QRD diffuser design: well depths, width and frequency range
A quadratic residue diffuser (QRD) is a row of wells of different depths, separated by thin fins. The depths follow the sequence n² mod N, scaled so that the deepest well suits a chosen design frequency. Three numbers then decide how it behaves: the well depth step, the well width and the prime N. Here is the design method with two worked examples.
1. The sequence
Pick a prime N (7, 11, 13, 17 and 23 are common). For each well n = 0, 1, 2 … N − 1 the sequence value is s = n² mod N: square the well number and keep the remainder after dividing by N. For N = 7:
| Well n | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| n² | 0 | 1 | 4 | 9 | 16 | 25 | 36 |
| n² mod 7 | 0 | 1 | 4 | 2 | 2 | 4 | 1 |
The pattern is symmetric (0, 1, 4, 2, 2, 4, 1) and repeats every N wells, so a panel is one or more periods side by side. For N = 13 the values are 0, 1, 4, 9, 3, 12, 10, 10, 12, 3, 9, 4, 1, with a largest value of 12.
2. The well depths
c is the speed of sound, 343 m/s at 20 °C. At the design frequency, the trip down each well and back delays its reflection by a different fraction of a wavelength, and the n² mod N pattern of those delays spreads the reflected sound evenly over many directions.
Example 1, N = 7 designed for 500 Hz: λ0 = 343 ÷ 500 = 0.686 m, so the depth step is 686 ÷ 14 = 49.0 mm and the depths are:
| Well n | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| Depth | 0 | 49 | 196 | 98 | 98 | 196 | 49 |
3. The frequency range
- Design frequency f0 (500 Hz in example 1): where the depths are exactly right. A QRD still scatters somewhat below it, but less evenly.
- Upper limit ≈ c ÷ (2 × well width): above it, sound no longer behaves as a plane wave in each well. 50 mm wells give about 3.4 kHz; narrower wells push it higher.
- Flat-plate frequency N × f0 (3500 Hz here): every depth becomes a whole number of half-wavelengths and the panel reflects like a flat board. A larger N moves this up.
- Listening distance: a diffuser needs room to work; about three wavelengths at the design frequency, here 2.1 m.
So a designer trades three things: deeper wells for a lower f0, narrower wells for a higher upper limit, and a larger N for a higher flat-plate frequency and finer scattering, at the cost of more wells and more depth variety to build.
The calculator works out the depths, the fins and the period for your panel width, and gives a printable build sheet and DXF files for the fins.
Open the N = 7, 500 Hz QRD in the calculatorExample 2: N = 13 with a 120 mm deepest well
Often the depth is fixed by the material instead: say the deepest well can be 120 mm. With N = 13 the largest sequence value is 12, so f0 = 343 000 × 12 ÷ (2 × 13 × 120) = 1319 Hz, and the depth step is 120 ÷ 12 = 10.0 mm. With 40 mm wells and 6 mm fins, the upper limit is about 4.3 kHz and the flat-plate frequency 17150 Hz. Open the N = 13 example.
Why calculators disagree about the frequency
Many DIY calculators quote c ÷ (2 × deepest well) as the lowest frequency, which ignores N and s_max. For example 2 that gives 1429 Hz, against the QRD design frequency of 1319 Hz. Neither is "the" bass limit; they are different definitions. The calculator shows both, so you can compare designs on the same basis.
Building it
- Fins should be thin and stiff (plywood or MDF of 3 to 6 mm is common) and run the full height of the panel.
- Make the well bottoms rigid and sealed; a rattling or leaky well bottom changes the effective depth.
- Repeat whole periods across the panel width so the pattern stays intact; the calculator reports how many periods fit.
- A skyline (2D block) diffuser uses the same idea in two directions, built from sticks of different heights. See skyline diffuser plans.
These are textbook relationships (Schroeder's sequences, c = 343 m/s). They describe the geometry, not a measured result; performance in a room depends on mounting, placement and the build.