A Schroeder diffuser is the wall of timber wells you have seen in every studio photograph since about 1985 — a row of narrow slots of different depths, usually separated by thin fins. It looks like a decorative device and is very often used as one. It is in fact one of the few pieces of acoustic treatment whose behaviour was derived from pure mathematics before it was ever built, and the depths are not chosen by eye.
The problem it solves
A wall can do three things with sound: absorb it, reflect it, or scatter it. Absorption removes energy. Reflection returns it in one direction, obeying the same law as a mirror. Scattering returns it in many directions at once, spread evenly over the hemisphere.
Very often scattering is what you want and the other two are what you can get. In a control room, a strong specular reflection from the rear wall arriving a few milliseconds after the direct sound produces comb filtering — a series of cancellations and reinforcements across the spectrum that colours everything the engineer hears. Absorbing the rear wall solves it, but at the cost of the room’s liveliness and the sense of space that makes it possible to work in for eight hours. In a concert hall, a plain side wall returns a single loud late reflection to a specific block of seats; the same energy spread evenly is heard as envelopment, which is one of the qualities that separates a good hall from a merely adequate one. Scattering keeps the energy in the room and takes the direction out of it.
The trick
Manfred Schroeder’s insight, published in 1975, was that the far-field scattering from a periodic surface is governed by the Fourier transform of the reflection coefficients along it. If you want energy spread evenly across the diffraction orders, you need the transform to be flat — and there is a family of number sequences whose discrete Fourier transform has constant magnitude.
The best known is the quadratic residue sequence. Take a prime number N and compute, for each n from 0 to N-1, the value of n² modulo N. For N = 7 that gives 0, 1, 4, 2, 2, 4, 1. For N = 17 it gives 0, 1, 4, 9, 16, 8, 2, 15, 13, 13, 15, 2, 8, 16, 9, 4, 1. Those integers become the well depths, scaled so that the deepest well is half a wavelength at the lowest design frequency: d(n) = s(n) · λ/(2N), where λ belongs to the lowest frequency you want to scatter.
Each well behaves as a short tube closed at the bottom. Sound enters, travels to the base, reflects, and comes back out having been delayed by twice the well depth — which means it re-emerges with a phase shift determined by that depth. The mouths of the wells therefore form a row of sources radiating at the same amplitude but with a pattern of phases set by the sequence. Because that sequence has a flat Fourier transform, the sources interfere in such a way that energy goes out equally into every direction the geometry supports. The wall has become a spatial phase grating.
What the numbers cost you
Two limits fall straight out of the design equations, and both are why real diffusers are the size they are.
The low-frequency limit is the depth of the deepest well, which must be a quarter wavelength at minimum and half a wavelength for full effect. Scattering down to 300 Hz needs about 570 mm of depth for the half-wavelength condition. That is why a diffuser that genuinely works into the lower midrange is a piece of structure and not a panel, and why most of the shallow ones sold as diffusers do nothing at all below about 1 kHz — where, incidentally, the room needed the help least.
The high-frequency limit is the well width. Once the wavelength approaches the width of a well the sound no longer sees a phase-shifted aperture, it sees a small flat surface, and the device reverts to a mirror with fins on it. A 25 mm well width puts that ceiling somewhere around 7 kHz. Narrower wells raise it and cost you more fins, and the fins themselves become an absorption mechanism through viscous losses in a very narrow slot — which is how a diffuser specified with too much enthusiasm turns into a mediocre absorber.
Where we use them, and where we do not
We use diffusion where the room needs to keep its energy but lose the direction of it: rear walls in control rooms, side walls in recital rooms, the wall opposite a stage in a rehearsal space. We do not use it as a general finish. A room lined in quadratic-residue wells is not a room with a flat frequency response; it is a room with a very distinctive one, because the design frequency and the periodicity of the array both leave audible signatures, and a repeated array produces lobes at the frequencies where the period matters.
And we do not use it as a substitute for shape. A well-proportioned room with modelled surfaces — box fronts, coffers, a canted ceiling, ordinary architectural relief at a scale of a few hundred millimetres — scatters perfectly well and costs nothing, because it is the architecture. The diffuser is a repair for a room that has already been drawn flat, and repairs are more expensive than getting it right on the section drawing. That is the argument we would always rather be having, and the one we try to have at Stage 2.
