Fresnelcut

Fundamentals

What is a quadratic-residue diffuser?

A quadratic-residue diffuser — QRD for short — is an acoustic panel that scatters sound reflections instead of bouncing them straight back. It's the striped, well-and-fin panel you see in studios and listening rooms.

Scattering, not soundproofing

A hard flat wall reflects sound like a mirror reflects light: one strong echo heads back toward you. A diffuser is more like frosted glass — it takes that same energy and spreads it across many directions, so no single reflection dominates. The room still sounds live, just even and spacious rather than harsh or echoey. A diffuser does not reduce how much sound is in the room; for that you need an absorber.

Why the wells vary in depth

The panel is a row of wells of different depths separated by thin fins. When a sound wave hits it, each well reflects with a slightly different delay, and those delayed reflections interfere so that energy fans out evenly. The magic is in the sequence of depths.

For a prime number of wells N, the depth of well number n is proportional to `n² mod N` — the remainder when n² is divided by N. This quadratic-residue sequence has a mathematically flat scattering response, which is exactly what you want.

s_n = n² mod N, d_n = s_n · λ₀ / (2N)
λ₀ is the wavelength at the design frequency f₀. A well's depth is its residue times a fixed step.

What band does it cover?

The design frequency f₀ sets the low edge — below it the wells are too shallow to do much. The high edge is roughly `c / (2·W)`, where W is the well width: above that the wells are wide compared with the wavelength and the panel starts acting like a flat plate. Narrower wells reach higher.

A single 1-D QRD scatters in one plane (perpendicular to its fins). Turn the fins vertical and it spreads sound side to side.

Related reading

How deep should diffuser wells be?Well depth is set by the design frequency and the number of wells. Here's the formula, a worked example, and how stock thickness limits your low end.QRD vs PRD vs skyline: which diffuser do you need?Compare the three main Schroeder diffusers — quadratic-residue, primitive-root, and 2-D skyline — and pick the right one for your room.Where to place diffusers in a roomFirst reflection points, the rear wall, and ceiling clouds — a practical guide to where diffusers help most, and where to use absorption instead.

Design one now

1D Quadratic-Residue Diffuser (QRD)The classic well-and-fin panel that scatters horizontally.2D Quadratic-Residue DiffuserA grid of square wells that scatters in both directions.

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