QRD Diffuser Calculator

How the QRD diffuser calculator works

Use this QRD diffuser calculator to calculate well depths, design frequency and diffusion range for a quadratic residue diffuser. It is intended for studio, listening room and home theater acoustic treatment planning.

This calculator sizes a quadratic residue diffuser (QRD) from your lowest design frequency, well count, well width, and speed of sound. Enter those four values and the tool outputs individual well depths following the quadratic residue sequence, total panel width, maximum well depth (which sets build clearance), the upper frequency limit of effective scattering, and a minimum listening distance rule of thumb. Studio designers and DIY acoustics builders use it to translate a target scattering band into a cut list before reaching for a saw.

Why diffusion instead of absorption?

Absorption removes acoustic energy from a room, which is useful for controlling reverberation time and eliminating discrete echoes. Diffusion redirects energy in multiple directions without removing it, preserving the sense of spaciousness and envelopment that makes a room feel natural rather than dead. In a well-designed studio, the front half of the room is typically treated with absorption at reflection points (ceiling, side walls, front wall) while the rear half uses a combination of diffusion and broadband absorption. An over-absorbed room loses low-level detail and sounds unpleasant to work in for long sessions.

How QRD well depths scatter sound

A quadratic residue diffuser works as a time-delay grating. Each well reflects incoming sound with a different phase delay proportional to its depth. When the reflected waves recombine, the phase differences spread energy evenly across a range of angles rather than returning it as a single specular reflection. The sequence of depths follows i² mod N (where N is the prime well count and i is the well index), a mathematical pattern that produces uniform energy distribution across the full hemisphere in front of the panel — at least within the design frequency band.

QRD diffuser well depth calculation

The well depth calculation starts with the design frequency, speed of sound and prime well count. Each well uses the quadratic residue value for its position, so deeper wells create longer reflection delays and shallower wells create shorter delays.

Design frequency, well count, and practical limits

The design frequency sets the deepest well: lower frequencies require longer path-length differences and therefore deeper wells. Targeting 500 Hz with a 7-well panel gives maximum depths around 50–70 mm, which fits easily in 18 mm MDF construction. Targeting 200 Hz pushes maximum depths beyond 400 mm — impractical for most builds. Well count must be a prime number; 7 and 11 are the most common choices for studio diffusers. More wells give finer angular resolution but wider panels. The upper frequency limit is set by well width: scattering degrades when the wavelength approaches the well width, typically limiting effective diffusion to below 4–8 kHz for typical 50 mm wells.

Prime number and number of wells

QRD well counts use prime numbers because the quadratic residue sequence repeats cleanly and distributes reflection delays evenly. Seven wells is a common compact build, while 11 or 13 wells give a wider panel and smoother scattering.

Maximum well depth and build example

Maximum well depth determines the minimum frame depth and often decides whether a design is practical. For example, an 11-well diffuser targeting the low midrange may need a deeper cabinet than a rear-wall panel can physically accept, so it is common to raise the design frequency or build multiple shallower modules.

QRD vs skyline diffuser

A QRD diffuser scatters mainly in one plane because the wells run in parallel slots. A skyline diffuser uses a grid of blocks at different heights, scattering in two dimensions but requiring more individual parts and more complex construction.

Key terms

• QRD (Quadratic Residue Diffuser) — a Schroeder diffuser whose well depths follow the sequence i² mod N, producing uniform hemispherical scattering within the design band.
• Well depth — depth_i = (c / (2 × f × N)) × (i² mod N); sets the phase delay for each well at the design frequency.
• Design frequency (f_low) — the lowest frequency the diffuser scatters effectively; deeper wells extend this lower.
• Well count (N) — must be a prime number; determines the length of the sequence before it repeats and the number of wells per period.
• Upper frequency limit — approximately c / (2 × well width); above this, the wavelength is too short for the well geometry to produce coherent scattering.
• Minimum listening distance — the distance at which the scattered wavefront has fully formed; roughly 2–3 times the total panel width.

Frequently asked questions

• What is a QRD diffuser? A quadratic residue diffuser (QRD) is a panel made of parallel wells with depths that follow a quadratic residue sequence. The varying depths introduce time delays that scatter reflected energy in multiple directions, reducing discrete echoes and flutter while preserving acoustic energy in the room — unlike an absorber, which removes energy.
• How is QRD well depth calculated? Each well depth is proportional to (i² mod N) × (c / (2 × f × N)), where i is the well index, N is the number of wells (a prime number), f is the design frequency, and c is the speed of sound. This sequence ensures the scattered energy is distributed uniformly across angles within the design band.
• What design frequency should I choose for a QRD diffuser? Choose the lowest frequency you want to scatter. Typical studio applications target 500 Hz–1 kHz for rear wall diffusion or 300–700 Hz for side walls. Lower design frequencies require deeper wells — targeting 200 Hz results in wells over 400 mm deep, which is impractical for most DIY builds.
• Where should QRD diffusers be placed in a room? The rear wall behind the listening position and the side walls at the first reflection point are the most common locations. Do not place diffusers directly behind speakers — at close range the scattered wavefront has not formed properly. Always ensure the listening position is at or beyond the minimum listening distance output by this calculator.
• How many wells should a QRD diffuser have? Well count must be a prime number: 5, 7, 11, 13, or 17 are typical. A 7-well panel is the most common starting point for studio use — it provides good angular distribution without requiring an overly wide panel. Higher prime counts give finer scattering but increase build complexity and panel width.
• What is the minimum listening distance for a diffuser? The minimum listening distance is typically the total panel width multiplied by 2–3. Below this distance, the scattered wavefront has not had room to develop and the panel behaves more like a reflective surface. This calculator outputs a rule-of-thumb minimum based on total panel width and design frequency.