2D Quadratic-Residue Diffuser

How it works

The depth of the well at grid position (i, j) is set by `(i² + j²) mod N` over an N×N grid of prime N. Because it scatters in two axes, one module already breaks up reflections well; tiling several modules across a wall benefits from modulation to avoid a repeating lattice — Fresnelcut turns that on automatically.

Each cell is a square block of a set footprint. The deepest cell drives how thick your stock needs to be, which Fresnelcut reports as you design.

Worked example: N = 5 at 500 Hz

Pick N = 5 and a 500 Hz design frequency. λ₀ ≈ 686.4 mm, so the per-cell step is u = λ₀/(2·5) ≈ 68.64 mm. The 1-D residues `i² mod 5` for i = 0…4 are `0, 1, 4, 4, 1`; the grid height at (i, j) is `(i² + j²) mod 5`, which for the first two rows looks like:

i \ j	0	1	2	3	4
0	0	1	4	4	1
1	1	2	0	0	2

Multiply each cell by u ≈ 68.64 mm to get its depth — the deepest cell in this 5×5 grid is height 4, so 4 × 68.64 ≈ 274.6 mm of stock. Open this exact panel in the designer to see the full 5×5 map and get the cut file.

Stock, kerf, and what you'll need to build it

A 2-D grid needs a solid backer sheet the wells are cut or routed into (through-cutting a grid the way a 1-D panel's wells can be through-cut isn't practical), so budget for a base panel plus the well depth. Tiling more than one module across a wide wall reimposes a repeating lattice on both axes at once — worse than a 1-D panel's single-axis repeat — so Fresnelcut turns on modulation automatically once a design spans multiple modules, alternating module orientation so the tiling doesn't add its own lobe.

2D QRD vs. skyline

Both scatter two axes. A skyline diffuser is built from a different sequence (primitive-root, folded by CRT into a non-square X×Y grid) and, block for block, suppresses the straight-back reflection the way a PRD does — a 2-D QRD does not. Pick 2-D QRD for the classic grid-of-wells look and the most published reference material; pick skyline for the block-city aesthetic and specular suppression.

Where a 2-D grid earns its keep over a 1-D strip

A 1-D QRD only scatters the axis perpendicular to its fins — mount it on a side wall and it does nothing for reflections arriving from the floor or ceiling. A ceiling cloud, a wide rear wall, or any surface catching reflections from more than one direction is where the extra build complexity of a 2-D grid pays for itself; on a narrow side-wall strip where reflections really do arrive along one axis, a 1-D panel does the same acoustic job for a simpler build.

The math

d(i,j) = [(i² + j²) mod N] · λ₀ / (2N)

An N×N grid (prime N). The additive residue array is what gives two-axis scatter.

Best for

Ceiling clouds and large rear walls
Rooms where reflections come from several directions

Not the tool for

Bass problems (use an absorber)
Tight side-wall strips where 1-D scatter is enough

Start from a common size

N is the number of wells (a prime). More wells scatter more smoothly but make a wider, deeper panel. Pick one to open the designer pre-set:

N = 5 N = 7 N = 11 N = 13

Frequently asked questions

Why is N limited to smaller primes like 5, 7, 11, 13 here?

The grid is N×N, so the well count grows with the square of N — a 2-D QRD gets large fast. Smaller primes keep the panel a manageable size; if you want a bigger single-axis spread instead, a 1D QRD supports larger N without the N² growth.

Do I need to route pockets, or can I through-cut the wells?

Most 2-D QRD builds route or CNC pocket each cell into a solid backer rather than through-cutting, since through-cutting would leave no floor for the shallower wells. Fresnelcut's cut list is set up for pocketing.

How is depth different from a 1-D QRD at the same N and frequency?

The per-cell step (u = λ₀/2N) is identical to a 1-D panel's, but the additive `(i²+j²) mod N` array can produce a different maximum residue than the 1-D `n² mod N` sequence at the same N — check the worked example above for the exact numbers at your settings.

What happens if I need a panel wider than one N×N module?

Fresnelcut tiles additional modules and automatically turns on two-shape modulation once more than one module is needed, so the repeat doesn't read as an obvious repeating grid.

Should I choose a 2-D grid over two crossed 1-D panels?

A true 2-D grid scatters both axes from every cell simultaneously; mounting two 1-D panels at right angles doesn't reproduce that — each still only scatters its own axis. For a ceiling or large rear wall catching reflections from multiple directions, the 2-D grid is the more correct choice.