Quadratic Diffuser Calculator
Diffuser Visualization
Results
Frequency Response
Effective Frequency Range: 0 - 0 Hz
This quadratic diffuser will:
- Begin scattering sound at 0 Hz
- Be most effective at the design frequency of 0 Hz
- Have reduced effectiveness above 0 Hz
For best results, place the diffuser at least 0 m away from the listening position.
Frequency Response Graph
Well Depths
About Quadratic Diffusers
A Quadratic Residue Diffuser (QRD) is an acoustic treatment that scatters sound waves in different directions, helping to reduce echoes and improve sound quality in a room.
How It Works
Quadratic diffusers consist of a series of wells of different depths separated by thin fins. When sound waves hit the diffuser, they reflect at different times based on the well depths, dispersing sound energy evenly across the room.
The Mathematics Behind It
The well depths in a QRD are calculated using a quadratic residue sequence, which is based on a prime number. For a given prime number N, the sequence is calculated as:
n² mod N (where n = 0, 1, 2, ... N-1)
The physical depth of each well is then determined by scaling this sequence to the maximum depth using the formula:
Well depth = (QR value / max QR value) × Maximum depth
Key Frequencies
- Design Frequency: The frequency where the diffuser is most effective. It's calculated as: f = c/(4×dmax), where c is the speed of sound and dmax is the maximum well depth.
- Scatter Frequency: Half the design frequency, representing the lowest frequency the diffuser effectively scatters.
- HF Cutoff: Highest usable frequency, determined by the well width: f = c/(2×w), where w is the well width.
This calculator helps you design a quadratic diffuser by automatically computing these values based on your desired dimensions.
Scientific References & Formulas
This calculator implements acoustic principles developed by Dr. Manfred R. Schroeder at Bell Laboratories in the 1970s. The specific formulas and algorithms used are based on established acoustic engineering research:
Core Formulas
- Quadratic Residue Sequence: s(n) = n² mod p
Source: Schroeder, M. R. (1975). "Diffuse Sound Reflection by Maximum-Length Sequences." Journal of the Acoustical Society of America, 57(1), 149-150. - Design Frequency: f₀ = c/(4×dmax)
Source: Cox, T. J., & D'Antonio, P. (2009). "Acoustic Absorbers and Diffusers: Theory, Design and Application." 2nd Ed., Taylor & Francis, p.285-289. - Scatter Frequency: fscatter = f₀/2
Source: D'Antonio, P., & Konnert, J. H. (1984). "The QRD Diffractal: A New One- or Two-Dimensional Fractal Sound Diffusor." Journal of the Audio Engineering Society, 32(3). - HF Cutoff: fmax = c/(2×w)
Source: Master Handbook of Acoustics, F. Alton Everest & Ken Pohlmann, 6th Ed., p.258-275.
Physical Construction Parameters
The typical constraints for QRD construction identified in the literature are:
- Minimum practical well width: approximately 1 inch (2.5 cm)
- Maximum practical well depth: approximately 16 inches (40 cm)
- Optimal prime numbers: 7, 11, 13, 17, 19, 23, 29, 31, etc.
The effectiveness of QRDs has been verified through extensive research, with diffusion coefficients typically 3-4 times higher than flat surfaces across the design frequency range (Cox & D'Antonio, 2009).
This calculator implements these principles to provide accurate design parameters for quadratic diffusers that can be used in recording studios, concert halls, home theaters, and other critical listening environments.
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