Hemisphärische Diffusion

2D-QRD-Diffusor-Design Zweidimensionaler quadratischer Rest

Zweidimensionale Quadratische-Rest-Diffusoren erweitern den bewährten QRD-Algorithmus zu einem 2D-Gittermuster und erzeugen hemisphärische Streuung, die Schall omnidirektional über horizontale und vertikale Ebenen verteilt.

What is a 2D QRD Diffuser?

Omnidirectional Scattering

While traditional 1D QRDs scatter sound in one plane (typically horizontal), 2D QRDs use a grid-based pattern that scatters sound equally in all directions, horizontal, vertical, and every angle in between.

The formula r(x,y) = (x² + y²) mod N generates a checkerboard-like pattern of varying well depths across a 2D grid. Each cell's depth is determined by its coordinates, creating a mathematically optimized surface that redirects sound energy uniformly across a hemisphere.

Grid-Based Design

A 2D QRD consists of an N×N grid (or N×M for rectangular variants), where N is typically a prime number like 7, 11, or 13. Each grid cell is a well with a specific depth calculated from the quadratic residue formula.

The 2D extension maintains the excellent mathematical properties of 1D QRDs, low autocorrelation, uniform energy distribution, and predictable performance, while adding the critical benefit of multidirectional scattering. This makes them ideal for corner treatments and ceiling applications.

Advanced Fabrication

2D QRDs are more complex to manufacture than 1D designs, typically requiring 3-axis CNC milling or modular construction with multiple depth-specific blocks. The 3D surface topology demands precise machining and careful assembly.

Use the Diffuser Designer preview to explore 2D patterns and plan overall geometry. Downloads and build outputs will be linked on the main Diffuser Designer page when available.

Anwendungsfälle

Room Corners & Tri-Corners. Useful for corner placements where sound arrives from multiple angles, scattering mid/high-frequency reflections and imaging artifacts omnidirectionally. Note: low-frequency modal buildup at tri-corners (wall-wall-ceiling junctions) is a separate problem that requires bass trapping, not diffusion.
Ceiling Applications. Excellent for ceiling-mounted diffusion where sound needs to scatter in all horizontal directions. Unlike 1D diffusers which create directional bias, 2D QRDs provide uniform coverage across the entire room from a single ceiling panel.
Critical Listening Positions. Ideal for placement directly behind or above critical listening positions (mix engineer, conductor podium). The hemispherical scattering ensures the listener receives diffuse energy from all directions without directional coloration.
Small Room Optimization. In compact spaces where wall area is limited, 2D diffusers maximize scattering efficiency per square meter. A single 1m² 2D panel can outperform multiple 1D panels due to its multidirectional coverage.
Focal Point Treatment. When treating specific focal points like rear walls behind speakers or reflection points at first-reflection zones, 2D QRDs ensure the scattered energy disperses evenly throughout the room rather than creating new reflection patterns.
Hybrid Acoustic Designs. Combine 2D QRDs with 1D diffusers for comprehensive treatment: use 2D panels at critical nodes (corners, ceiling) and 1D panels for wall coverage. This hybrid approach optimizes both coverage and cost-effectiveness.

Vorteile

Omnidirectional Scattering: Scatters sound uniformly in all directions (hemispherical pattern), unlike 1D diffusers which are directionally biased.
Superior Corner Performance: Ideal for room corners and tri-corners where sound approaches from multiple angles. Addresses all planes simultaneously.
Ceiling Optimized: Perfect for ceiling mounting where scattering must cover the entire room below without directional preference.
Mathematically Optimized: Inherits the proven QRD mathematical foundation with excellent autocorrelation properties, now extended to two dimensions.
Space Efficient: Provides maximum scattering per unit area. One 2D panel can replace multiple strategically angled 1D panels in some applications.
Predictable Performance: Well-established theory and numerous published studies validate performance characteristics and design guidelines.
Aesthetic Impact: The 3D checkerboard pattern creates a striking visual appearance that can serve as both acoustic treatment and architectural feature.
Scalable Grids: Available in multiple grid sizes (7×7, 11×11, 13×13, etc.) to match different room sizes, frequency requirements, and fabrication capabilities.

Überlegungen

Complex Fabrication: Requires 3-axis CNC milling or intricate modular assembly. Significantly more difficult to manufacture than 1D diffusers, increasing cost and lead time.
Higher Cost: Machining time, material usage, and complexity result in 2-3× higher fabrication costs compared to equivalent 1D QRD panels.
Large Surface Area Required: A 7×7 grid with 8cm elements requires a 56×56cm panel minimum. Larger grids (11×11, 13×13) demand substantial wall or ceiling space.
Weight Considerations: Deep 2D patterns require thick material stock, resulting in heavy panels (10-30kg depending on size). Ceiling mounting requires robust structural support.
Limited Bandwidth: Typically effective across ~2.5 octaves (e.g., 600Hz-3kHz), narrower than some 1D designs or MLS diffusers. Design frequency must be carefully chosen.
Not DIY-Friendly: Difficult for home builders without access to CNC equipment. Modular approaches are possible but require precise fabrication of many unique block sizes.
Installation Complexity: Heavy, bulky panels require professional installation, especially for ceiling mounting. Alignment and secure attachment are critical for safety.
Limited Low-Frequency: Like all diffusers, performance below 400-600Hz is limited by practical depth constraints. Low-frequency room modes still require bass trapping.

Designparameter

Grid Size (N). The grid dimensions determine the number of unique well depths and the overall panel size. Use prime numbers for optimal scattering:7×7: 49 wells, compact design, good for small spaces11×11: 121 wells, balanced performance, most common13×13: 169 wells, maximum variation, large roomsRectangular: 7×11, 11×13 for asymmetric coverage
Design Frequency (f₀). Sets the center frequency and determines maximum well depth, as a planning estimate. For 2D diffusers, typically lower than 1D equivalents:800Hz: d_max ≈ 21cm (deep for this format, larger panel)1000Hz: d_max ≈ 17cm (standard, balanced)1200Hz: d_max ≈ 14cm (shallower, easier mounting)1500Hz: d_max ≈ 11cm (minimal depth, ceiling-friendly)
Element Size (Cell Width). Each grid cell's width mainly constrains the high-frequency limit and overall panel dimensions, as planning estimates:5cm: ~3.5kHz planning limit, 11×11 = 55cm panel7cm: ~2.5kHz planning limit, 11×11 = 77cm panel8cm: ~2.1kHz planning limit, 11×11 = 88cm panel10cm: ~1.7kHz planning limit, 11×11 = 110cm panel
Well Depth Distribution. The quadratic residue formula produces N unique depths (for N=7, depths range from 0 to 6 units):Depth Scale: d_i = (i / N) × d_maxFor N=7, d_max=17cm: depths are 0, 2.4, 4.9, 7.3, 9.7, 12.1, 14.6, 17cmFor N=11, d_max=17cm: 11 unique depths from 0 to 17cm
Modulus Selection. The modulus N should be a prime number for good autocorrelation properties. Common choices:N=7: 7 unique depths, smallest practical 2D QRDN=11: 11 unique depths, widely used in practiceN=13: 13 unique depths, maximum complexityN=17: 17 depths, research/specialty applications
Panel Thickness & Weight. Total material thickness must accommodate the deepest well plus backing structure:Material Thickness: d_max + 2cm minimum backingFor d_max=17cm: 19cm thick material requiredWeight: MDF/plywood at 650kg/m³, 11×11 at 8cm cells ≈ 15-25kg

How to Design a 2D QRD Diffuser

Choose your placement and size. Decide where the diffuser will go and what footprint you can accommodate (width × height).
Select grid dimensions. Choose an N×N grid size. Larger grids provide more spatial variation but require more surface area and fabrication effort.
Choose maximum depth. Set the maximum well depth based on the lowest target frequency and your practical depth constraints.
Set element (cell) size. Choose cell width and the overall panel size to balance build complexity and desired high-frequency behavior.
Build and install. 2D diffusers can be fabrication-intensive. Use rigid materials, keep tolerances consistent, and install securely at the planned location.

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