ISO 9613-2 is the international standard for calculating outdoor sound propagation. It accounts for geometric divergence, atmospheric absorption, ground effects, barrier diffraction, and other attenuation factors. GeoNoise implements the core ISO 9613-2 attenuation terms (geometric divergence, atmospheric absorption, ground effect, barrier diffraction), with documented approximations for thick-barrier and multi-edge geometries. Miscellaneous attenuation, long-term meteorological correction, and source directivity are not modeled (see in-app About → Limitations).

How does barrier diffraction work?

Barrier diffraction uses the Maekawa formula: A_bar = 10·log₁₀(3 + 20N) where N is the Fresnel number. Sound bends over and around barriers, with higher frequencies attenuated more than lower frequencies. GeoNoise calculates both over-top and side diffraction paths. Thick-barrier and multi-edge cases use documented approximations (see in-app About → Limitations).

What is the ground dip phenomenon?

The ground dip is a frequency-dependent reduction in sound level caused by destructive interference between direct and ground-reflected sound waves. It typically occurs around 200-500 Hz depending on geometry. GeoNoise models this using coherent two-ray phasor summation.

What is the Delany-Bazley model?

The Delany-Bazley model calculates ground surface impedance from flow resistivity: Z = 1 + 9.08(f/σ)^(-0.75) - j·11.9(f/σ)^(-0.73). It's used to determine how much sound is reflected vs absorbed by the ground. GeoNoise supports both Delany-Bazley and the Miki extension.

GeoNoise - Environmental Noise Modeling & Acoustic Propagation Software

GeoNoise implements ISO 9613-based outdoor sound propagation with coherent phasor summation for multi-path interference modeling. All calculations are performed per octave band (63 Hz – 16 kHz). A binaural listener with analytical HRTF (head-related transfer function) provides per-ear spectral readout — ITD and ILD — for spatial hearing analysis. Press L to add a binaural listener.

L_p(f) = L_W(f) - A_{div} - A_{atm}(f) - A_{gr}(f) - A_{bar}(f)

geometric divergence (A_div) Spherical spreading: $A = 20\log_{10}(d) + 10\log_{10}(4\pi)$

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Point source spherical spreading models how sound energy spreads over an expanding sphere.

A_{div} = 20\log_{10}(d) + 10\log_{10}(4\pi)

where d = 3D distance (m), 10·log₁₀(4π) ≈ 10.99 dB for full spherical radiation.

Derivation: Power flux Φ = W/(4πr²), therefore L_p = L_W - 10·log₁₀(4πr²)

Cylindrical spreading (optional propagation setting): applies the line-source divergence law to all sources — energy spreads over a cylinder's circumference rather than a sphere's surface area. Formula: A_div = 10·log₁₀(d) + 10·log₁₀(2π), where 10·log₁₀(2π) ≈ 7.98 dB. All GeoNoise sources are point emitters; this setting changes only the divergence law.

atmospheric absorption (A_atm) ISO 9613-1: $A_{atm} = \alpha(f) \cdot d / 1000$

ISO 9613-1

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Air absorbs sound energy through viscous, thermal, and molecular relaxation processes.

A_{atm} = \alpha(f) \cdot \frac{d}{1000} \text{ dB}

ISO 9613-1: Calculates α from temperature, humidity, and pressure.

α = α_cl + (b_N·f²)/(f_r,N + f²/f_r,N) + (b_O·f²)/(f_r,O + f²/f_r,O)

The coefficient $\alpha$ combines classical losses with N₂ and O₂ molecular relaxation.

Simple: Uses the lookup table below (20°C, 50% RH baseline) with temperature/humidity correction factors applied.

63 Hz	125 Hz	250 Hz	500 Hz	1 kHz	2 kHz	4 kHz	8 kHz	16 kHz
0.1	0.3	1.0	2.0	4.0	8.0	20	60	200

Values in dB/km — the engine's approximation table; the full ISO 9613-1 calculation is the "ISO 9613-1" mode above

ground effect (A_gr) $A_{gr} = A_s + A_r + A_m$ · Table 3 exponentials or two-ray phasor

ISO 9613-2

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Ground effect accounts for sound interaction with the ground surface. GeoNoise offers two models:

ISO 9613-2 §7.3.1 Table 3 (ground effect):

Divides the path into source, middle, and receiver regions. Each region uses height-dependent exponential functions per octave band.

  S ──────────────────────────────────────── R
  │←── As ──→│←────── Am ────────→│←── Ar ──→│
  │  source  │      middle        │ receiver │
  └──────────┴────────────────────┴──────────┘

A_{gr}(f) = A_s(f) + A_r(f) + A_m(f)

A_s, A_r per band (use h=h_s or h_r; d_p = projected source–receiver distance):

Band	Formula
63 Hz	$A = -1.5$ (fixed)
125 Hz	$A = -1.5 + G \cdot a'(h, d_p)$
250 Hz	$A = -1.5 + G \cdot b'(h, d_p)$
500 Hz	$A = -1.5 + G \cdot c'(h, d_p)$
1 kHz	$A = -1.5 + G \cdot d'(h, d_p)$
2–8 kHz	$A = -1.5(1-G)$

where the Table 3 functions are:

a'(h,d_p) = 1.5 + 3.0·e^-0.12(h-5)²·(1-e^-d_p/50) + 5.7·e^-0.09h²·(1-e^{-2.8×10⁻⁶d_p²})

b' = 1.5 + 8.6·e^-0.09h²·(1-e^-d_p/50) · c' = 1.5 + 14.0·e^-0.46h²·(…) · d' = 1.5 + 5.0·e^-0.9h²·(…)

A_m: 63 Hz → $-3q$ · 125–8 kHz → $-3q(1-G_m)$ · where $q = \max(0, 1 - 30(h_s+h_r)/d_p)$

Two-Ray Phasor Model (Physically Accurate):

Models coherent interference between direct and ground-reflected paths:

A_{gr} = -20\log_{10}\left|1 + \Gamma \cdot \frac{r_1}{r_2} \cdot e^{-jk(r_2 - r_1)}\right|

Γ via Delany-Bazley impedance: Z_n = 1 + 9.08(f/σ)^-0.75 − j·11.9(f/σ)^-0.73 (Miki extension for f/σ ≥ 1; the grid engine treats f/σ < 0.01 as near-rigid, Z_n = 100 — the operative branch for low bands over soft/mixed ground). Captures constructive/destructive interference (ground dip).

Model Comparison:

Aspect	ISO 9613-2	Two-Ray Phasor
Standards	ISO 9613-2 methods	Physics-based
Frequency	Exponential h/d functions	Wavelength-based phase
Interference	Empirical (Table 3)	Full wave interference
Ground dip	Not modeled	Naturally captured

ISO 9613-2:2024 Corrections:

The 2024 revision adds K_geo geometric correction (§7.3) to scale A_gr down at short distances where source and receiver regions overlap, and barrier ground cancellation (§7.4) to zero positive A_gr for barrier-blocked paths. See the ISO 9613-2:2024 revisions section below.

sommerfeld correction (probe) Spherical wave: $Q = \Gamma + (1-\Gamma)\,F(\nu)$

Weideman

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Corrects the plane-wave Fresnel coefficient for spherical wave spreading near the ground. Transitions from Q → 1 (near-field) to Q → Γ_plane (far-field).

Q = \Gamma_{\text{plane}} + (1 - \Gamma_{\text{plane}})\,F(\nu)

Numerical distance (complex):

\nu = \frac{1+i}{2}\sqrt{k\,r_2}\left(\cos\theta + \frac{1}{Z_n}\right)

Boundary loss factor via Faddeeva function:

F(\nu) = 1 + i\sqrt{\pi}\,\nu\;\text{wofz}(\nu) \qquad \text{wofz}(z) = e^{-z^2}\,\text{erfc}(-iz)

wofz computed via Weideman (1994) N=32 rational approximation — O(1), stable across the entire complex plane. No Taylor series or region switching.

Note: The grid engine uses a simplified asymptotic approximation (valid for |w| ≥ 4). The full Weideman N=32 computation is used in the probe engine only.

Limits: ν → 0 ⇒ F → 1 ⇒ Q → 1 (total reflection) · |ν| → ∞ ⇒ F → 0 ⇒ Q → Γ_plane · Passive surfaces: |Q| ≤ 1 enforced.

barrier diffraction Maekawa: $A_{bar} = 10\log_{10}(3 + 20N)$

Maekawa

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Sound diffracts over and around barriers. Attenuation depends on the path difference δ.

Over-Top Diffraction (vertical):

    S                    R
     \                  /
      \   δ = A+B-d   /
       A ──→ ● ←── B
             │
      ═══════╧══════   Barrier

\delta = A + B - d \quad | \quad N = \frac{2\delta f}{c}

Insertion loss:

A_{bar} = 10\log_{10}(3 + 20N) \;\text{thin, cap 20 dB}; \quad 10\log_{10}(3 + 40N) \;\text{thick, cap 25 dB}

Side Diffraction (horizontal):

Sound can also bend around the ends of finite-length barriers:

      ════════════════  Barrier (top view)
     ●                  ●
   Left               Right
   Edge               Edge
     ↑                  ↑
  S ─┘                  └─ R
(around left)      (around right)

Path difference for side: $\delta = |S \to Edge| + |Edge \to R| - |S \to R|$

The path with minimum δ (over-top or around sides) determines the effective attenuation.

Side Diffraction Toggle:

Mode	Behavior
Off	ISO 9613-2 infinite barrier assumption. Only over-top diffraction.
Auto	Enable side diffraction for barriers < 50m (recommended default).
On	Always compute side diffraction for all barriers.

Low frequencies diffract easily → less attenuation. High frequencies are blocked effectively.

ISO 9613-2:2024 Alternative:

The 2024 revision replaces the Maekawa formula with D_z = 10·log₁₀(3 + C₂/λ·C₃·δ·K_met), which includes explicit meteorological correction for wind/temperature gradient effects on barrier effectiveness. See the ISO 9613-2:2024 revisions section below.

ISO 9613-2:2024 revisions K_geo correction, barrier cancellation, revised D_z + K_met

ISO 9613-2:2024

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The 2024 revision (ISO 9613-2:2024 — "Engineering method") supersedes ISO 9613-2:1996 with three corrections to the ground effect and barrier diffraction models. All three are optional in GeoNoise — select the ISO 9613-2:2024 calculation profile or enable individually.

1. K_geo Geometric Correction (§7.3)

At short source–receiver distances, the three-region ground model (A_s + A_r + A_m) becomes geometrically invalid because source and receiver regions overlap. K_geo reduces A_gr in this regime:

K_{geo} = \frac{d_p^2 + (h_s - h_r)^2}{d_p^2 + (h_s + h_r)^2}

where d_p = horizontal distance [m], h_s = source height [m], h_r = receiver height [m]. Applied as: A_gr,2024 = K_geo × A_gr.

Behavior: K_geo → 1.0 for d_p >> h_s + h_r (1996 and 2024 agree at distance). Significant only below ~20 m. Dimensionless: [m²]/[m²].

d_p	h_s	h_r	K_geo	Effect
100 m	1 m	1.5 m	0.999	<0.1% — negligible
20 m	1 m	1.5 m	0.985	~2% reduction
5 m	1 m	1.5 m	0.808	~19% reduction

2. Barrier Ground-Effect Cancellation (§7.4)

When a barrier blocks the direct path, the ground reflection geometry is disrupted. The 2024 revision cancels positive ground attenuation for blocked paths:

A_{gr} = 0 \quad \text{when } A_{gr} > 0 \text{ and path is barrier-blocked}

This is the conservative choice — it predicts higher noise levels at the receiver. Ground boosts (A_gr < 0 from constructive interference) are preserved.

Algebraically: the standard writes A_bar = D_z − A_gr. In the total sum A_div + A_atm + A_gr + A_bar, the A_gr terms cancel. GeoNoise implements this equivalently by setting A_gr = 0.

3. Revised Barrier D_z + K_met (§7.4)

Replaces the 1996 Maekawa formula with a revised insertion loss that includes explicit meteorological correction:

D_z = 10\log_{10}\!\left(3 + \frac{C_2}{\lambda}\,C_3\,\delta\,K_{met}\right)

Condition: δ > z_min. If δ ≤ z_min, D_z = 0.

z_{min} = -\frac{2\lambda}{C_2 \cdot C_3}

where C₂ = 20 (thin/single-edge) or 40 (thick/building), C₃ = 1 (single diffraction), λ = c/f [m], δ = path difference [m].

Meteorological correction:

K_{met} = \exp\!\left(-\frac{1}{2000}\sqrt{\frac{(\max(d_{ss},d_{sr})+e)\cdot\min(d_{ss},d_{sr})\cdot d}{2(\delta - z_{min})}}\right)

where d_ss, d_sr = source/receiver to diffraction edge [m], e = edge separation [m] (0 for thin barriers), d = total distance [m]. For lateral (horizontal) diffraction: K_met = 1.

Why revised: The 1996 K_met over-corrected at large distances with low barriers, predicting near-zero barrier attenuation that contradicted field measurements. The 2024 formula uses (δ − z_min) in the denominator and max/min ordering to prevent this.

D_z produces ~2–3 dB less attenuation than 1996 Maekawa for the same geometry, consistent with the revision intent. Caps: 20 dB (thin), 25 dB (thick).

Dimensional Verification:

K_geo: [m²]/[m²] = dimensionless ✓
z_min: [m]/dimensionless = [m] ✓
K_met exponent: [1/m]·√([m³]/[m]) = [1/m]·[m] = dimensionless ✓
D_z inner: dimensionless + [1/m]·[m]·dimensionless = dimensionless ✓

wall reflections Image source method: $|\Gamma| \approx 0.9$

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Reflections from building walls computed using virtual image sources:

         Wall
           │
   S ──────●────── R
    \      │      /
     \     │     /
      S'   │   (S' = mirror of S across wall)

|\Gamma| \approx 0.9 \text{ (pressure amplitude) per reflection} \;\; (\text{energy absorption } \alpha \approx 0.19)

Reflections are validated for line-of-sight and checked against other buildings.

coherent phasor summation $p_{total} = \sum_i p_i \cdot e^{j\phi_i}$

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Multiple paths from the same source are summed with phase (capturing interference):

p_{total} = \sum_i p_i \cdot e^{j\phi_i} \quad \text{where } \phi_i = -k \cdot d_i + \phi_{reflection}

This captures:

Constructive interference (paths in-phase → +6 dB)
Destructive interference (paths out-of-phase → deep nulls)
Ground dip phenomenon at specific frequencies
Comb filtering from wall reflections

Multiple sources are summed energetically (incoherent):

L_{p,total} = 10\log_{10}\left[ \sum_j 10^{L_{p,j}/10} \right]

source templates ASHRAE · BS 5228 · FHWA RCNM

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GeoNoise includes a library of 40 pre-defined noise source templates based on standardized emission data from ASHRAE Handbook, BS 5228, and FHWA RCNM.

Categories:

Industrial (15) — compressors, pumps, generators, fans, transformers
HVAC (10) — rooftop units, chillers, AHU, heat pumps, VRF
Construction (10) — excavators, pile drivers, jackhammers, bulldozers
Generic (5) — white/pink noise, speech, tonal alarm, subwoofer

Data sourced from ASHRAE Handbook, BS 5228, and FHWA RCNM.

accuracy & limitations Expected precision and model assumptions

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Accuracy pending measurements and validation

Known Limitations:

Flat terrain assumed (no topography effects)
Neutral atmosphere (no temperature/wind refraction)
First-order wall reflections only (higher orders not yet implemented; the reflection-order selector is hidden until they are)
2.5D geometry (vertical plane diffraction)
Homogeneous ground (single type per scene)
All sources are omnidirectional (no directivity pattern support yet)
Miscellaneous attenuation (A_misc — foliage, industrial sites, dense housing) is not modeled
Long-term meteorological correction (C_met) is not applied; output is short-term downwind propagation
One dominant obstacle per path in the grid engine (worst-case max δ; buildings use a two-edge thick-barrier form). Cascaded barriers are not combined per ISO 9613-2 §7.4 multiple-screen — see the Limitations tab.
Moving source animation caps: 3 entities (CPU), 16 entities (GPU)
Quasi-static motion: physics is recomputed from frozen geometry at each sample — no waveform-level pitch bend. The optional Doppler setting applies an octave-band spectral shift from the 3D radial closing speed to probe, binaural, and flight exports; the noise-map colormap stays quasi-static (typically ≲0.3 dB, worst-case ~0.5 dB head-on at drone speeds).
Grid engine Sommerfeld correction uses asymptotic approximation (|w| ≥ 4 only)
Ground-reflection coherence: GPU heatmap matches CPU within engineering tolerance as of v0.11.0 — wave-cancellation null cells may show small (<0.5 dB) numerical differences due to GPU floating-point precision. Most cells match within a hundredth of a decibel.

ISO 9613-2:2024 Support:

K_geo geometric correction (§7.3) — optional
Barrier ground-effect cancellation (§7.4) — optional
Revised D_z + K_met barrier formula (§7.4) — optional
All default OFF for backward compatibility with 1996

Numerical Guards:

Distance floor: d_min = 0.1m to prevent division by zero
Angle limits: cos θ clamped for grazing incidence
Impedance cap: |Z| bounded to avoid runaway coefficients
Level floor: -100 dB (acoustic silence threshold)

Map Integration:

Real-world map backgrounds via Mapbox GL JS with synchronized pan/zoom

references Standards and academic sources

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ISO 9613-1:1993 — Acoustics — Attenuation of sound during propagation outdoors — Part 1: Calculation of absorption by atmosphere
ISO 9613-2:1996 — Acoustics — Attenuation of sound during propagation outdoors — Part 2: General method of calculation
Maekawa, Z. (1968). "Noise reduction by screens." Applied Acoustics, 1(3), 157-173
Delany, M.E. & Bazley, E.N. (1970). "Acoustical properties of fibrous absorbent materials." Applied Acoustics, 3(2), 105-116
Pierce, A.D. (1981). "Acoustics: An Introduction to Its Physical Principles and Applications." McGraw-Hill
Chien, C.F. & Soroka, W.W. (1975). "Sound propagation along an impedance plane." JASA, 58(6). Defines Q = Γ + (1−Γ)F(w) and the boundary-loss factor
Kearns, J.A. (1989). NASA report — asymptotic spherical-wave over impedance plane, numerical distance, grazing-incidence limits
Weideman, J.A.C. (1994). "Computation of the Complex Error Function." SIAM J. Numer. Anal., 31(5). N=32 rational approximation for wofz(z)
Poppe, G.P.M. & Wijers, C.M.J. (1990). "Algorithm 680: Evaluation of the Complex Error Function." ACM TOMS. Stable wofz(z) reference implementation
Taraldsen, G. (2005). "A note on reflection of spherical waves." JASA. Chien & Soroka lineage, Sommerfeld F context
Al Azah, M. & Chandler-Wilde, S.N. (2020). "Computation of the Complex Error Function using Modified Trapezoidal Rules." Integral representations + symmetry
Gautschi, W. (1969). NASA report — early wofz(z) computational reference for complex arguments
Miki, Y. (1990). "Acoustical properties of porous materials — Modifications of Delany-Bazley models." J. Acoust. Soc. Jpn., 11(1)
Shaw, E.A.G. (1974). "Transformation of sound pressure level from the free field to the eardrum in the horizontal plane." JASA, 56(6), 1848-1861. Measured head-related transfer functions showing 10-15 dB ILD at 6-8 kHz
Kuhn, G.F. (1977). "Model for the interaural time differences in the azimuthal plane." JASA, 62(1), 157-167. ILD increases linearly 1-4 kHz, plateaus at 10 dB
Blauert, J. (1997). "Spatial Hearing: The Psychophysics of Human Sound Localization." MIT Press. Standard reference for ILD of 8-10 dB at 4-8 kHz for rigid sphere model matching average human head
Woodworth, R.S. (1938). "Experimental Psychology." Holt. ITD formula for rigid sphere: ITD = (a/c)(theta + sin(theta)), basis for analytical HRTF model
Algazi, V.R. et al. (2001). "The CIPIC HRTF Database." IEEE WASPAA. Measured HRTF sets at 25 azimuths x 50 elevations for 45 subjects
ISO 9613-2:2024 — Acoustics — Attenuation of sound during propagation outdoors — Part 2: Engineering method for the prediction of sound pressure levels. Supersedes ISO 9613-2:1996. K_geo geometric correction (§7.3), barrier ground cancellation (§7.4), revised D_z + K_met (§7.4)

CORE LEVELS

Overall level at receiver:

L_p = L_W - A_div - A_atm - A_gr - A_bar

Energetic summation of multiple sources:

L_sum = 10 log₁₀( Σ 10^L_i/10 )

Sound power conversions:

L_W = 10 log₁₀(P / 10^-12)

P = 10^-12 · 10^L_W/10

GEOMETRY

Plan distance:

d_2D = √((x₂-x₁)² + (y₂-y₁)²)

3D distance:

d_3D = √((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²)

Direct path used for diffraction reference:

d_direct = √(d_2D² + (h_s - h_r)²)

SPREADING

Point source (spherical):

A_div = 20 log₁₀(d) + 10 log₁₀(4π)

Cylindrical option (line-source divergence law, global setting):

A_div = 10 log₁₀(d) + 10 log₁₀(2π)

ATMOSPHERIC ABSORPTION

Path loss:

A_atm = α · d / 1000

Where α is the atmospheric attenuation coefficient in dB/km and d is the distance in m

Speed of sound and wavelength:

c = 331.3 + 0.606 T_C

λ = c / f

ISO 9613-1 absorption coefficient:

α = 8.686 f² [ (1.84e-11 (p/p₀)^-1 (T/T₀)^1/2)

+ (T/T₀)^-2.5 (0.01275 e^-2239.1/T (f_r,O/(f_r,O² + f²))

+ 0.1068 e^-3352/T (f_r,N/(f_r,N² + f²))) ]

C = -6.8346 (T₀₁/T)^1.261 + 4.6151

p_sat = p₀ · 10^C, h = RH · p_sat / p

f_r,O = (p/p₀) [24 + 4.04e4 h ((0.02 + h)/(0.391 + h))]

f_r,N = (p/p₀) (T/T₀)^-1/2 [9 + 280 h e^{-4.17((T/T₀)^-1/3 - 1)}]

GROUND EFFECTS

ISO 9613-2 Tables 3-4 (per-band, primary model):

A_gr = A_s + A_r + A_m

A_s, A_r per band (Table 3 exponential functions):

63 Hz: A = −1.5 (fixed)

125 Hz: A = −1.5 + G·a'(h, d_p)

250 Hz: A = −1.5 + G·b'(h, d_p)

500 Hz: A = −1.5 + G·c'(h, d_p)

1 kHz: A = −1.5 + G·d'(h, d_p)

2–8 kHz: A = −1.5(1 − G)

a' = 1.5 + 3.0·e^{−0.12(h−5)²}·(1−e^−d_p/50) + 5.7·e^−0.09h²·(1−e^{−2.8×10⁻⁶d_p²})

A_m (63 Hz) = −3q

A_m (125–8 kHz) = −3q(1 − G_m)

q = max(0, 1 − 30(h_s+h_r)/d_p)

G = ground factor (0=hard, 0.5=mixed, 1=soft)

Mixed-ground σ: the grid engine interpolates per the selected model (σ_soft/G or log-interpolation); the probe engine always log-interpolates between preset hard (2×10⁶) and soft (2×10⁴ Pa·s/m²) resistivities.

Two-ray phasor geometry:

r₁ = √(d² + (h_s - h_r)²), r₂ = √(d² + (h_s + h_r)²)

Normalized impedance — Delany-Bazley (0.01 < f/σ < 1.0):

ζ = 1 + 9.08 (f/σ)^-0.75 - j 11.9 (f/σ)^-0.73

Miki extension (f/σ > 1.0):

ζ = 1 + 5.50 (f/σ)^-0.632 - j 8.43 (f/σ)^-0.632

Rigid cap — grid engine (f/σ < 0.01, e.g. low bands over soft/mixed ground):

ζ = 100 (near-rigid surface, |Γ| → 1)

Reflection coefficient (locally reacting):

Γ = (ζ cosθ - 1) / (ζ cosθ + 1)

Two-ray ratio and attenuation:

ratio = 1 + Γ (r₁/r₂) e^{-jk(r₂ - r₁)}

A_gr = -20 log₁₀(max(|ratio|, 10^-6))

BARRIER DIFFRACTION

Thin barrier (single edge, cap 20 dB):

A = √(d(S,I)² + (h_b - h_s)²)

B = √(d(I,R)² + (h_b - h_r)²)

δ = A + B - d_direct

Thick barrier / building (double edge, cap 25 dB):

A = √(d(S,I_in)² + (h_b - h_s)²)

B = d(I_in, I_out) (roof width)

C = √(d(I_out,R)² + (h_b - h_r)²)

δ = A + B + C - d_direct

Maekawa-style insertion loss:

N = 2δ / λ

Thin: A_bar = 10 log₁₀(3 + 20N) cap 20 dB

Thick: A_bar = 10 log₁₀(3 + 40N) cap 25 dB

Thick-barrier formula is an approximation in the C₃≈2 limit — see Limitations tab above.

Blocked path total (ISO 9613-2 §7.4 additive):

A_total = A_div + A_atm(d_actual) + A_bar + A_gr

ISO 9613-2:2024 revised barrier (§7.4):

D_z = 10 log₁₀(3 + C₂/λ · C₃ · δ · K_met)

z_min = −2λ / (C₂ · C₃) — below this, D_z = 0

C₂ = 20 (thin), 40 (thick) C₃ = 1 (single diffraction)

C₃=1 is a strict-ISO under-prediction for thick (multi-edge) barriers — see Limitations tab above.

K_met meteorological correction:

K_met = exp(−(1/2000) · √((max(d_ss,d_sr)+e) · min(d_ss,d_sr) · d / (2(δ−z_min))))

Lateral diffraction: K_met = 1

ISO 9613-2:2024 ground corrections (§7.3):

K_geo = (d_p² + (h_s−h_r)²) / (d_p² + (h_s+h_r)²)

Barrier cancellation: A_gr = 0 when A_gr > 0 and barrier-blocked

WEIGHTING

Octave-band weighting:

L_A = L_band + A(f)

L_C = L_band + C(f)

MOVING SOURCES

Waypoint-based animation for sources, probes, and binaural listeners with real-time noise map recompute.

Animation System:

Waypoint tool (W key): Click-to-place waypoints on canvas — committed live; per-waypoint speed (m/s) and height (z) overrides in the movement drawer
Catmull-Rom spline: Smooth interpolation between waypoints
Dual-rate loop: 60 Hz visual rendering, adaptive physics recompute (20 Hz single entity, 5 Hz multi-entity CPU, 60 Hz GPU)
Transport controls: Play/pause, scrub, speed (m/s), loop, ping-pong, reverse, live Leq map averaging (cumulative ∞ or rolling LAeq,T)

Capacity:

CPU backend: max 3 concurrent animated entities
GPU backend: max 16 concurrent animated entities

Flight Exports:

Σ Flight Leq Map: exact offline pass — samples the whole flight and energy-averages the noise map
Flight Receiver CSV: per-receiver Leq / SEL / Lmax
Flight Binaural CSV: per-ear LAeq, signed ILD/ITD time series, per-ear 9-band spectra, per-ear SEL / LAmax, cadence metadata

Noise map, receivers, and probes recompute in real-time as entities move along their paths. Incremental recompute for single-source movement; full recompute for multi-source.

BINAURAL LISTENER

Virtual binaural listener with L/R ear separation and head-related transfer function modeling.

Head Geometry:

L/R ears: default 17.5 cm separation (editable), rotatable head azimuth
Dual probes: Independent coherent ray-tracing per ear
Placement: L key to place, drag to move, lollipop to rotate

HRTF Model (Analytical):

Per-path processing: Arrival angle computed for each traced path
Head shadow: Frequency-dependent attenuation — low frequencies pass through; attenuation grows with frequency up to a 15 dB cap above 4 kHz on the shadow side (Shaw 1974), scaled by cos(elevation) of the arriving path so the shadow vanishes for overhead arrivals
ITD/ILD readout: Signed Interaural Time Difference (+ = left ear leads) and A-weighted Interaural Level Difference (+ = left louder), shown as "(+L)" in the inspector

References: Shaw (1974), Kuhn (1977), Woodworth & Schlosberg (1954)

GeoNoise uses two calculation engines optimized for different use cases. Understanding when each is applied helps interpret results correctly.

GRID ENGINE

Used for: Mic Grids, Receivers, Noise Maps

Path Finding:

Single direct path per source-receiver pair
Checks for barrier occlusion → computes diffraction
Ground effect via A_gr model (ISO or Two-Ray)

Ground Effect Models:

ISO 9613-2: Empirical per-band coefficients (no interference)
Two-Ray: Phasor interference between direct + ground reflection

Summation:

L = 10·log₁₀(Σ 10^Lᵢ/10)

Energy (power) summation across sources.

Trade-off: Fast computation for heat maps. With Two-Ray enabled, captures ground dip. No wall reflections or multi-path beyond ground.

The grid engine supports WebGPU acceleration for real-time noise map computation at 60 Hz during animation. GPU↔CPU fallback is automatic — receivers and panels compute on CPU while grid maps leverage GPU parallel processing.

PROBE ENGINE

Used for: Probe microphones and binaural listeners (per-ear)

Path Finding (Ray Tracing):

Direct path
Ground-reflected path (specular reflection)
Wall reflections (first-order image sources)
Side diffraction around barrier edges
Over-top diffraction

Ground Effect Models:

Physical Impedance: Complex reflection coefficient from surface impedance
ISO 9613-2: Tabulated absorption coefficients (0.01–0.60)

Impedance Models (when Physical Impedance selected):

Delany-Bazley:

$Z_n = 1 + 9.08(f/\sigma)^{-0.75} - j \cdot 11.9(f/\sigma)^{-0.73}$

Delany-Bazley-Miki (piecewise hybrid, X = f/σ):

$Z_n = 1 + 9.08X^{-0.75} - j \cdot 11.9X^{-0.73} \; (X < 1); \quad 1 + 5.5X^{-0.632} - j \cdot 8.43X^{-0.632} \; (X \ge 1)$

Reflection Coefficient:

$\Gamma = \frac{Z_n \cos\theta - 1}{Z_n \cos\theta + 1}$

Summation:

p = Σ pᵢ·e^jφᵢ where φ = -k·d

Phasor (coherent) summation with phase from path length.

Trade-off: Full multi-path interference (ground dip, wall comb filtering, diffraction patterns). Slower, used only for point analysis.

CALCULATION ARCHITECTURE

┌─────────────────────────────────────────────────────────────────┐
│                      GeoNoise Calculation Flow                  │
├─────────────────────────────────────────────────────────────────┤
│                                                                 │
│  ┌──────────────┐              ┌──────────────┐                 │
│  │   Sources    │              │  Receivers   │                 │
│  │ (with Lw/f)  │              │    Grids     │                 │
│  └──────┬───────┘              │    Probes    │                 │
│         │                      └──────┬───────┘                 │
│         ▼                             │                         │
│  ┌──────────────────────────────────────────────────────────────┤
│  │                    PATH FINDING                              │
│  ├──────────────────────────────────────────────────────────────┤
│  │                                                              │
│  │   ┌─────────────────────┐    ┌─────────────────────────────┐ │
│  │   │   GRID ENGINE       │    │      PROBE ENGINE           │ │
│  │   │                     │    │                             │ │
│  │   │  • Single path      │    │  • Ray tracing (multi-path) │ │
│  │   │  • Direct + best    │    │  • Ground reflection rays   │ │
│  │   │    diffraction      │    │  • Wall reflection rays     │ │
│  │   │  • A_gr handles     │    │  • Side diffraction paths   │ │
│  │   │    ground effect    │    │  • Used for: Probes only    │ │
│  │   └─────────┬───────────┘    └─────────────┬───────────────┘ │
│  │             │                              │                 │
│  └─────────────┼──────────────────────────────┼─────────────────┤
│                ▼                              ▼                 │
│  ┌──────────────────────────────────────────────────────────────┤
│  │                 ATTENUATION (ISO 9613-2)                     │
│  ├──────────────────────────────────────────────────────────────┤
│  │   For each path, apply per-band:                             │
│  │   • A_div  (geometric spreading)                             │
│  │   • A_atm  (atmospheric absorption)                          │
│  │   • A_gr   (ground effect — ISO or Two-Ray phasor)           │
│  │   • A_bar  (barrier diffraction if blocked)                  │
│  └──────────────────────────────────────────────────────────────┤
│                              │                                  │
│                              ▼                                  │
│  ┌──────────────────────────────────────────────────────────────┤
│  │                    SUMMATION                                 │
│  ├──────────────────────────────────────────────────────────────┤
│  │                                                              │
│  │   ┌─────────────────────┐    ┌─────────────────────────────┐ │
│  │   │   INCOHERENT        │    │      COHERENT (Phasor)      │ │
│  │   │                     │    │                             │ │
│  │   │  L = 10·log₁₀(Σ10^  │    │  p = Σ pᵢ·e^(jφᵢ)          │ │
│  │   │          (Lᵢ/10))   │    │  L = 20·log₁₀|p|            │ │
│  │   │                     │    │                             │ │
│  │   │  • Power summation  │    │  • Phase-aware interference │ │
│  │   │  • Fast, stable     │    │  • Captures nulls & peaks   │ │
│  │   │  • Used by: Grids   │    │  • Used by: Probes          │ │
│  │   └─────────────────────┘    └─────────────────────────────┘ │
│  │                                                              │
│  └──────────────────────────────────────────────────────────────┘
└─────────────────────────────────────────────────────────────────┘

COMPARISON SUMMARY

Aspect	Grid Engine	Probe Engine
Path count	1 per source	Multiple (ray traced)
Ground effect	ISO or Two-Ray phasor	Physical Impedance or ISO 9613-2
Wall reflections	No	Yes (first-order)
Summation	Incoherent (power)	Coherent (phasor)
Ground interference	With Two-Ray only	Always (if enabled)
Multi-path interference	No (walls, etc.)	Yes (walls, diffraction)
Ground dip	With Two-Ray enabled	Naturally emerges
Performance	Fast (grid-parallel)	Slower (per-probe)
Output	Heatmap, single dB value	Frequency response curve

WHY TWO ENGINES?

Grids/Receivers need to compute thousands of points quickly for heat maps. Using multi-path ray tracing everywhere would be prohibitively slow. The ISO 9613-2 single-path approach is designed for this: it's an engineering standard optimized for outdoor prediction across large areas with acceptable accuracy.

Probes are point measurements where you want detailed frequency response. Ray tracing + coherent summation reveals interference effects invisible to incoherent methods: the ground dip phenomenon, wall reflection comb filtering, and barrier edge diffraction patterns.

Both engines share the same ISO 9613-2 attenuation functions for A_div, A_atm, A_gr, and A_bar. The difference is in how many paths are traced and how they're combined at the receiver.

GeoNoise's diffraction models are physically defensible for the most common use cases (single thin barriers, free-field, audio frequencies, audio screens up to ~15 m) but contain four documented approximations. Each is enumerated below with the drift bound, what it means for your results, and the planned closure — followed by feature-scope caveats that sit outside the diffraction path.

MULTI-EDGE CASCADE TREATMENT

Sound diffracting over two or more obstacles in series (e.g. a noise wall behind a hedge) is summed as independent single-edge paths, coherently (Epstein-Peterson-style, not the sequential screen-to-screen construction, in the probe engine; max(δ) heuristic in the grid engine). The strict acoustic prescription — ISO 9613-2:1996/2024 §7.4 multiple-screen treatment with CNOSSOS-EU equivalent-screen reduction — is not yet implemented. The underlying wave-diffraction theory of multi-edge cascade was originally developed for RF / microwave propagation (Deygout 1966, ITU-R P.526) and adapted in acoustic research; see PRIMARY REFERENCES below.

Drift bound: ~5–10 dB at low frequencies (250 Hz–1 kHz) for thrice-cascaded geometries. Single-barrier scenes are unaffected.

Closure: A future release will surface a user-toggleable “Diffraction model” selector in Advanced Settings. Default stays Maekawa single-edge per ISO 9613-2 §7.4. Opt-in modes target ISO 9613-2 §7.4 multiple-screen / CNOSSOS-EU equivalent-screen for the probe engine; full ISO 9613-2:2024 strict (with C₃(e/λ) coupling) is gated as Pro and requires building roof-depth input. The Deygout (RF) cascade is preserved as a research / cross-validation option.

THICK-BARRIER C₃ APPROXIMATION (e ≈ 3 m BAKED IN)

The thick-barrier formula A_bar = 10·log₁₀(3 + 40·N) capped at 25 dB is not a direct ISO 9613-2:1996 §7.4 thick-obstacle citation — strict ISO requires the building roof depth e as input and computes C₃ = (1 + (5λ/e)²) / (⅓ + (5λ/e)²) per band. The current implementation bakes a single e ≈ 3 m (the 1-kHz reference) into the constant 40 and ships it for all bands.

Drift bound: within ±2 dB of strict ISO for typical buildings (3–10 m roof depth, 250 Hz–4 kHz). Worst-case under-prediction at 4 kHz with 10 m roof: −1.7 dB. Still well inside the ISO 9613-2:1996 §9 stated ±3 dB(A) overall prediction accuracy, and comparable to the class-1 SLM tolerance per IEC 61672-1:2013 §5 (±1.5 dB at 1 kHz). Note that such approximation errors add to the overall prediction uncertainty rather than being absorbed by the ±3 dB envelope.

Closure: v0.11.0 surfaces the model knob; full ISO 9613-2:2024 D_z with C₃(e/λ) per band is gated behind a Pro “ISO 9613-2 strict” option that requires building roof depth as input (deferred to v0.12+).

GRID BACKEND max(δ) HEURISTIC

The grid (heatmap) compute backend selects the worst single-edge δ over all candidate barriers between source and receiver, then runs that single δ through barrierAttenuation(). This is the classic max(δ) heuristic — it does not implement the ISO 9613-2 §7.4 multiple-screen treatment (which would combine the multiple shadowing edges into an equivalent screen, per the CNOSSOS-EU equivalent-screen reduction).

Drift bound: probe and grid engines disagree by 1–4 dB for single-barrier scenes and can disagree by 5–8 dB for cascaded scenes. Heatmap users see the grid result; probe users see the phasor-sum result — probe is the recommended ground truth for compound-barrier compliance work today.

Closure: A future release ports the ISO 9613-2 §7.4 multiple-screen / CNOSSOS-EU equivalent-screen treatment into the grid backend (CPU + WGSL shader).

MAEKAWA DUPLICATE FUNCTION (DEPRECATION SEAM)

The probe engine ships two byte-identical functions: maekawaDiffraction() and singleEdgeDiffraction(). Both implement A_bar = 10·log₁₀(3 + 20·N) capped at 20 dB. Production callers were swapped to maekawaDiffraction on 2026-04-29; singleEdgeDiffraction is retained as a deprecation seam for test fixtures and downstream consumers.

Drift bound: none — functions are bit-identical and produce identical numerical output for all inputs.

Closure: v0.11.0 may delete after surveying any downstream consumers.

FACADE PROBE — THE “+3 dB (APPROX.)” REFLECTION TOGGLE

A facade probe samples the incident sound field by default — the reference condition of ISO 1996-2:2017 §9.2.1.2 b), location 1. The “+3 dB facade reflection (approx.)” toggle switches the reading to location 3 of the same clause: a microphone 0,5 m to 2 m in front of the reflecting surface, whose correction to free field is 3 dB when the Annex B.5 criteria are met. The offset is applied at display time only (10·log₁₀(2) ≈ 3,01 dB, energetic — not the coherent +6 dB); the stored curve is always raw incident level. The probe's default standoff is 2 m, inside the standard's band — other offsets remain an exact energetic doubling but no longer map to the standard's measurement position.

Drift bound: within the model the +3 dB is exact. Against real facade measurements it is an approximation, and the deviations are signed and named in the standard: in dense building density / canyon streets the true facade gain can be much higher than 3 dB (§9.2.1.2 NOTE 3); at near-grazing incidence the standoff position is not recommended; the Annex B validity data (Table B.1, u_loc = 0,4 dB all-angle / 1,0 dB grazing) cover A-weighted noise from moving traffic only; for stationary sources and low frequencies Annex B.2 recommends the flush-mounted 5,7 dB convention instead of the 3 dB standoff — and GeoNoise's industrial sources are stationary; and per §9.3.2.8 NOTE 1 the 0,5–2 m standoff position is undefined for low-frequency measurement, so per-band facade values at 63 Hz carry the weakest support.

Closure: none needed for the default — the incident field is the standard's own reference condition and carries no correction. If comparison against measured facade levels becomes a workflow, the natural extension is a selectable flush-mounted 5,7 dB mode (Annex B.4) alongside the 3 dB standoff. Distilled clause notes: docs/references/ISO_1996_2_2017_notes.md.

FACADE PROBE — EDGE RE-INDEXING ON VERTEX EDITS

A facade probe follows its host building live: it stores the host's id plus the polygon edge indices it spans, and re-derives its sample line from the host's current geometry on every move, rotate and resize. Indices that fall out of range are caught — the probe detaches, renders distinctly, and its cached result is invalidated. An index that stays in range but comes to point at a different edge (a vertex inserted or deleted mid-polygon) would instead re-attach the probe to the wrong facade silently.

Drift bound: unreachable through the UI as of 2026-07-29 — no in-app operation inserts or deletes a building vertex, so indices can only shift by hand-editing a saved scene file. Impact if reached: the curve and stats describe a different facade than the one drawn.

Closure: store an edge fingerprint (the endpoint coordinates) beside the indices and validate it on re-derive, detaching on mismatch. Deferred until a vertex-editing operation actually ships — it is the introduction of that operation, not the probe, that opens the hole.

PRIMARY REFERENCES

The four diffraction approximations above derive from the following published literature — anyone implementing or verifying GeoNoise can consult these primary sources directly:

Acoustic standards (the audience for ISO 9613-2 compliance work):
• ISO 9613-2:1996 §7.4 — outdoor sound propagation, barrier diffraction including the multiple-screen treatment for cascaded edges.
• ISO 9613-2:2024 §7.4 — revised D_z formulation with C₃(e/λ) thick-barrier coupling and the K_met meteorological correction.
• CNOSSOS-EU — Commission Directive (EU) 2015/996, Annex II Part VI (“Common Noise Assessment Methods in Europe”) — equivalent-screen reduction for multiple barriers along a propagation path.
• ISO/TR 17534-3:2015 — quality assurance test cases for ISO 9613-2 / CNOSSOS-EU implementations including multi-barrier geometries.

Acoustic chart-fit lineage (the formulas in our code):
• Maekawa, Z. (1968). “Noise reduction by screens.” Applied Acoustics 1(3), 157–173.
• Kurze, U. J. & Anderson, G. S. (1971). “Sound attenuation by barriers.” Applied Acoustics 4(1), 35–53 — closed-form fit to the Maekawa chart.
• Pierce, A. D. (1974). “Diffraction of sound around corners and over wide barriers.” JASA 55(5), 941–955 — coherent-sum theory used in the probe-engine phasor summation.
• Salomons, E. M. (2001). Computational Atmospheric Acoustics. Springer — multi-edge geometric extensions for outdoor propagation.

Underlying wave-diffraction theory (originally developed for RF / microwave propagation; cited for cross-validation, not as the audience-expected acoustic standard):
• Deygout, J. (1966). “Multiple knife-edge diffraction of microwaves.” IEEE Trans. Antennas Propag. AP-14(4), 480–489.
• Bullington, K. (1947). “Radio propagation at frequencies above 30 megacycles.” Proc. IRE 35(10), 1122–1136.
• Epstein, J. & Peterson, D. W. (1953). “An experimental study of wave propagation at 850 Mc.” Proc. IRE 41(5), 595–611.
• ITU-R P.526-15 (2019) §4.5 — Bullington / Epstein-Peterson / Deygout / Vogler comparison + worked examples.

Measurement-uncertainty context:
• ISO 9613-2:1996 §9 — stated ±3 dB(A) prediction accuracy for d < 1000 m.
• IEC 61672-1:2013 §5 — class-1 sound-level-meter tolerance ±1.5 dB at 1 kHz.
• ISO 1996-2:2017 Annex A — outdoor measurement uncertainty ±2–3 dB(A).