QRD N7, N11, N13 and N17: Well Depth Tables and Formula

Every dimension of a quadratic residue diffuser follows from three numbers: the prime that sets the number of wells, the design frequency that sets the depths, and the well width that sets the top of the working range. Get those three right and the rest of the build is joinery. Get the first one wrong and you end up with a panel that is deeper, heavier and more expensive than it needed to be, scattering a band the room never had a problem with.

This is the depth reference for that decision, with the sequences written out so nothing has to be taken on trust.

The well depth formula

Each well in a period is indexed by n, counting from zero. The depth of well n is

dₙ = sₙ λ / (2N), where sₙ = n² mod N

N is a prime number, λ is the wavelength of the design frequency, and sₙ is the quadratic residue sequence that gives the device its name. The design wavelength comes from the speed of sound, taken here as 343 m/s at 20 degrees Celsius, so λ = 343 / f.

Two properties of that sequence matter for a build. Residues at indices n and N − n match; with the zero-depth well at one end, the printed list is not a simple palindrome. The sequence never reaches the value N itself, which is why the deepest well is always shallower than half the design wavelength, and by different amounts for different primes.

The factor of two represents sound travelling into the well and back out. Depth is the air path inward from the common opening plane. For the phase explanation, see how QRD scattering works.

That second property is the one most depth charts skip, and it is where the money is.

The four sequences worth knowing

Prime	Quadratic residue sequence sₙ	Deepest well	Depth as a fraction of λ
N = 7	0, 1, 4, 2, 2, 4, 1	4	0.286
N = 11	0, 1, 4, 9, 5, 3, 3, 5, 9, 4, 1	9	0.409
N = 13	0, 1, 4, 9, 3, 12, 10, 10, 12, 3, 9, 4, 1	12	0.462
N = 17	0, 1, 4, 9, 16, 8, 2, 15, 13, 13, 15, 2, 8, 16, 9, 4, 1	16	0.471

Read the last column as the multiplier on the design wavelength that gives the deepest well. An N7 panel needs 0.286 wavelengths of depth. An N13 panel tuned to the same frequency needs 0.462 wavelengths, roughly 60 percent more, for the same low-frequency reach.

Depths at a 500 Hz design frequency

At 500 Hz the wavelength is 68.6 cm. Depths in centimetres, rounded to one decimal:

QRD N7 wells at 500 Hz

Each residue unit is 68.6 / 14 = 4.9 cm. These are calculated dimensions, using 343 m/s for sound speed.

Well index n	Residue n² mod 7	Depth (cm)
0	0	0.0
1	1	4.9
2	4	19.6
3	2	9.8
4	2	9.8
5	4	19.6
6	1	4.9

N11, N13 and N17 depths at 500 Hz

Prime	Well depths, first well to last
N = 11	0.0, 3.1, 12.5, 28.1, 15.6, 9.4, 9.4, 15.6, 28.1, 12.5, 3.1
N = 13	0.0, 2.6, 10.6, 23.7, 7.9, 31.7, 26.4, 26.4, 31.7, 7.9, 23.7, 10.6, 2.6
N = 17	0.0, 2.0, 8.1, 18.2, 32.3, 16.1, 4.0, 30.3, 26.2, 26.2, 30.3, 4.0, 16.1, 32.3, 18.2, 8.1, 2.0

The first well is always zero depth, which is a flat section of the panel face, not an open hole through the backing. Keep the sequence order. Adding an equal offset to every well preserves relative phases in the ideal model, but also changes cavity lengths and possible losses in a constructed panel.

Choosing the prime from the depth you can live with

Most small room projects start from the opposite end. There is a wall, there is a limit on how far a panel can protrude into the room before it becomes a hazard or an eyesore, and the question is what that depth buys.

Lowest design frequency reachable, in hertz, for a given deepest well:

Deepest well	N = 7	N = 11	N = 13	N = 17
10 cm	980	1403	1583	1614
15 cm	653	935	1055	1076
20 cm	490	702	792	807
25 cm	392	561	633	646
30 cm	327	468	528	538

These values use f₀ = c × max(sₙ) / (2N × d_max), with depth in metres. Use the actual maximum residue; it is 4 for N7, not N − 1. The equation gives a design frequency, not a certified lower operating limit.

Among these four primes, a lower prime reaches a lower design frequency for a fixed depth budget. A 20 cm deep N7 panel is designed at 490 Hz; the same 20 cm spent on N13 lands at 792 Hz. A larger prime does not automatically improve low-frequency reach at a given depth.

What the larger prime buys is a longer period before the pattern repeats, and therefore fewer artefacts from periodicity when panels are tiled across a wall. It also gives a smoother scattered polar response at the design frequency, because there are more phase steps available. If the wall is wide and the depth is unconstrained, higher primes are worth the material. If depth is the binding constraint, which it usually is in a domestic room, N7 or N11 is the rational choice.

Well width sets the ceiling

Depth sets the design-frequency scale. Clear well width sets an approximate upper limit on the plane-wave model inside each well. Above this limit, transverse modes make the simple depth equation a less reliable description of the response.

The working upper limit is approximately f = c / (2w), where w is the well width:

Well width	Upper limit
2.5 cm	6860 Hz
3.8 cm	4513 Hz
5.0 cm	3430 Hz
7.0 cm	2450 Hz

A 5 cm clear well width gives a sizing estimate of about 3.4 kHz, not an abrupt measured cutoff. Narrower wells raise this estimate and make a narrower period at the same prime, while increasing the proportion of the face occupied by fins.

For N clear wells of width w, internal fins occupy (N − 1) × t. If two outer cheeks also have thickness t, overall width is N × w + (N + 1) × t. An N7 period with 3.8 cm wells and 6 mm fins and cheeks is 31.4 cm; N13 is 57.8 cm. A thicker outer case must be allowed for separately.

Running the same arithmetic across primes and design frequencies by hand gets tedious. The QRD diffuser sizer on this site does the full sequence, the deepest well, the upper limit and the period width from the three inputs.

Bandwidth sizing beyond well width

The ideal well phases also repeat at N × f₀. At that frequency every round trip becomes an integer number of cycles, so the ideal grating returns in phase. The optimization thesis discusses this limitation alongside well-width sizing.

For N7 at 500 Hz, this phase repetition is at 3500 Hz. With 50 mm wells the width estimate is 3430 Hz. Reducing width to 38 mm raises that estimate to 4513 Hz but leaves phase repetition at 3500 Hz. Both limits belong on a design sheet.

Neither number establishes a continuous useful band. Finite panel width, incident angle and the number of repeated periods also affect the result. Treat the design frequency as an input to a geometry model; assess the intended assembled surface across frequency before claiming performance.

The metric that matters

The directional diffusion coefficient describes the uniformity of reflected sound across angle. ISO 17497-2 specifies its free-field measurement. A lower level at one listening-position microphone cannot establish this: energy could have been absorbed or redirected somewhere else.

Compare polar responses and diffusion curves across frequency, using the same incident angle and measurement geometry. Include a flat reference of the same overall dimensions so that edge diffraction does not get mistaken for the benefit of the well pattern. Geometry calculations alone cannot supply a measured diffusion coefficient.

What the formula does not tell you

The depth equation describes an idealised surface of rigid, lossless wells. Two effects push a real panel away from that model.

The first is absorption. The cited 1992 Applied Acoustics paper reports absorption from constructed N7 diffusers. Narrow, deep cavities can lose energy near resonance, and the later review discusses the effects of flexing panels, porous well bottoms and leaking joints. A rigid, sealed, non-porous assembly is closer to the lossless model, but its absorption still needs measurement.

The practical reading is that a diffuser is not an acoustically neutral substitute for bare wall, but it is also not the broadband absorber those early numbers suggest, provided it is built properly.

The second is periodicity. Repeating identical periods across a wall creates a grating, which produces lobes at predictable angles rather than an even spread. The cited review covers modulation of arrays. A mirrored QRD alone is not a reliable cure because its residue pattern has cyclic symmetry; evaluate the complete arrangement.

Neither effect invalidates the depth table. Both mean the table is the starting point of a design rather than the whole of it.

FAQ

What is a QRD N7 diffuser?

A QRD N7 diffuser has seven wells per period, with depths proportional to the quadratic residues 0, 1, 4, 2, 2, 4, 1. N7 describes the sequence length; design frequency and clear well width determine its dimensions.

What depth are N7 wells at 500 Hz?

At a sound speed of 343 m/s, N7 wells at 500 Hz are 0, 49, 196, 98, 98, 196 and 49 mm deep, measured inward from the opening plane. The deepest well is 196 mm; add backing and mounting space separately.

Where to go next

If the decision is still between device types rather than primes, the comparison of QRD, skyline and BAD panel diffusers covers what each geometry does to the scattered field and what it costs in depth. If the prime is settled and the next step is a cut list, building a QRD diffuser works through material choice, fin seating and mounting distance. For the background on why a room wants scattering at all, start with how acoustic diffusers work.