12.2 Sound Propagation, Octave Bands, and Frequency Weighting

Sound Propagation, Octave Bands, and Frequency Weighting

Decibel arithmetic tells you how to combine levels. Propagation and frequency analysis tell you what level to expect at a worker position and which part of the spectrum is doing the damage — the two things an engineering noise control decision actually depends on.

1. Geometric Acoustic Spreading and Inverse Square Law

In a free acoustic field (an open space free of reflecting boundaries), sound energy propagates outward from the source, spreading its power across an expanding geometric wavefront surface area.

   +-------------------------------------------------------------------------+
   |                  POINT SOURCE VS. LINE SOURCE PROPAGATION               |
   +-------------------------------------------------------------------------+
   |                                                                         |
   |  A. SPHERICAL POINT SOURCE (3D Spreading):                              |
   |     - Area of sphere = 4πr²                                             |
   |     - Doubling distance spreads energy over 4× the surface area.        |
   |     - Attenuation rate = -6 dB per distance doubling!                   |
   |                                                                         |
   |          Source (*) ===> Area A (r=1) ====> Area 4A (r=2)               |
   |                          Lp1                Lp2 = Lp1 - 6 dB            |
   |                                                                         |
   |  B. CYLINDRICAL LINE SOURCE (2D Spreading):                             |
   |     - Area of cylinder = 2πrL                                           |
   |     - Doubling distance spreads energy over 2× the surface area.        |
   |     - Attenuation rate = -3 dB per distance doubling!                   |
   |                                                                         |
   |          Line [=======] ====> Area A (r=1) ====> Area 2A (r=2)          |
   |                               Lp1                Lp2 = Lp1 - 3 dB       |
   +-------------------------------------------------------------------------+

Point Sources (Spherical Wave Propagation)

For an isolated, compact machine whose dimensions are small relative to measurement distance and wavelength (a point source), sound radiates as expanding concentric spheres (A = 4π r²):

$L_{p2} = L_{p1} - 20 \log_{10}\left(\frac{r_2}{r_1}\right)$

$\mathbf{\text{Point Source Rule: Sound pressure level drops by } 6\text{ dB per doubling of distance.}}$

Line Sources (Cylindrical Wave Propagation)

For continuous, elongated acoustic emitters such as high-velocity industrial pneumatic conveyor lines, continuous overhead duct runs, piping headers, or dense continuous highway traffic (a line source), sound radiates as expanding concentric cylinders (A = 2π r L):

$L_{p2} = L_{p1} - 10 \log_{10}\left(\frac{r_2}{r_1}\right)$

$\mathbf{\text{Line Source Rule: Sound pressure level drops by } 3\text{ dB per doubling of distance.}}$

Directivity Factor (Q) and Directivity Index (DI)

Real-world industrial machines operate near reflecting room surfaces (floors, walls, corners), which constrain sound radiation into partial spheres, concentrating sound intensity:

   +-------------------------------------------------------------------------+
   |                     ACOUSTIC DIRECTIVITY FACTOR (Q)                     |
   +-------------------------------------------------------------------------+
   |                                                                         |
   |  Q = 1: Full Sphere (Free space / suspended)     -> DI = 0 dB           |
   |  Q = 2: Hemisphere (Machine on hard floor)       -> DI = +3 dB          |
   |  Q = 4: Quarter Sphere (Floor-wall junction)     -> DI = +6 dB          |
   |  Q = 8: Octant (Machine in trihedral corner)     -> DI = +9 dB          |
   +-------------------------------------------------------------------------+

The relationship between Sound Power Level (Lw), directivity (Q), distance (r in meters), and Sound Pressure Level (Lp) in a free field is:

$L_p = L_w + 10 \log_{10}\left( \frac{Q}{4 \pi r^2} \right)$

$DI = 10 \log_{10}(Q)$

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2. Frequency Analysis and Octave Bands

Human auditory sensitivity and physical engineering control mechanics (such as barrier transmission loss and silencer design) vary dramatically across the acoustic frequency spectrum (20 Hz to 20,000 Hz). Broadband sound is divided into standardized frequency intervals known as Octave Bands and 1/3-Octave Bands.

Octave Band Center Frequencies and Bandwidths

An octave band represents a frequency interval where the upper cutoff frequency (f2) is exactly twice the lower cutoff frequency (f1):

$f_2 = 2 \times f_1$

The geometric center frequency (fc) is the geometric mean of the cutoff frequencies:

$f_c = \sqrt{f_1 \times f_2} \implies f_1 = \frac{f_c}{\sqrt{2}} = 0.7071 f_c, \quad f_2 = f_c \times \sqrt{2} = 1.4142 f_c$

$\text{Bandwidth (}\Delta f\text{)} = f_2 - f_1 = 1.4142 f_c - 0.7071 f_c = 0.7071 f_c$

1/3-Octave Bands: For pinpointing pure-tone whine, blade-pass frequencies, or resonant structural vibration, each octave band is split into three sub-bands where f2 / f1 = 2(1/3) = ∛2 ≈ 1.2599.

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3. Frequency Weighting Networks: A, C, and Z Weightings

The human auditory system does not possess a flat frequency response. The human ear is biologically tuned to be most sensitive in the speech intelligibility frequencies between 1,000 Hz and 4,000 Hz (peaking around the ear canal resonance at 3,000--4,000 Hz) and is remarkably insensitive to low-frequency acoustic energy.

   +-------------------------------------------------------------------------+
   |                  FREQUENCY WEIGHTING PROFILES (A, C, Z)                 |
   +-------------------------------------------------------------------------+
   |                                                                         |
   |  Gain (dB)                                                              |
   |    +10 |                                  .. C-Weighting (Flat)         |
   |      0 |---------------------------------''-------------------- Z-Flat  |
   |    -10 |                       .---.                                    |
   |    -20 |                     .'     '.  A-Weighting (40-phon contour)   |
   |    -30 |                   .'                                           |
   |    -40 |       .----------'                                             |
   |    -50 |     .'                                                         |
   |        +--------------------------------------------------------------  |
   |            31.5   63   125   250   500   1k   2k   4k   8k   16k (Hz)  |
   +-------------------------------------------------------------------------+

The Standard Weighting Networks

1. A-Weighting (extdBA):
   • Designed to mimic the 40-phon equal-loudness contour (Fletcher-Munson curve).
   • Heavily penalizes (attenuates) low frequencies (e.g., -39.4 dB at 31.5 Hz; -16.1 dB at 125 Hz) and slightly boosts frequencies around 2.5 kHz.
   • Regulatory Mandate: Mandated by OSHA (29 CFR 1910.95), ACGIH, and NIOSH for all occupational noise exposure monitoring, dose calculations, and hearing conservation risk assessments.
2. C-Weighting (extdBC):
   • Designed to mimic the 100-phon equal-loudness contour for high-intensity acoustic fields.
   • Provides an essentially flat response across the audible spectrum (31.5 Hz to 8 kHz), attenuating only -3.0 dB at 31.5 Hz and -3.0 dB at 8 kHz.
   • Primary Uses: Peak/impact noise measurement (140 dBC ceiling), evaluating low-frequency noise (e.g., diesel engines, large fans), and calculating real-world Hearing Protection Device (HPD) attenuation.
3. Z-Weighting (extdBZ / Linear / Flat):
   • "Zero-weighting" (IEC 61672-1), providing a flat, unattenuated acoustic response from 10 Hz to 20 kHz ± 1.5 dB.
   • Primary Uses: Engineering frequency analysis, octave band sound power characterization, and acoustic enclosure design.

Standard Octave Band Weighting Correction Factors

Converting Octave Bands to Overall dBA: To calculate the overall A-weighted sound level from octave band measurements: add the A-weighting correction Δ LA to each octave band level, then sum the resulting levels logarithmically.

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4. Instrument Dynamic Response Settings and Standards

Sound Level Meters (SLMs) and personal noise dosimeters incorporate standardized dynamic ballistic response settings that govern how the meter's detector averages fluctuating acoustic signals over time.

   +-------------------------------------------------------------------------+
   |                  INSTRUMENT DYNAMIC TIME CONSTANTS                      |
   +-------------------------------------------------------------------------+
   |                                                                         |
   |  1. SLOW RESPONSE (τ = 1000 ms = 1.0 s):                                |
   |     - Heavily damped exponential time-averaging.                        |
   |     - Smooths out fluctuating sound levels to yield a steady reading.   |
   |     - MANDATED BY OSHA 1910.95 for occupational noise surveys!          |
   |                                                                         |
   |  2. FAST RESPONSE (τ = 125 ms = 1/8 s):                                 |
   |     - Rapid response tracking transient sound level variations.         |
   |     - Used for identifying cycling machinery and brief task events.     |
   |                                                                         |
   |  3. PEAK / IMPULSE DETECTOR (Rise Time < 35 µs to 100 µs):              |
   |     - Captures instantaneous unweighted or C-weighted pressure peaks.   |
   |     - Required for impact noise (punch presses, explosions, gunshots).  |
   +-------------------------------------------------------------------------+

Dynamic Response Specifications

• Slow Response: Exponential time constant τ = 1,000 ms (1.0 second). Mandated by OSHA for occupational compliance monitoring because it dampens visual meter fluctuations, enabling reproducible human reading.
• Fast Response: Exponential time constant τ = 125 ms (1/8 second). Four times faster than slow response; tracks rapidly changing acoustic events.
• Peak Response: Rise time constant < 35 microseconds (µs) (ANSI S1.43 / IEC 61672) to capture the true instantaneous acoustic peak pressure (Lpeak), holding the peak value for decay evaluation. Mandated by OSHA for verifying compliance with the 140 dB peak sound pressure limit.

Sound Level Meter (SLM) Classifications (ANSI S1.4 / IEC 61672)

• Class 1 / Type 1 (Precision): Laboratory and precision field grade instrument with tight tolerance limits (± 1.0 dB across 10 Hz to 20 kHz). Required for complex acoustic research, legal testimony, and environmental boundary compliance.
• Class 2 / Type 2 (General Purpose): General field survey grade instrument with broader tolerance limits (± 2.0 dB across 100 Hz to 8 kHz). Accepted by OSHA for routine workplace occupational noise surveys and dosimeter compliance.

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5. Worked Step-by-Step Calculation Examples

Worked Example 11.1.1: Logarithmic Summation of Multiple Machine Sources

Problem: An industrial hygiene survey in a machining shop measures the sound pressure levels of four machines operating independently at a workstation:

• Machine A (Stamping press): 91.0 dBA
• Machine B (CNC milling center): 88.0 dBA
• Machine C (Hydraulic power pack): 88.0 dBA
• Machine D (Parts tumbler): 82.0 dBA

1. Calculate the total combined sound pressure level (Ltotal) when all four machines operate simultaneously using the exact logarithmic addition formula.
2. Cross-verify the calculation using the CIH rule-of-thumb shortcut addition method.

Solution Steps:

1. Apply the exact logarithmic summation formula (Ltotal = 10 log10[Σ 10(Li / 10)]): $10^{91.0 / 10} = 10^{9.10} = 1,258,925,412$ $10^{88.0 / 10} = 10^{8.80} = 630,957,344$ $10^{88.0 / 10} = 10^{8.80} = 630,957,344$ $10^{82.0 / 10} = 10^{8.20} = 158,489,319$ $\sum 10^{L_i / 10} = 1,258,925,412 + 630,957,344 + 630,957,344 + 158,489,319 = 2,679,329,419$ $L_{\text{total}} = 10 \log_{10}(2,679,329,419) = 10 \times 9.4280 = 94.28\text{ dBA} \approx \mathbf{94.3\text{ dBA}}$

2. Shortcut Verification:

   • Combine two identical 88.0 dBA sources: 88.0 + 3.0 = 91.0 dBA.
   • Combine this with Machine A (91.0 dBA): 91.0 + 91.0 = 91.0 + 3.0 = 94.0 dBA.
   • Combine 94.0 dBA with Machine D (82.0 dBA): Difference is 94.0 - 82.0 = 12.0 dB (difference ≥ 10 dB → +0.0 dB). Combined total = 94.0 dBA (exact 94.3 dBA).

Result: The total workstation noise level with all machines active is 94.3 dBA.

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Worked Example 11.1.2: Correcting for Background Ambient Noise

Problem: An industrial hygienist tests a newly installed exhaust blower. With the blower running, the sound level meter measures a total sound pressure level of Ltotal = 89.0 dBA. When the blower is switched off, the ambient plant background noise level is measured at Lbg = 85.0 dBA. What is the true acoustic sound pressure level generated by the exhaust blower alone (Lblower)?

Solution Steps:

1. Identify the measured levels: Ltotal = 89.0 dBA, Lbg = 85.0 dBA.

2. Evaluate difference: Δ L = 89.0 - 85.0 = 4.0 dB. (Since Δ L ≥ 3 dB, background correction is valid).

3. Apply the logarithmic subtraction formula (Lsource = 10 log10[10^Ltotal/10 - 10^Lbg/10]): $10^{89.0 / 10} = 10^{8.90} = 794,328,235$ $10^{85.0 / 10} = 10^{8.50} = 316,227,766$ $10^{L_{\text{blower}}/10} = 794,328,235 - 316,227,766 = 478,100,469$ $L_{\text{blower}} = 10 \log_{10}(478,100,469) = 10 \times 8.6795 = 86.795\text{ dBA} \approx \mathbf{86.8\text{ dBA}}$

4. Shortcut Verification: For a 4 dB difference (89 - 85), the table instructs subtracting 2.0 dB from Ltotal: 89.0 - 2.0 = 87.0 dBA (in close agreement with 86.8 dBA).

Result: The true sound pressure level generated exclusively by the exhaust blower is 86.8 dBA.

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Worked Example 11.1.3: Inverse Square Law Propagation for a Point Source

Problem: A high-pressure air compressor operates as an omnidirectional point source on an open concrete slab (Q = 2, hemispherical radiation). At a distance r1 = 3.0 meters from the compressor, the sound pressure level is measured as Lp1 = 96.0 dBA.

1. Calculate the sound pressure level (Lp2) at a boundary fence line located r2 = 24.0 meters away, assuming free-field conditions.
2. Determine the total Sound Power Level (Lw) of the compressor.

Solution Steps:

1. Apply the Point Source Inverse Square Law (Lp2 = Lp1 - 20 log10[r2 / r1]): $\frac{r_2}{r_1} = \frac{24.0\text{ m}}{3.0\text{ m}} = 8.0$ $20 \log_{10}(8.0) = 20 \times 0.90309 = 18.06\text{ dB}$ $L_{p2} = 96.0\text{ dBA} - 18.06\text{ dB} = 77.94\text{ dBA} \approx \mathbf{77.9\text{ dBA}}$ (Shortcut check: Distance doubles from 3 to 6 m [-6 dB], 6 to 12 m [-6 dB], and 12 to 24 m [-6 dB]. Total attenuation = 3 × 6 = 18 dB → 96 - 18 = 78 dBA).

2. Calculate Sound Power Level (Lw = Lp - 10 log10[Q / (4 π r²)]): $L_p = 96.0\text{ dB at } r = 3.0\text{ m}, \quad Q = 2$ $\frac{Q}{4 \pi r^2} = \frac{2}{4 \pi (3.0)^2} = \frac{2}{113.097} = 0.017684$ $10 \log_{10}(0.017684) = 10 \times (-1.7524) = -17.52\text{ dB}$ $L_w = 96.0 - (-17.52) = 96.0 + 17.52 = \mathbf{113.5\text{ dB}}$

Result: The sound level at 24 meters is 77.9 dBA, and the compressor's sound power level is 113.5 dB.