This is a real Quadratic Residue Diffuser (QRD). Well n of N total wells gets depth proportional to the quadratic-residue sequence s(n) = n² mod N — a deterministic but non-repeating-looking pattern that, unlike a flat wall, sends an incoming wave back in many directions instead of one specular bounce.
Each well acts as a tiny resonant duct: reflecting off its bottom adds an extra round-trip path of 2×depth, delaying that well's re-radiated wave in phase relative to its neighbours. The scattered field at every outward angle is the Huygens sum of all N well-mouth wavelets, each carrying its own depth-phase and position-phase:
- phase(n, θ) = (2π/λ) · [ x(n)·(sin θ − sin θin) + 2·depth(n) ]
- energy(θ) = |Σₙ ei·phase(n,θ)|² / N²
Summed over many outward angles θ this gives the polar scattering pattern and the diffusion coefficient (Cox & D'Antonio formula — 1 means perfectly uniform scattering, 0 means all energy stays in one lobe).
Designed bandwidth — a QRD only diffuses well over a limited frequency range set by its own geometry: below the design low frequency the wells are too shallow (less than roughly a quarter wavelength) to add a useful phase delay, and above the design high frequency the well width itself becomes comparable to the wavelength and the array starts producing grating-lobe artefacts instead of smooth diffusion. Drag the frequency slider through the shaded band on the response curve to see the diffusion coefficient rise inside it and fall outside it.
Simplification: reflection inside each well is treated as a simple round-trip phase delay (normal-incidence duct resonance), which is the standard first-order QRD approximation — real wells also have finite-width diffraction inside the slot that this model does not resolve.