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 full ISO 9613-2 model for noise mapping.

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.

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).

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 (line sources): Uses 2π instead of 4π, as 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.

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 at standard conditions (20°C, 70% RH).

63 Hz	125 Hz	250 Hz	500 Hz	1 kHz	2 kHz	4 kHz	8 kHz	16 kHz
0.1	0.4	1.0	1.9	3.7	9.7	33	117	392

Values in dB/km

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 Table 3 (Standard Compliant):

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=min(30h, d)):

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. Captures constructive/destructive interference (ground dip).

Model Comparison:

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

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.

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) \quad \text{capped at 20-25 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.

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{ per reflection (10% absorption)}

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]

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
2.5D geometry (vertical plane diffraction)
Homogeneous ground (single type per scene)

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)

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)

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)

Line source (cylindrical):

A_div = 10 log₁₀(d) + 10 log₁₀(2π)

ATMOSPHERIC ABSORPTION

Path loss:

A_atm = α · d

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'(f) + b'(f)·G·log(h_s) + c'(f)·G·log(d_p,s) + d'(f)·G

A_r = a'(f) + b'(f)·G·log(h_r) + c'(f)·G·log(d_p,r) + d'(f)·G

A_m = a(f)·q + b(f)·(1-G)·q where q = max(0, 1 - 30(h_s+h_r)/d)

d_p = min(30·h, d) G = ground factor (0=hard, 0.5=mixed, 1=soft)

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

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

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

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

WEIGHTING

Octave-band weighting:

L_A = L_band + A(f)

L_C = L_band + C(f)

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

GRID ENGINE

Used for: Measure 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.

PROBE ENGINE

Used for: Probe microphones only

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:

$Z_n = 1 + 5.5(f/\sigma)^{-0.632} - j \cdot 8.43(f/\sigma)^{-0.632}$

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	Two-ray phasor
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.