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

where d = 3D distance (m), 11 dB = 10·log₁₀(4π) for hemispherical radiation.

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

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

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

The coefficient α 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

Sound reaches the receiver via direct and ground-reflected paths, creating frequency-dependent interference.

      S (source)
     /|\
    / | \  r₁ = direct path
   /  |  \
  /   |hs \ ─────────────────── R (receiver)
 /    |    \                  /
/     |     \      r₂       / hr
──────┴──────●─────────────/──────  Ground (z=0)
             reflection point

Path lengths:

Delany-Bazley normalized impedance:

where σ = flow resistivity (hard: 200,000, soft: 20,000 Pa·s/m²)

Coherent ground effect:

This captures constructive/destructive interference (ground dip phenomenon).

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

Insertion loss:

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: δ = |S→Edge| + |Edge→R| - |S→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.

For thick obstacles, sound diffracts twice (entry + exit edges):

    S                              R
     \                            /
      A₁ ──→ ●────────● ←── A₂
             │  ROOF  │
      ═══════╧════════╧═══════   Building

Path: Total = A₁ + T (roof width) + A₂

Note: coefficient is 40 (not 20) for double-edge diffraction.

Horizontal diffraction: Also computed around building corners using single-edge Maekawa. The minimum-loss path is selected.

Reflections from building walls computed using virtual image sources:

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

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

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

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):

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)

• 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