Acoustics Noise Decibels

Overview

Engineering reference data for Acoustics Noise Decibels in acoustics.

Key Formulas

Speed of Sound

$c = \sqrt{\gamma R T}$

Speed of sound in an ideal gas.

Sound Level

$L = 10 \log_{10}(I/I_0)$

Decibel level.

Wavelength

$\lambda = c / f$

Wavelength = speed / frequency.

Variables

Sound Level Weighting Filters

Sound pressure filters (A, B, and C weightings) compensate for the frequency response of the human ear at different sound levels.

Noise Rating Systems

Several standardized systems are used to evaluate and specify acceptable background noise levels in buildings:

Noise Rating (NR) Curves

NR curves define acceptable indoor noise environments for:

• Hearing preservation
• Speech communication
• Annoyance reduction

Comparison of Rating Systems

Sound Propagation

Indoor Sound Propagation

In an enclosed space, sound reaches a receiver as both direct and reverberant sound. The room constant $R$ governs the reverberant field:

$R = \frac{S \bar{\alpha}}{1 - \bar{\alpha}}$

where $S$ is total surface area and $\bar{\alpha}$ is average absorption coefficient.

Outdoor Sound Attenuation

For a point source in free field, sound level decreases with distance:

$\Delta L = 20 \log_{10}\left(\frac{r_2}{r_1}\right)$

For a line source (e.g., road traffic):

$\Delta L = 10 \log_{10}\left(\frac{r_2}{r_1}\right)$

Sound Barriers and Fresnel Number

The effectiveness of a partial barrier is characterized by the Fresnel number:

$N = \frac{2\delta}{\lambda}$

where $\delta$ is the path length difference (direct path minus diffracted path) and $\lambda$ is the wavelength.

Noise Exposure Standards

NIOSH Recommended Exposure Limit (REL)

• 85 dBA for 8-hour time-weighted average (TWA)
• Exchange rate: 3 dB (exposure time halves for each 3 dB increase)

Maximum Permissible Exposure Duration (NIOSH)

$T_{max} = \frac{8}{2^{(L-85)/3}} \text{ hours}$

where $L$ is the sound level in dBA.

Speed of Sound in Various Media

The speed of sound depends on the medium's elastic properties and density:

$c = \sqrt{\frac{K}{\rho}}$

where $K$ is the bulk modulus and $\rho$ is the density.

In Air (at 1 atm)

$c_{air} = 331.3 + 0.606 \cdot T \quad \text{[m/s]}$

where $T$ is temperature in °C.

Typical Values

Speech Interference Levels

Speech Interference Level (SIL) quantifies background noise frequencies that interfere with speech communication, typically evaluated in the 500–4000 Hz octave bands.

Required Voice Level at Distance

Adding Decibels from Multiple Sources

Since decibels follow a logarithmic scale, they cannot be added arithmetically. For $n$ equal sound sources:

$L_{total} = L_{single} + 10 \log_{10}(n)$

For two sources with levels $L_1$ and $L_2$:

$L_{total} = 10 \log_{10}\left(10^{L_1/10} + 10^{L_2/10}\right)$

References

• Original Source

Acoustic Impedance

Acoustic impedance is a fundamental property describing how a medium resists sound wave propagation. It combines the medium's density and sound speed.

Specific Acoustic Impedance relates sound pressure to particle velocity: $Z_s = \frac{p}{v} = \rho c$ where $Z_s$ is specific acoustic impedance (Pa·s/m or rayl), $p$ is sound pressure (Pa), $v$ is particle velocity (m/s), $\rho$ is medium density (kg/m³), and $c$ is speed of sound (m/s).

Characteristic Acoustic Impedance for a medium is: $Z_c = \rho c$ For air at 20°C: $Z_c \approx 415$ Pa·s/m For water: $Z_c \approx 1.5 \times 10^6$ Pa·s/m

Impedance mismatch between media affects sound transmission. Large mismatches (like air-to-water) cause high reflection.

Floor Vibrations

Human activities and machinery can induce floor vibrations through resonance effects.

Natural Frequency of Floor Systems can be estimated by: $f_n = \frac{\pi}{2L^2} \sqrt{\frac{EI}{m}}$ where $f_n$ is natural frequency (Hz), $L$ is span length (m), $E$ is modulus of elasticity (Pa), $I$ is moment of inertia (m⁴), and $m$ is mass per unit length (kg/m).

Peak Particle Velocity for vibration assessment: $v = 2\pi f A$ where $v$ is velocity (m/s), $f$ is frequency (Hz), and $A$ is displacement amplitude (m).

Vibration Acceptability Criteria for human comfort typically follow guidelines like ISO 2631 or BS 6472, with thresholds varying by building use and frequency.

Fan Noise Characteristics

Blade Pass Frequency is a primary tonal component in fan noise: $f_{BPF} = N \cdot \frac{RPM}{60}$ where $f_{BPF}$ is blade pass frequency (Hz), $N$ is number of blades, and $RPM$ is rotations per minute.

Sound Power Level generated by a fan: $L_W = 10 \log_{10}\left(\frac{Q \cdot \Delta P}{10^{-12}}\right) + C$ where $L_W$ is sound power level (dB re 10⁻¹² W), $Q$ is volume flow rate (m³/s), $\Delta P$ is total pressure rise (Pa), and $C$ is fan-specific constant (typically 5-15 dB).

Fan Noise Reduction with Silencers: $\Delta L = 10 \log_{10}\left(\frac{A_2}{A_1}\right) + \alpha \cdot L$ where $\Delta L$ is insertion loss (dB), $A_1$ and $A_2$ are cross-sectional areas, $\alpha$ is attenuation coefficient (dB/m), and $L$ is length (m).

Speed of Sound in Air vs Temperature

The speed of sound in air varies with temperature approximately by: $c_{air} = 331.3 \sqrt{1 + \frac{T}{273.15}}$ where $c_{air}$ is speed of sound (m/s) and $T$ is temperature (°C).

For engineering calculations:

• At 0°C: $c \approx 331$ m/s
• At 20°C: $c \approx 343$ m/s
• At 30°C: $c \approx 349$ m/s

This relationship is valid for temperatures between -40°C and 1000°C at standard atmospheric pressure.

Additional Noise Rating Systems

Room Criteria (RC) measures background noise across 16-4000 Hz octave bands: $RC = \frac{1}{N} \sum_{i=1}^{N} L_{pi} - 10 \log_{10}\left(\frac{f_i}{1000}\right)$ where $RC$ is room criteria level, $L_{pi}$ is octave band sound pressure level (dB), $f_i$ is center frequency (Hz), and $N$ is number of bands (typically 10).

Preferred Noise Criterion (PNC) curves provide stricter limits than NC curves for critical listening spaces.

RC vs NC vs dB(A):

• RC evaluates low-frequency rumble and hiss characteristics
• NC is a single-number rating for background noise
• dB(A) is A-weighted overall level, less sensitive to low frequencies

Maximum Sound Pressure Levels in Rooms

Maximum recommended sound pressure levels for various room types to ensure acoustic comfort and speech intelligibility.

Acceptable Noise Levels at Common Locations

Typical acceptable A-weighted sound pressure levels (L_{Aeq}) in different environments for health and well-being.

Noise from Road Traffic

Noise generated by road traffic is a major source of environmental pollution. The estimated sound level depends on several factors:

Key Factors:

• Traffic Volume: Number of vehicles per hour.
• Vehicle Type Mix: Percentage of heavy trucks, buses, and light vehicles.
• Average Vehicle Speed.
• Road Surface: Porous asphalt can reduce noise by 3-6 dBA compared to dense asphalt.
• Distance from Road: Sound attenuates with distance.

A simplified empirical formula for estimating the hourly equivalent sound level L_{eq,1h} at a distance d from a road is: L_{eq,1h} = L_{E} + 10 \cdot \log_{10}(N) - 10 \cdot \log_{10}(d) - \Delta L_{surface} + C Where:

• L_{E} = Mean energy level per vehicle (depends on speed and vehicle type).
• N = Number of vehicles per hour.
• d = Perpendicular distance from the road center (m).
• \Delta L_{surface} = Surface correction (dB).
• C = Constant combining other factors.

Sound Attenuation in Lined Ducts

Duct lining (fiberglass, foam, etc.) is a common method for attenuating noise in HVAC systems. The attenuation (insertion loss) depends on lining thickness, duct dimensions, and frequency.

Typical Attenuation in Rectangular, Straight, Sheet-Metal Ducts with 25mm Fiberglass Lining: The attenuation increases with frequency. For a standard rectangular duct:

• Low Frequencies (125 Hz): 0.1 - 0.3 dB/m
• Mid Frequencies (500 Hz): 0.5 - 1.5 dB/m
• High Frequencies (2000 Hz): 1.0 - 4.0 dB/m

General Principle: Thicker lining and smaller duct cross-sections provide greater attenuation, especially at higher frequencies.

EPA Protective Noise Levels

The U.S. Environmental Protection Agency (EPA) identified levels of environmental noise necessary to protect public health and welfare, including preventing hearing loss, ensuring adequate sleep, and allowing for speech communication.

Maximum Daily Noise Dose

Occupational noise exposure is often regulated by a dose limit. The Noise Dose (D) is a function of the actual exposure time at a given level and the duration permitted at that level.

NIOSH and OSHA use a 3-dB exchange rate: For every 3 dB increase in sound level, the permissible exposure time is halved.

Formula for Calculating Noise Dose (D): D = 100 \times \left( \frac{C_1}{T_1} + \frac{C_2}{T_2} + ... + \frac{C_n}{T_n} \right) Where:

• C_i = Actual time of exposure at a specific sound level (hours).
• T_i = Permissible exposure time at that sound level (hours).
• The total dose D of 100% corresponds to the maximum permissible exposure limit (e.g., 85 dBA for 8 hours under NIOSH REL).

Permissible Exposure Time (T) for an 8-hr workday (3-dB exchange rate): T = \frac{8}{2^{(L - L_{ref}) / 3}} Where L is the measured sound level (dBA) and L_{ref} is the reference level (e.g., 85 dBA for NIOSH REL, 90 dBA for OSHA PEL).