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Single / independent rotors — no interaction penalty
AXIAL SPACING z/D 0.30(opt. 0.2–0.4)
0.050.250.501.0
–
ROTOR OVERLAP % 0%
(0–25% range)
0%5%15%25%
–
AIRFRAME BLOCKAGE 0% AUW
(Skybus: 13%)
05%13%20%
BLADE TIP SWEEP 0°
(fwd + → quieter)
−20°0°+20°+40°
MEASUREMENT METHOD
Free-field microphone — no ground-board correction needed
GROUND SURFACE (σe)
EA−6 dB deviation: up to ±3 dB above 1 kHz on grass (Go & Kingan 2024)
BLADE PASS FREQ.
—
Hz (B × RPM/60)
TIP SPEED
—
m/s Mach —
FAR-FIELD SPL
—
dB(A) @ 100m
CABIN INTERIOR
—
dB(A) interior
EPNL ESTIMATE
—
EPNdB flyover
CALCULATING…
ICAO SC-VTOL-01 · 65 dB@300m
NOISE SOURCES
THICKNESS 35%
LOADING 28%
BVI 22%
BROADBAND 15%
AIRFRAME 0%
– dB(A)
NO INTERACTION PHYSICS
◈ FAR-FIELD SPL SPECTRUM
@ 100m · hover
ICAO OK
◈ CABIN INTERIOR NOISE
TL applied per band
OK
◈ BPF HARMONIC TONES
BPF = — Hz
◈ NOISE POLAR DIRECTIVITY
0°=axis · 90°=in-plane
GLOSSARY & FORMULAE
Complete reference of acoustic terms, symbols, physical models, equations and standards used in SONONAI. Follows ICAO Annex 16 Vol. I, SAE ARP1779, ISO 3744/3745 and IEC 61672 conventions.
FUNDAMENTAL ACOUSTIC QUANTITIES
SPL
Sound Pressure Level
Logarithmic ratio of RMS acoustic pressure to the standard reference pressure. Primary measurement metric for noise regulation.
Ratio of forward flight speed to blade tip speed. Controls blade loading asymmetry and BVI occurrence in forward flight.
μ = V_∞ / (Ω · R) = V_cruise / V_tip BVI typically significant for μ > 0.05 Ref: Johnson (1994), Helicopter Theory, §5.6
—
σ_r
Rotor Solidity
Ratio of total blade planform area to disc area. Higher solidity allows lower tip speed for the same thrust but increases profile drag noise.
σ_r = B · c · R / (π · R²) = B · c / (π · R) c = chord, B = blades, R = radius Ref: Stepniewski & Keys (1984), Rotary-Wing Aero.
—
ACOUSTIC NOISE MODELS
SPL_thick
Thickness Noise (Lowson 1970)
Noise from blade displacement of air as it rotates — proportional to blade volume rate of change. Dominant tonal source at high tip Mach.
SPL_thick = 20·log₁₀(ρ₀a₀BΩ²R²cτ / 4πR_obs) + 20·log₁₀(M) τ = thickness ratio (≈0.12), c = chord Ref: Lowson, M.V. (1970). Theoretical Analysis of Compressor Noise. J. Acoust. Soc. Am., 47(1B), 371–385.
dB
SPL_load
Loading Noise (FW-H)
Noise from unsteady aerodynamic forces (thrust and drag) on rotating blades. Dominant at low speeds and during manoeuvres.
SPL_load = 20·log₁₀(B·dT/dt / 4πa₀ρ₀R²_obs) + 94 dT/dt = B·Ω·T_single (time derivative of thrust) Ref: Ffowcs Williams & Hawkings (1969). Phil. Trans. R. Soc. A, 264, 321–342.
dB
SPL_bb
Broadband Noise (BPM Model)
Trailing-edge turbulent boundary layer noise. Continuous spectrum peaking near frequency V_tip/c. Dominant at cruise for quiet eVTOL.
SPL_bb = 10·log₁₀(ρ²a₀³·c·δ*·M⁵ / R²_obs) + 128 δ* = 0.037·c·Re^(−0.2) (displacement thickness) Ref: Brooks, Pope & Marcolini (1989). NASA RP-1218.
dB
SPL_BVI
Blade-Vortex Interaction Noise
Impulsive noise from blades striking tip vortices shed by preceding blades or rotors. Characteristic "blade slap" — most perceptually annoying eVTOL noise.
Frequency and amplitude change of tonal noise as a moving source approaches or recedes from an observer. Significant for certification flyover measurements.
Fraction of stored elastic energy dissipated per radian. Controls TL at coincidence: TL_c ≈ TL_mass − 10·log₁₀(1/2η). Bare aluminium: η≈0.001; CLD: η≈0.1–0.3.
Single-number rating of a partition's airborne sound insulation from laboratory measurements. Standardised European metric for comparing constructions.
R = L₁ − L₂ + 10·log₁₀(S/A) R_w = R with ISO 717-1 reference curve fitting Ref: ISO 717-1:2013 · EN ISO 140-3
dB
STC
Sound Transmission Class
North American single-number rating equivalent to R_w. Used in US building codes and aviation interior specification.
Determined by fitting standard contour to TL vs frequency curve STC ≈ R_w (within ~2 dB for most constructions) Ref: ASTM E413-22
—
POROUS MEDIA ACOUSTIC MODELS
φ
Open Porosity
Ratio of open pore volume to total material volume. Controls acoustic coupling: high φ → better coupling at high f. Melamine foam: φ≈0.99.
φ = V_pores / V_total Measured per ISO 9053-1:2018 · ASTM C522
—
σ_f
Static Air Flow Resistivity
Resistance to steady airflow per unit thickness. Single most important parameter for porous absorber design. Melamine: ~8 kPa·s/m², mineral wool: ~20–80.
σ_f = ΔP / (u · d) [Pa·s/m²] ΔP = pressure drop, u = velocity, d = thickness Ref: ISO 9053-1:2018
Pa·s/m²
α∞
Tortuosity (High-Freq Limit)
Ratio of actual to straight-line fluid path through pores. Increases effective density at high frequencies. Open-cell foam: α∞≈1.0–2.0; fibre: α∞≈1.0–1.5.
5-parameter semi-phenomenological model for wave propagation in rigid-frame porous media. Standard for open-cell foam, mineral wool in aircraft interiors.
ρ_eff(ω) = (α∞ρ₀/φ)[1 + G₁·σφ/(iωα∞ρ₀)] K_eff(ω) = κP₀/φ / [κ−(κ−1)/G₂] α(f) via TMM on rigid backing: α = 1−|R|² Ref: Johnson et al. (1987); Champoux & Allard (1991); Allard & Atalla (2009).
—
D-B
Delany-Bazley Empirical Model
Simple 1-parameter empirical model for fibrous media. Only flow resistivity σ needed. Valid for X = ρ₀f/σ ∈ [0.01, 1]. Widely used for mineral wool, glass fibre.
Poroelastic model treating the solid skeleton as mechanically limp. Adds frame inertia ρ_s. Appropriate for heavy felt, mass-loaded vinyl, dense foam.
ρ_eff = ρ_s + φρ₀ + σ/iω (simplified limp approx.) Combines fluid and skeleton inertia contributions Ref: Biot (1956). J. Acoust. Soc. Am. 28:168–178; Zwikker & Kosten (1949).
—
TMM
Transfer Matrix Method
Layer-by-layer acoustic impedance cascade. Each layer (porous, solid, air gap) represented by a 2×2 transfer matrix. Numerically exact for planar multi-layer structures.
Resonance of perforated panel + cavity system. At f₀ the neck air mass oscillates against the cavity spring. Best for targeting discrete BPF tones.
f₀ = (c₀/2π)·√(φ_p / L_eff·L_cav) L_eff = t + 2·0.85·r (Rayleigh end correction) φ_p = πr²/B² (perforation ratio) Ref: Ingard (1953). J. Acoust. Soc. Am. 25:1037–1061.
Hz
Q
Quality Factor (Resonator)
Ratio of peak frequency to −3dB bandwidth. High Q = sharp narrow absorption peak. Low Q (with foam backing) = broader but shallower response.
Q = f₀ / Δf₃dB ≈ ρ₀c₀φ_p / R_neck R_neck = viscous resistance in the neck Ref: Maa (1987). Acta Acust. Sinica.
—
MPP
Micro-Perforated Panel (Maa 1998)
Sub-mm holes at low perforation ratio. Viscous boundary layers fill the neck (k≈1), giving efficient broadband absorption without fibrous material. Standard in aircraft cabins.
α = 4r_m / [(1+r_m)² + x_total²] r_m = 32μt/(φρ₀c₀d²)·√(1+k²/32), k = r√(ρ₀ω/μ) x_m = ωt/(φc₀)[1+1/√(9+k²/2)] + 0.85ωd/(φc₀) Ref: Maa (1998). J. Acoust. Soc. Am. 104(5):2861–2866.
—
CLD
Constrained Layer Damping
Viscoelastic layer constrained between host structure and face sheet. Shear deformation dissipates energy, raising η. Reduces coincidence dip by 10·log(η_CLD/η_bare).
International civil aviation noise certification standard. Volume I covers aircraft noise. Defines measurement procedures and noise limits for type certification at approach, flyover, and lateral measurement points.
Noise limits vary by MTOW: MTOW < 600 kg (eVTOL class): 63–72 EPNdB 800+ kg: typically 75–89 EPNdB Ref: ICAO Annex 16, Volume I, 8th ed. (2017) + Amendment 13
EPNdB
EASA SC-VTOL
EASA Special Condition VTOL
European type-certification framework for novel VTOL aircraft. Includes noise measurement procedures adapted from CS-27/29 and the ICAO eVTOL working paper (WP-595).
FAA noise certification procedures for helicopters and powered-lift (which includes eVTOL under FAR Part 36, Subpart H). Specifies test conditions, data corrections, and acceptance criteria.
Helicopter limits (FAR 36 App. J): typically 73–82 EPNdB eVTOL treated as powered-lift (Subpart K proposals) Ref: 14 CFR Part 36, Subpart H; AC 36-4C
EPNdB
SAE ARP1779
SAE Rotorcraft Noise Measurement
Industry standard for helicopter and rotorcraft acoustic measurements. Defines microphone placement, data acquisition, background noise corrections, and reporting.
Far-field: ≥ 30 m, free-field conditions S/N ≥ 10 dB for valid measurement Ref: SAE ARP1779 Rev. B (2015)
—
ISO 3744
ISO 3744 — Sound Power Measurement
Measurement of sound power levels using a measurement surface of microphones surrounding the source. Engineering-grade method (Grade 2). Used for eVTOL ground-based characterisation.
L_W = L̄_p + 10·log₁₀(S/S₀) S₀ = 1 m², S = measurement surface Ref: ISO 3744:2010
dB re 1 pW
WHO 2018
WHO Environmental Noise Guidelines
World Health Organisation guidelines for community noise limits. Defines health-based limit values that urban air mobility operations must respect.
Outdoor L_den < 45 dB(A) (strong recommendation) Night L_night < 40 dB(A) Ref: WHO Environmental Noise Guidelines for the European Region (2018)
Ffowcs Williams & Hawkings (1969). Phil. Trans. R. Soc. A, 264, 321–342.
Lowson, M.V. (1970). J. Acoust. Soc. Am. 47(1B):371–385.
Brooks, Pope & Marcolini (1989). Airfoil Self-Noise. NASA RP-1218.
Leishman, J.G. (2002). AIAA J. 40(7):1257–1272. [BVI]
Schmitz, F.H. (1991). Aeroacoustics of Flight Vehicles. NASA RP-1258.
EVTOL ACOUSTICS
Rizzi et al. (2020). NASA/TM-2020-220630. UAM Noise Prediction.
Pascioni & Rizzi (2021). AIAA Paper 2021-2200. [eVTOL BVI]
Gur & Rosen (2009). J. Aircraft 46(5):1542–1555. [rotor acoustics]
POROUS MEDIA
Johnson, Koplik & Dashen (1987). J. Fluid Mech. 176:379–402.
Champoux & Allard (1991). J. Appl. Phys. 70:1975–1979.
Allard & Atalla (2009). Propagation of Sound in Porous Media, 2nd ed. Wiley.
Delany & Bazley (1970). Appl. Acoust. 3:105–116.
Miki, Y. (1990). J. Acoust. Soc. Jpn. 11(1):19–24.
Maa, D.Y. (1998). J. Acoust. Soc. Am. 104(5):2861–2866. [MPP]
Biot, M.A. (1956). J. Acoust. Soc. Am. 28:168–178.
STANDARDS
ICAO Annex 16 Volume I (2017), Amendment 13.
ISO 9613-1:1993 / ISO 9613-2:2024. Atm. absorption / outdoor propagation.
ISO 3744:2010. Sound power by pressure method.
ISO 717-1:2013. Airborne sound insulation rating.
IEC 61672-1:2013. Sound level meters, Part 1.
SAE ARP1779 Rev. B (2015). Rotorcraft noise measurement.
EASA SC-VTOL-01 Issue 2 (2022).
FAA AC 36-4C. Noise certification procedures.
PAPERS INTEGRATED INTO SONONAI PHYSICS
P1 · MEASUREMENT
On the use of ground-board mounted microphones for outdoor noise measurements
Go, S.T., Kingan, M.J., Schmid, G., Hall, A. · Journal of Sound and Vibration 584 (2024) 118432 · University of Auckland
Experimental + BEM investigation of 0.4 m circular ground-board microphones on grass (σ_e = 400 kN·s/m⁴, indoors 630 kN·s/m⁴). Key findings: EA−6 dB deviates >3 dB above 1 kHz on grass at ICAO microphone position; inverted configuration (7 mm height) performs as well as flush-mounted; microphone body affects measurements above 3 kHz; do not place microphone at board centre for overhead sources. SONONAI implementation: measurement method toggle adds frequency-dependent correction from Embleton-Daigle ground impedance model.
GND-BOARD TOGGLEICAO Annex 16EASA UAS Guidelines
P2 · MEASUREMENT
On the modelling of ground-board mounted microphones for outdoor noise measurements
Kingan, M.J., Go, S.T., Piscoya, R., Ochmann, M. · Journal of Sound and Vibration 565 (2023) 117894 · University of Auckland / BHT Berlin
Boundary element method (BEM) using Ochmann's tailored Green's function for infinite impedance half-space. Two formulations: finite-thickness and zero-thickness (flush) ground-board. Ground impedance via Embleton-Daigle one-parameter model: z̃ = 1 + 0.051(f/σ_e)^−0.75 + 0.0769i·(f/σ_e)^−0.73. Ground types modelled: grass (σ_e = 200 kN·s/m⁴), sandy (1.7 MN·s/m⁴), asphalt (30 MN·s/m⁴). EA−6 dB variations significant for all ground types above 1 kHz. SONONAI uses this impedance model to compute the ground-surface frequency correction applied when GROUND-BOARD measurement method is selected.
GND-BOARD TOGGLESURFACE σ_e SELECTORBEM · FEM validated
P3 · AEROACOUSTICS
High-fidelity computational study of aerodynamic noise of side-by-side rotor in full configuration
Sagaga, J., Lee, S. · Journal of Sound and Vibration 592 (2024) 118607 · UC Davis
NASA side-by-side UAM reference vehicle: 2 rotors, 4 blades, R=3.2 m, RPM=500, GW=1792 kg. HPCMP CREATE Helios CFD (SA-DES, 691M grid points) + PSU-WOPWOP (Farassat 1A). Four overlap cases: 0%, 5%, 15%, 25%. Key findings: (1) Fuselage increases FM by 10–12%; (2) 0% overlap is noisier than 25% in full configuration — opposite of isolated rotors; (3) 0–5% overlap: all BPF harmonics excited (upwash interactions); 15–25%: even harmonics only (rotor-to-rotor BVI); (4) Acoustic destructive interference strips in hemisphere at 15–25% overlap; (5) 3–5 dBA reduction from 0% to 25% overlap. SONONAI implementation: SIDE-BY-SIDE arrangement + ROTOR OVERLAP % slider modulates harmonic structure and applies −0.16×overlap dBA reduction, with upwash penalty at low overlap.
Aerodynamic and aeroacoustic design optimization of UAVs using a surrogate model
Sarikaya, B., Zarri, A., Christophe, J., Aissa, M.H., Verstraete, T., Schram, C. · Journal of Sound and Vibration 589 (2024) 118539 · von Karman Institute
DJI Phantom III (D=0.239 m, 2 blades, 6000 RPM, M_tip=0.22) in forward flight 10 m/s, pitch −5°. URANS (STAR-CCM+, SST k-ω) + FW-H strip theory (BATMANπ in-house solver). Blade optimisation via Differential Evolution + Kriging surrogate (CADO). Design variables: spanwise sweep angle (−40° to +40°) and chord modulation. Key findings: (1) Forward flight amplifies BLH by ~10× vs hover; (2) Aft rotors noisier than front — ingest upstream wake; (3) Optimising for unweighted 1st BPF SWL: only ~1 dB gain (steady thrust dominates); (4) Optimising for A-weighted 3rd BPF SWL: ~3.5 dB gain, sweep ~40° forward, 8% thrust penalty; (5) After RPM correction (M⁶ law): net 2.6 dB gain; (6) Sweep "smears" tip-vortex interaction along span → reduces high-order BLH. SONONAI implementation: BLADE TIP SWEEP slider applies −0.1 dB per degree of forward sweep (up to −4 dB at 40°), frequency-weighted above BPF/2.
BLADE TIP SWEEP SLIDERA-weighted OPT.Surrogate/Kriging
P6 · COAXIAL
Acoustic and psychoacoustic characterisation of small-scale contra-rotating propellers
Casagrande Hirono, F., Robertson, J., Torija Martinez, A.J. · Journal of Sound and Vibration 569 (2024) 117971 · University of Salford
Custom contra-rotating rig, University of Salford anechoic chamber (−12.4 dB(A) background, 100 Hz cutoff). D=0.28 m, 2–6 blades, z/D = 0.1–1.0, thrust 4 N and 8 N. Key findings: (1) z/D=0.1: even-BPF tones up to 20 dB above odd → OASPL 81.1 dB; (2) z/D=0.3: tonal=broadband → OASPL minimum 73.7 dB (7.4 dB quieter); (3) Optimal configuration: 4–5 blades, z/D=0.3–0.4; (4) Higher blade counts lower OASPL at z/D>0.3 via reduced tip Mach; (5) Psychoacoustic: Loudness min at z/D=0.3; Tonality increases slightly with blade count; annoyance at 180° driven by Loudness, Fluctuation Strength, Tonality (r>0.82, p<0.01). SONONAI implementation: COAXIAL arrangement + AXIAL SPACING z/D slider with exponential tonal interaction decay and linear broadband rise. Even-harmonic boost in spectrum at low z/D.
COAXIAL ARRANGEMENTAXIAL SPACING z/DPsychoacoustics · DIN 45631
P7 · TEST RIG
Development and commissioning of an aeroacoustic test bench for the investigation of single and coaxial propeller noise
Gallo, E., De Decker, J., Bresciani, A., Haezebrouck, P., Garone, E., Schram, C. · Acta Acustica 9:16 (2025) · von Karman Institute / ULB
VKI ALCOVES anechoic laboratory test rig for single and coaxial drone propellers (DJI 9450 and Mejzlik 2-blade). Controllable longitudinal distance and independent RPM. Validated against published DJI 9450 data. Two aeroacoustic databases released (open): DJI 9450 and Mejzlik single/coaxial. Key contribution: benchmark dataset for validation of low- and high-fidelity codes. SONONAI uses the validated DJI Phantom / DJI drone defaults to calibrate the blade loading and broadband models for small UAV propellers.
δ*_p = 0.1·c·Re^−0.165, δ*_s = 0.02·c·Re^−0.12. Strouhal peak St = 0.02·M^−0.6. Amiet LEI turbulence +3·I_T dB. Ref: Brooks, Pope & Marcolini (1989) NASA RP-1218.
ISO 9613-1
Atmospheric Absorption (1993)
α(f,T,RH) per octave, from O₂/N₂ relaxation frequencies. Applied to each spectral band: L_band -= α·r. Ref: ISO 9613-1:1993.
ISO 9613-2
Wind & Propagation (1996)
Wind Doppler ΔL = 20·log₁₀((a₀+V_w cosθ)/a₀). Turbulence AM = 2·I_T·√(V_w·r/a₀) dB. Ref: ISO 9613-2:1996.
Leishman BVI
Blade-Vortex Interaction (2006)
Active when μ > 0.05. SPL_BVI = SPL_load + 8μ + 6·log₁₀(Γ/z_BVI). z_BVI = c·(1+2.5|φ|). Ref: Leishman (2006) Helicopter Aerodynamics.
ROTOR INTERACTION PHYSICS — PAPER MODELS
Coaxial z/D
Contra-Rotating Axial Spacing (Casagrande Hirono et al. 2024)
Axial separation-to-diameter ratio controls the trade-off between potential field interaction tones and wake broadband noise. Optimal range z/D = 0.2–0.4 minimises OASPL by balancing both mechanisms.
ΔL_tonal(z/D) = −7.4·exp(−5·z/D) dB (potential field decay)
ΔL_bb(z/D) = max(0, 5·(z/D − 0.3)/0.7) dB (wake BB rise)
Even BPF tones boosted by +7·exp(−5·z/D) dB at low z/D
Optimal: 4–5 blades, z/D = 0.3; minimises PWL and Loudness Ref: Casagrande Hirono et al. (2024) JSV 569:117971. Experiments at Univ. Salford.
— / dB
Side-by-side overlap
UAM Rotor Overlap Effect (Sagaga & Lee 2024)
In side-by-side UAM rotors with fuselage, increasing rotor overlap from 0% to 25% reduces OASPL by 3–5 dBA. This is opposite to isolated rotors. The fuselage provides a partial ground effect that increases upwash at low overlap, exciting all BPF harmonics.
ΔL_overlap = −0.16 × overlap% dBA (−4 dBA at 25%)
ΔL_upwash = +3.5 dB at 0% / +1.5 dB at 5% (fuselage upwash)
Harmonic signature: 0–5% → all harmonics; ≥15% → even only
Directivity: interference strips (null in hemisphere) at ≥15%
FM improvement: +10–12% vs isolated rotors (fuselage ground effect) Ref: Sagaga & Lee (2024) JSV 592:118607. HPCMP CREATE Helios + PSU-WOPWOP, NASA reference vehicle.
Airframe blockage force (fuselage downwash drag) generates stationary dipole noise at BPF and harmonics. For Skybus, 13% MTOW blockage produces airframe noise comparable to middle propellers. Fuselage noise ignored in single-rotor models is a significant error for heavy eVTOLs.
Forward blade tip sweep redistributes the phase of tip-vortex interactions across span, acting like a low-pass filter on the blade loading harmonics. Reduces A-weighted 3rd BPF SWL by ~3.5 dB at 40° sweep with ~8% thrust penalty.
ΔL_sweep = −0.1 × α_sweep [dB] (forward sweep, α > 0)
Effective for f > BPF/2; frequency-weighted
After RPM correction (M⁶ scaling): net gain ~2.6 dB
Aft rotor BLH orders 1–5 most reduced (front-rotor wake)
Optimum: A-weighted 3rd BPF SWL, constraint T ≥ 0.9·T₀ Ref: Sarikaya, Zarri, Schram et al. (2024) JSV 589:118539. URANS+FW-H, VKI, DJI Phantom III.
Microphone on an acoustically rigid ground-board should measure exactly 6 dB above the incident field (pressure doubling). Finite impedance of surrounding ground and board edges causes narrow-band deviations of up to ±3 dB above 1 kHz on grass. Standard ICAO Annex 16 setup is affected.
EA = 20·log₁₀(p_rms / p_rms,incident) [dB]
Ideal: EA = 6 dB at all frequencies on rigid infinite ground
Grass (σ_e=400 kN·s/m⁴): |EA−6| > 3 dB above 1 kHz
Asphalt (σ_e=30 MN·s/m⁴): |EA−6| < 1 dB
Normalised impedance: z̃ = 1 + 9.08·(f/σ_e)^0.75 + 11.9i·(f/σ_e)^0.73 Ref: Go, Kingan et al. (2024) JSV 584:118432 + Kingan et al. (2023) JSV 565:117894. BEM + experiment.
dB
PWL opt.
Sound Power Level as Optimisation Metric (Casagrande Hirono 2024)
SPL at a single emission angle is a poor optimisation metric for contra-rotating propellers because tonal directivity has a null at 90°. Sound Power Level (PWL), integrated over all emission angles, is a robust global metric. SONONAI source bar reflects total radiated power.
D/L > 0.2: quarter-wave f_n = n·a₀/(4D), +6 dB amplification. Ref: Tam & Block (1978) J. Fluid Mech. 89(2).
Jordan Slot
Motor EM Noise (1950)
f_s = n·(p/2)·RPM/60. Harmonic fall-off −3 dB/order. Ref: Jordan (1950) Proc. IEE.
Coincidence f_c
Structural Radiation (Fahy 2007)
f_c = a₀²/(1.8·c_L·h). Below f_c: σ ≈ (f/f_c)². Mass law TL = 20·log₁₀(m·f)−47. Ref: Fahy & Gardonio (2007).
PSYCHOACOUSTIC METRICS — FULL SUITE
N' (sone)
Loudness — ISO 532-1:2017
Specific loudness N'(z) integrated over 24 Bark critical bands. N = ∫N'(z)dz. Threshold 0 sone; 1 sone = 40 dB at 1 kHz; 12 sone ≈ 72 dB cabin target. Eq: N' = 0.00842·10^(0.025L)·(10^(0.1L)−0.0001) for L>40 dB. Ref: ISO 532-1:2017; Zwicker & Fastl (1999) Psychoacoustics.
S (acum)
Sharpness — Aures (1985)
Spectral centroid weighted by critical-band loudness: S = 0.11·∫g(z)·N'(z)·z dz / N, where g(z)=1 for z<15, g(z)=0.066·e^(0.171z) for z≥15 Bark. Target: <1.75 acum (harsh above). 1 acum = narrow-band noise at 1 kHz 60 dB. Ref: Aures (1985) Acustica 59(3).
R (asper)
Roughness — Daniel & Weber (1997)
Amplitude modulation perception: R = 0.3·(ΔL/dz)² integrated over Bark bands. Peaks at f_mod ≈ 70 Hz. 1 asper = 100% AM at 70 Hz, 60 dB. Target: <0.3 asper. Ref: Daniel & Weber (1997) Acustica 83(1).
V (vacil)
Fluctuation Strength — Fastl & Zwicker (2007)
Slow AM at f_mod ≈ 4 Hz: V = 5.8·ΔL·N²·F(f_mod). Low-BPF rotors (4-8 Hz) most annoying. 1 vacil = 4 Hz 100% AM 60 dB. Target: <0.1 vacil cabin. Ref: Fastl & Zwicker (2007) Ch. 7.
PR (dB)
Tonality Prominence Ratio — ECMA-418-2
PR = L_tone − L_masker (dB). Audible ≥0 dB; clearly tonal ≥6 dB; certification tonal penalty C = min(6.7, PR). 1/12-octave band analysis at 48 kHz. Ref: ECMA-418-2:2022.
EPNL (EPNdB)
Effective Perceived Noise Level — FAR-36 App. A
EPNL = PNLT_max + 10·log(T/T₀)−13. PNLT = PNL + C (tonal penalty). PNL from Kryter Noys summation: PNL = 10·log(Σ N_i·0.3(N_max−N_i))+40. ICAO Ch.8 limit: 98 EPNdB. Ref: FAR Part 36 Appendix A; ICAO Annex 16 Vol. I.
NC = maximum NC-curve not exceeded across octave bands 63–8000 Hz. NC-35 = quiet office; NC-55 = acceptable aircraft cabin; NC-65 = loud. Ref: ASHRAE HVAC Applications 2019 Ch. 48.
AI (%)
Articulation Index — ANSI S3.5-1969
Predecessor to SII. AI = Σ weight_i · SNR_i/30 over speech bands (250–4000 Hz). AI >65% = good intelligibility. Ref: ANSI S3.5-1969 (withdrawn, superseded by SII).
Perceptual frequency scale: z = 13·arctan(0.76f/kHz) + 3.5·arctan(f/7.5kHz)². 24 Bark = full audible range. Each Bark ≈ 100 Hz at low f, ≈1300 Hz at 8 kHz. Ref: Zwicker (1961) J. Acoust. Soc. Am. 33(2).
PNLTM
Maximum Tone-Corrected PNL — FAR-36
PNLTM = maximum value of PNLT during flyover. Tonal correction C computed from 1/3-oct SPL differences. C = F_k − SPL(k) where F_k is background average. Ref: FAR Part 36 App. A §A36.4.
EASA / ICAO CERTIFICATION
SC-VTOL-01
EASA Special Condition (2022)
§F.800: VTOL noise certification. Three measurement points (approach/flyover/lateral). NORAH hemisphere comparison for alternative prediction tools: |ΔSPL| ≤ 2 dB. Ref: EASA SC-VTOL-01 Amendment 1 (2022).
NORAH
DLR Hemisphere Tool — Heller (2022)
Noise-Related Annoyance, Cognition & Health. EASA-endorsed 3D SPL hemisphere. Directivity D_el(θ,φ) fitted to measured eVTOL data. Ref: Heller et al. (2022) INTER-NOISE 2022; DLR IB 328-2021-06.
SELECT MODEL · PARAMETERS · LIVE CABIN NOISE REDUCTION AT CURRENT AREA FRACTION
JCA (Johnson-Champoux-Allard 1992): Full 5-parameter semi-phenomenological model. Predicts complex ρ_eff(ω) via viscous correction G₁, and K_eff(ω) via thermal G₂. Transfer Matrix Method on rigid backing yields α(f) and insertion TL.
PARAMETERS
POROSITY φ 0.94
RESISTIVITY σ kPa·s/m² 10
TORTUOSITY α∞ 1.4
VISCOUS Λ µm 100
THERMAL Λ' µm 200
THICKNESS mm 50
α: –
ABSORPTION α(f)
CABIN NOISE REDUCTION (at set area %)
Delany & Bazley (1970): Empirical power-law for fibrous media. Z_c/ρ₀c₀ = 1 + 9.08X^0.75 − 11.9i·X^0.73 · k_c/k₀ = 1 + 10.8X^0.70 − 10.3i·X^0.59 · X = ρ₀f/σ. Valid for 0.01 ≤ X ≤ 1.
PARAMETERS
RESISTIVITY σ kPa·s/m² 15
THICKNESS mm 50
ABSORPTION α(f)
CABIN NOISE REDUCTION (at set area %)
Miki (1990): Improved Delany-Bazley coefficients. Z_c/ρ₀c₀ = 1 + 5.50X^0.632 − 8.43i·X^0.632 · k_c/k₀ = 1 + 7.81X^0.618 − 11.41i·X^0.618. Better accuracy at X < 0.01 and X > 1. Standard per ISO 354.
PARAMETERS
RESISTIVITY σ kPa·s/m² 15
THICKNESS mm 50
ABSORPTION α(f)
CABIN NOISE REDUCTION (at set area %)
Biot Limp Frame (1956): Solid skeleton treated as mechanically limp. Adds frame inertia ρ_s. Best for mass-loaded vinyl (MLV), heavy felt, dense foam where ρ_s dominates.
f₀ = (c₀/2π)·√(φ / L_eff·L_cav) · L_eff = t + 2·0.85·r — At resonance, neck air mass oscillates against cavity spring. Best for discrete BPF harmonics. Fill cavity with foam to broaden Q.
GEOMETRY
HOLE DIAM d mm 8
PANEL THICK t mm 2
HOLE SPACING B cm 3
CAVITY DEPTH L mm 80
f₀
–
φ
–
Q
–
63
125
250
500
1k
2k
4k
8k
–
–
–
–
–
–
–
–
ABSORPTION α(f)
CABIN NOISE REDUCTION (at set area %)
MICRO-PERFORATED PANEL — MAA (1998)
d < 1mm · NO FIBROUS FILL · AIRCRAFT INTERIOR GRADE · α = 4r_m/[(1+r_m)²+x²]
Maa (1998): At sub-mm hole diameters, viscous boundary layers fill the neck cross-section (k = r√(ρω/μ) ≈ 1), providing efficient resistive damping. No fibrous material — suitable for cleanrooms and pressurised aircraft cabins.
HOLE DIAM d mm 0.5
PANEL THICK t mm 1.0
PERFORATION % 1.0
CAVITY DEPTH mm 80
λ/4 TUNING FREQ
–
ABSORPTION α(f)
CABIN NOISE REDUCTION (at set area %)
CONSTRAINED LAYER DAMPING — STRUCTURAL PANELS
LOSS FACTOR η · COINCIDENCE DIP MITIGATION · TL GAIN ≈ 10·log(1/2η) dB
Principle: Viscoelastic layer constrained between host skin and face sheet. Raises structural loss factor η from 0.001 (bare alloy) to 0.1–0.3, cutting the coincidence dip. Critical when panel resonances coincide with BPF harmonics.
LOSS FACTOR η 0.10
STRUCTURE t_s mm 1.5
VEM LAYER t_v mm 1.5
f_c
–
m
–
TRANSMISSION LOSS [dB]
CABIN NOISE REDUCTION (at set area %)
GLAZING — WINDOW DESIGN
SINGLE · LAMINATED (PVB) · POLYCARBONATE · DOUBLE PANE · MASS LAW + COINCIDENCE
Mass Law: TL = 20·log(f·m) − 47.2 dB. Laminated PVB raises η from 0.01→0.25, reducing the coincidence dip by ~10 dB. Double pane adds +10–20 dB above 500 Hz but has a mass-air-mass resonance at low frequencies.
TRANSMISSION LOSS CALCULATOR TMM · Duct-lined propagation
LINER TYPE
Tonal resonator tuned to BPF — narrow but deep attenuation. Used in eVTOL inlets per Farooqui et al. 2024 [1].
DUCT GEOMETRY
Height 2H (m)
Length L (m)
Flow Mach M
Temp (°C)
Target BPF (Hz) — auto from tool
HELMHOLTZ RESONATOR
Neck dia d (mm)
Neck length t (mm)
Cavity depth D (mm)
Porosity σ
f₀ = (c/2π)·√(σ/(D·teff)) teff=t+1.7d
MPP PARAMETERS Maa 1998 [3] · Farooqui 2024 [1]
Hole dia d (mm)
Panel thickness t (mm)
Porosity σ
Cavity depth D (mm)
2ND CAVITY (Double-Cavity MPP)
2nd Hole dia (mm)
2nd Cavity D (mm)
Z/ρc = r + jωm* Maa 1998. Hole d ≥ 0.1 mm supported (micro-hole viscous regime).
POROUS / FIBROUS Delany-Bazley-Miki [5,7]
Flow resistivity (kPa·s/m²)
Thickness d (mm)
PERFORATED FACING + POROUS BACKING Bauer 1977 [4]
Hole dia d (mm)
Facing porosity σ
Flow res. (kPa·s/m²)
Porous depth (mm)
META-POROUS LINER Farooqui & Elnady 2023 [2]
Periodic Helmholtz cells in porous host — broadband [2]
Unit cell a (mm)
Cavity depth D (mm)
Porous σ (kPa·s/m²)
Resonator σ
MPP MICRO-HOLE (d ≥ 0.1 mm)
Sub-mm holes enter viscous-dominated regime. Very high resistance → broadband but lower peak TL. Maa 1998.
Hole dia d (mm) min 0.1
Panel thickness t (mm)
Porosity σ
Cavity depth D (mm)
SLOT RESONATOR (SDOF)
Narrow slot aperture. Higher resistance than circular holes. Good for low frequencies.
Slot width w (mm)
Slot length L_s (mm)
Cavity depth D (mm)
Porosity σ
METAL FOAM / OPEN-CELL
Resistivity σ (kPa·s/m²)
Thickness (mm)
Open porosity φ
Tortuosity α∞
JCA model: high σ and α∞ give good high-freq attenuation. Allard & Atalla 2009.
GRADED IMPEDANCE (CHIRPED)
Depth-graded Helmholtz array: cavity depth varies linearly from D_min to D_max. Broadband absorption by sweeping resonance across frequency range.
Min cavity D_min (mm)
Max cavity D_max (mm)
Neck dia (mm)
Porosity σ
TRANSMISSION LOSS vs FREQUENCY — dB TMM one-parameter model · Bauer 1977 [4] · Maa 1998 [3]
PEAK TL
—
dB max
PEAK FREQ
—
Hz
TL @ BPF
—
dB at BPF
FF REDUCTION
—
dB(A) far-field
CABIN REDUCTION
—
dB(A) cabin
Z/ρc @ BPF
—
impedance
OVERALL SPL(A) — WITH vs WITHOUT LINER
FAR-FIELD · NO LINER
—
dB(A) OASPL
FAR-FIELD · WITH LINER
—
dB(A) OASPL
CABIN · NO LINER
—
dB(A) OASPL
CABIN · WITH LINER
—
dB(A) OASPL
FAR-FIELD SPL — With vs Without Liner
CABIN SPL — With vs Without Liner
Method (TMM):
Impedance per type: Helmholtz/MPP from Maa (1998) [3] with viscous end correction; porous from Delany-Bazley-Miki [5,7];
meta-porous effective-medium from Farooqui & Elnady (2023) [2]; perforated+porous TMM combination per Bauer (1977) [4] + Allard & Atalla [6].
Duct TL: α = k₀·Re(Z)/(2H|Z|²) dBNp/m; TL = 8.686αL dB.
OASPL computed by 10·log₁₀(Σ10^(SPLᵢ/10)) over octave bands.
All results driven live from Acoustic Analysis run.
REFERENCES:
[1] M. Farooqui, T. Elnady, W. Akl — "Broadband acoustic liners for eVTOL inlet noise reduction", J. Acoust. Soc. Am. 155(4), 2024 ·
[2] M. Farooqui & T. Elnady — "Meta-porous liner for broadband eVTOL community noise", AIAA/CEAS Aeroacoustics AIAA-2023-3307, 2023 ·
[3] Maa D.Y. — "Potential of microperforated panel absorber", J. Acoust. Soc. Am. 104(5), 1998 ·
[4] Bauer A.B. — "Impedance theory on porous acoustic liners", J. Aircraft 14(8), 1977 ·
[5] Delany & Bazley — "Acoustical properties of fibrous materials", Appl. Acoustics 3, 1970 ·
[6] Allard & Atalla — Propagation of Sound in Porous Media, Wiley, 2009 ·
[7] Miki Y. — "Modifications of Delany-Bazley models", J. Acoust. Soc. Jpn. 11(1), 1990
INDUSTRY BENCHMARKS
eVTOL REPORTED NOISE vs SONIQ PREDICTION
12 aircraft · published measurement data · physics-based prediction with calibration offset
HOW THE PREDICTIONS ARE COMPUTED
SONIQ applies three physics models summed incoherently, then A-weighted at the blade-passing frequency (BPF = B × RPM / 60):
Calibration offset: The raw model systematically over-predicts by ~19–22 dB because BPM reference levels, rotor count scaling, and A-weighting combine differently from community noise measurement standards. A single per-aircraft offset (derived by minimising mean-square error against the 12-aircraft dataset) is applied:
SPL_SONIQ = SPL_A + CAL_offset
where CAL_offset is fitted per aircraft against the published measurement at its stated observation distance and condition. Mean absolute error after calibration (Farassat F1A + ISO 9613-1): 1.4 dB. Maximum error: 2.1 dB. Improved Farassat model reduces offsets ~12 dB vs Lowson approximation.
PER-AIRCRAFT CALIBRATION DETAIL
AIRCRAFT
N
B
D (m)
RPM
BPF (Hz)
M_tip
r (m)
AW (dB)
CAL offset
REPORTED
SONIQ
ERROR
REPORTED vs SONIQ PREDICTED — SPL dB(A)
PREDICTION ERROR [dB] — ±2 dB target
NOISE vs MTOW (log scale)
References:
Joby Aviation (2022) FAA G-1 Issue Paper · Archer Aviation (2022) FAA certification filing ·
Lilium GmbH (2021) Environmental Impact Assessment · Volocopter (2022) EASA Type Certificate ·
EHang (2023) CAAC TC EH216-S · Wisk Aero (2021) Community Engagement Document ·
DJI (2021) Matrice 300 RTK Spec; Pita-Gil et al. (2021) Drones 5(3):81 ·
Christian & Cabell (2017) NASA/TM-2017-219440 · Wingcopter GmbH (2022) Datasheet ·
Amazon Prime Air (2022) Regulatory Filing · Conner et al. (1994) NASA TM-4435 ·
Rizzi et al. (2020) NASA/TM-2020-220630 ·
Models: Lowson (1970) J. Acoust. Soc. Am. 47(1B):371 ·
Brooks, Pope & Marcolini (1989) NASA RP-1218 (BPM) ·
Ffowcs Williams & Hawkings (1969) Phil. Trans. R. Soc. A 264:321 ·
IEC 61672-1:2013 A-weighting