Benchmark Case Study: Comparing 3D Voxel Modal Simulation Predictions Against REW (Room EQ Wizard) Measurements

1. Overview & Objective

This case study evaluates the predictive fidelity of a web-based 3D Voxel Finite-Difference Helmholtz eigensolver against physical acoustic measurements gathered using a UMIK-1 calibrated microphone and Room EQ Wizard (REW).

The goal is to determine the accuracy of low-frequency ($20\text{ Hz} - 200\text{ Hz}$) modal frequency identification, standing wave pressure distributions, and reverberation decay ($T_{60}$) estimations in small listening environments under real-world boundary conditions.

2. Test Environment Setup & Boundary Conditions

Room Geometry & Construction

• Dimensions: $L = 5.24\text{ m}$, $W = 3.82\text{ m}$, $H = 2.45\text{ m}$ (Volume $V \approx 49.0\text{ m}^3$).
• Construction: Double-layer drywall over wooden studs (walls/ceiling), carpet over concrete slab (floor). Single door at rear right corner, standard double-pane glass window on left wall.
• Calculated Schroeder Cutoff Frequency ($f_s$): $f_s \approx 2000 \sqrt{\frac{T_{60}}{V}} = 2000 \sqrt{\frac{0.42}{49.0}} \approx 185\text{ Hz}$

Measurement Rig

• Microphone: MiniDSP UMIK-1 (individually calibrated 0° text file loaded).
• Audio Interface & Output: RME Babyface Pro FS, driving a pair of Neumann KH 120 A active studio monitors.
• Sweep Parameters: REW 5.30 log sine sweep ($20\text{ Hz} - 20,000\text{ Hz}$, $256\text{k}$ length, $85\text{ dB SPL}$ reference level at listening position).

Simulation Parameters (roomtreatment.diy Engine)

• Grid Discretization: 3D uniform spatial voxel grid pitch $h = 0.18\text{ m}$ ($N_{\text{AIR}} = 7,420$ interior air cells).
• Boundary Condition: Robin impedance mapping for drywall ($Z_{\text{wall}} \approx 8.5 \rho_0 c$) and concrete slab ($Z_{\text{floor}} \approx 50 \rho_0 c$).
• Eigensolver: SciPy scipy.sparse.linalg.eigsh (Shift-and-Invert Arnoldi/Lanczos, $\sigma = 10^{-5}$, $M = 45$ eigenmodes computed).

3. Empirical Results: Simulated vs. Measured Modal Frequencies

The table below compares the calculated room mode frequencies from the 3D Voxel PDE solver against the physical resonant peaks identified via REW high-resolution FFT measurement at the listening chair ($x = 1.91\text{ m}, y = 2.00\text{ m}, z = 1.15\text{ m}$).

Note: Minor down-shifts in measured real-world modal frequencies relative to pure analytical shoebox numbers are attributed to wall compliance (drywall flexing at low frequencies increases effective acoustic volume).

4. Frequency Response & Null Alignment

Below $100\text{ Hz}$, boundary-influenced phase cancellation produces sharp notches in spatial SPL.

In this test setup, the primary left speaker placement ($x = 1.10\text{ m}, y = 1.00\text{ m}, z = 1.20\text{ m}$) created a measured Speaker-Boundary Interference Response (SBIR) null at 58 Hz dropping -18.5 dB below baseline.

• REW Measured Null: $58.2\text{ Hz}$ (Depth: $-18.5\text{ dB}$)
• Simulated PDE Null (roomtreatment.diy): $57.8\text{ Hz}$ (Depth: $-16.8\text{ dB}$)
• Predictive Frequency Delta: $\mathbf{0.4\text{ Hz}}$ alignment.

5. First-Order Perturbation Theory vs. Post-Treatment REW Decay

Four 100mm porous fiberglass panels ($1.2\text{ m} \times 0.6\text{ m}$, flow resistivity $\sigma \approx 16,000\text{ Pa}\cdot\text{s/m}^2$) were installed at front wall-ceiling dihedral junctions.

Instead of full matrix re-computation, the simulation updated modal decay rates using 1st-order surface integral perturbation:

$d_{\text{diss}, m}^{(p)} = \text{Re}(Y_p(f_m)) \frac{c \cdot h^2}{V} \sum_{k \in \text{panel}} \phi_m^2(k)$

$1/3$-Octave Low-Frequency $T_{60}$ Comparison:

• 63 Hz Octave Band:

  • Bare Room (Measured): $0.58\text{ s}$ | Bare Room (Simulated): $0.56\text{ s}$
  • Treated Room (Measured REW): $0.41\text{ s}$ | Treated Room (Simulated): $0.43\text{ s}$
  • Delta: $\mathbf{+0.02\text{ s}}$

• 125 Hz Octave Band:

  • Bare Room (Measured): $0.46\text{ s}$ | Bare Room (Simulated): $0.45\text{ s}$
  • Treated Room (Measured REW): $0.29\text{ s}$ | Treated Room (Simulated): $0.31\text{ s}$
  • Delta: $\mathbf{+0.02\text{ s}}$

6. Limitations & Edge Cases

1. Flexible Boundaries: Low-frequency membrane flexing in thin drywall non-rigidly alters acoustic impedance, causing $\sim 1\text{ Hz}$ downward modal shifts not fully captured by static Robin boundary coefficients.
2. High-Frequency Transition: Above $200\text{ Hz}$, the solver transitions from wave PDEs to Image Source Method (ISM) and stochastic ray tracing. Diffusion coefficients for highly textured surfaces require manual approximation.

7. Community Feedback Invitation

We developed this numerical solver pipeline into an open web tool at roomtreatment.diy to make fast 3D acoustic simulation accessible directly in the browser without requiring desktop CAD/FEA installs.

We welcome feedback from the Audio Science Review community regarding boundary admittance assumptions, perturbation error limits, and comparison against your own REW impulse response measurements.