, f₀ = 500 Hz | Deepest well | c / (4·d_max) |
|---|---|---|
| N7 | 19.6 cm | 438 Hz |
| N11 | 28.1 cm | 306 Hz |
| N13 | 31.7 cm | 271 Hz |
Same design frequency, three different “cutoffs.” Quote f₀ as the design floor and treat anything below it as unproven until a boundary element prediction or a measurement backs it.
Why does well width set the upper cutoff?
Because the depth formula assumes sound travels straight down each well and back as a plane wave. Once half a wavelength fits across the clear width w, a cross-mode can form between the fins and the simple phase model stops describing the well. That happens at f = c/(2w), the same cut-on rule that governs any rigid duct.
Cox and Lam’s 1994 JASA paper tested this directly. They found the phase-change, locally reacting model of each well face “good over a wide range of frequencies,” defined where that approximation runs out, and developed a boundary integral method that works “below and above the well cutoff frequency.” The ceiling is a modelling boundary, not a point where the panel goes silent.
| Clear well width | c / (2w) |
|---|---|
| 1 in (25.4 mm) | 6,752 Hz |
| 1.5 in (38.1 mm) | 4,501 Hz |
| 2 in (50.8 mm) | 3,376 Hz |
| 3 in (76.2 mm) | 2,251 Hz |
Measure w as the open gap between fins, not centre to centre. Narrower wells raise the ceiling but add fin area and viscous losses, which turn some scattering into absorption. Fin seating and joint sealing are covered in building a QRD diffuser.
What is the flat-plate frequency?
It is the frequency at which every well’s round trip equals a whole number of wavelengths, so all wells reflect in phase and the panel acts like a flat surface of the same size. For a QRD it first occurs at N × f₀ and recurs at multiples of that. Setting f = N·f₀ in the phase equation gives φₙ = 2π·sₙ: a full cycle for every well.
Dadiotis, Angus and Cox (JASA, 2008) put numbers on it: a typical diffuser “using p=7 and f(0)=500 Hz, has an upper frequency limit of only 3.5 kHz.” Pushing the first flat-plate frequency above 20 kHz “requires a prime equal to at least 41,” which they note makes diffusers too big to be practical in most applications. Their answer was a different sequence family; power residue sequences moved the flat-plate frequency outside the audible range.
At a 500 Hz design frequency, the first flat-plate frequency is 3.5 kHz for N7, 5.5 kHz for N11, 6.5 kHz for N13 and 8.5 kHz for N17. Between f₀ and N·f₀, integer multiples of f₀ are not a problem: in the ideal model the residue sequence still spreads equal energy into its lobes there.
So the usable top is the lower of c/(2w) and N·f₀. For N7 at 500 Hz with 5 cm wells, the two ceilings sit at 3,430 Hz and 3,500 Hz. Narrowing the wells to 2.5 cm lifts the plane-wave limit to 6,860 Hz and buys nothing; the flat-plate frequency still arrives at 3.5 kHz.
Why does period width matter at the low end?
Because a repeating array scatters only into discrete grating-lobe directions, and at normal incidence the first lobe beside specular exists only when the wavelength is shorter than the period width N × w. Below c/(N·w), a long periodic QRD wall sends its far-field energy back specularly, however deep its wells are.
The grating equation is sin θₘ = sin θᵢ + m·λ/(N·w). Oblique incidence lowers the threshold, down to half at grazing angles, but a rear-wall panel behind the listener often sees sound arriving close to normal. Lateral size matters in research too: the Scientific Reports team evaluated their small QRD as six repeated unit cells “in order to clearly generate the characteristic N diffraction grating lobes.”
For an N7 array to put its first lobe at 500 Hz, the period has to be 68.6 cm, so each well must be 9.8 cm wide. That drops the plane-wave ceiling to 1,750 Hz. The prime is a three-way trade:
| Option at f₀ = 500 Hz | Deepest well | First lobe (normal) | c/(2w) | N·f₀ | Usable top |
|---|---|---|---|---|---|
| N7, 5 cm wells | 19.6 cm | 980 Hz | 3,430 Hz | 3,500 Hz | about 3.4 kHz |
| N7, 9.8 cm wells | 19.6 cm | 500 Hz | 1,750 Hz | 3,500 Hz | about 1.75 kHz |
| N13, 5 cm wells | 31.7 cm | 528 Hz | 3,430 Hz | 6,500 Hz | about 3.4 kHz |
No row wins outright. The N13 panel is the only one that covers roughly 500 Hz to 3.4 kHz as a repeating array, and it costs 12.1 cm of extra depth. If depth is fixed, a modulated, non-repeating layout lengthens the effective period instead; the Cox and D’Antonio review surveys the original design’s limitations and the fixes developed for them, and the comparison of QRD, skyline and BAD panels covers the alternative geometries.
How to spec a panel from its limits
Work through the four numbers in order:
- Decide the band the room actually needs diffused; how acoustic diffusers work covers when diffusion beats absorption.
- From the depth you can tolerate, compute f₀ for each candidate prime.
- If panels will repeat, check c/(N·w) against f₀.
- Take the lower of c/(2w) and N·f₀ as the top of the band.
These are geometric limits for an idealised rigid, lossless surface. Finite panel edges diffract, real wells absorb, and incident angle shifts every lobe. Cox and D’Antonio’s textbook covers the prediction and measurement methods that turn these estimates into performance claims. A single quoted “cutoff frequency” for a QRD is summarising at least two of the numbers above.
FAQ
does a qrd diffuser work below its design frequency
Partly. Below f₀ the well phases shrink in proportion to frequency, so the panel scatters less and behaves more like a flat surface of the same size as frequency falls. The fade is gradual rather than a cliff, but any performance claimed below f₀ should come from a boundary element prediction or a measurement.
how do you calculate the upper frequency limit of a qrd diffuser
Calculate two numbers and take the lower one: c/(2w), where w is the clear well width, and N × f₀, the first flat-plate frequency. At 343 m/s, 5 cm wells give 3,430 Hz, and an N7 panel designed at 500 Hz goes flat at 3,500 Hz, so its practical ceiling is about 3.4 kHz.
which qrd prime number has the widest frequency range
No prime wins on every axis. At a fixed design frequency, a higher prime raises the flat-plate ceiling and widens the period, which helps repeating arrays reach lower. It also deepens the deepest well: at 500 Hz, N7 needs 19.6 cm and N13 needs 31.7 cm. Well width still caps the top at c/(2w).
is the qrd design frequency the same as the schroeder frequency
No. The Schroeder frequency is a room property, the approximate crossover between modal and statistically diffuse behaviour, set by reverberation time and room volume. A QRD’s design frequency is a panel property set by well depth. Both carry Manfred Schroeder’s name, which is why they get confused, but neither determines the other.
Related across the network
- Room Modes: How to Calculate and Fix Bass Problems — studiomonitorguide.com
- Studio Monitor Setup: Placement and Room Acoustics — studiomonitorguide.com
