Occupational Soundscapes – Part 3: The Decibel Scale

     In all likelihood, readers of this series have encountered the decibel scale many times.  It may have been used in the specifications of new machinery or personal electronic devices.  Some may be able to intuit the practical application of these values, but it is likely that many lack knowledge of the true meaning and implications of the decibel scale.
     This installment of the “Occupational Soundscapes” series introduces the decibel (dB) and its relevance to occupational noise assessment and hearing conservation.  Those with no exposure to the scale and those that have a functional understanding, but lack foundational knowledge, benefit from understanding its mathematical basis.  The characteristics of sound to which it is most-often applied is also presented to continue developing the knowledge required to effectively support a hearing loss prevention program (HLPP).

The Decibel
     Two key characteristics of the decibel scale define its use and contribute greatly to its lack of common understanding.  First, it is a logarithmic scale.  Linear scales are more common, which may lead those unfamiliar with the decibel scale to assume it, too, is linear.
     Second, the decibel scale is a comparative measure, incorporating the ratio of the measured quantity to a reference value.  Absolute scales are more common, potentially leading to another erroneous assumption.
     Making either assumption leads to gross misinterpretation of the information provided by cited values.  Mathematically, the general expression of the decibel scale is:

(all logarithms cited are base 10, log10, unless otherwise specified).  The multiplication factor of 10 converts Bels to decibels.  One Bel is defined as the increase corresponding to a tenfold increase in the ratio of values.  A decibel (dB) is, therefore, one tenth of a Bel.  The nature of the scale yields a dimensionless value that is valid for any system of units.

Sound Parameters
     To use the decibel scale effectively, in the context of occupational soundscapes, the interrelationships of power, intensity, and pressure must be understood.  Differentiating these measures is critical to understanding the true nature of the sound environment under scrutiny.
     Sound power (W), measured in watts (W), is the amount of acoustical energy produced by a sound source per unit time.  It is a characteristic of the source and is, therefore, independent of its location or surroundings.  In this discussion, it is assumed that sounds are generated by point sources, with sound dispersing spherically; variations will be introduced later.
     Sound intensity (I), measured in watts per square meter (W/m^2), is the sound power per unit area.  It is dependent on location, as it accounts for the dispersion of sound energy at a specified radial distance from the source:

The equation reveals that intensity decreases with the square of the distance from the source.  This inverse square law is depicted in Exhibit 1.

     Sound intensity, I, is a vector quantity.  In free-field conditions, however, the lack of obstructions and reflecting surfaces renders the specification of direction moot.  The intensity at a given distance from the source is equal in all directions.
     Sound pressure (P), measured in newtons per square meter (N/m^2) or, equivalently, Pascal (Pa), is the variable air pressure (force per unit area) superimposed on atmospheric pressure.  Propagation of pressure fluctuations as sound waves was introduced in Part 2; root mean square pressure (PRMS) is typically used.  Sound pressure is an effect of sound power generated by a source; it is influenced by the surrounding environment and distance from the source.
     Of the three parameters described, only pressure can be measured directly.  With adequate pressure data, however, it is possible to work backwards to obtain intensity and power values.  To do this, first calculate the RMS pressure of the sound wave.
     Sound intensity is calculated using the following formula:

where P is the RMS pressure (Pa), ρ is the density of air, and c is the speed of sound in air.
     At standard conditions, ρ = 1.2 kg/m^3; though the density of air varies, this approximation provides sufficient accuracy for most purposes.  Likewise, the approximation of c = 343 m/s will typically suffice.
      With the intensity at a known distance from the source, calculating sound power is simple:

where A is the spherical area at distance r (A = 4 Π r^2).
 
Sound Levels
     In the previous section, sound power, intensity, and pressure were discussed in absolute terms.  More often, however, these measures are referenced by their levels, using the decibel scale.  Doing so makes the very wide range of values encountered more manageable.
     The sound power level (LW or PWL) is calculated using the general expression of the decibel scale, rewritten as:

where Wref is the reference power value; Wref = 10^-12 W.
     Likewise, the sound intensity level (LI or SIL) is calculated with the general expression rewritten as:

where Iref is the reference intensity value; Iref = 10^-12 W/m^2.
     Using the expression for LI and the inverse square law, it can be shown that 6 dB of attenuation is attained by doubling the distance from the source.  Choosing an arbitrary value, (I/Iref) = 40, at distance r, we get LI(r) = 10 log (40) = 16 dB.  Doubling the distance increases the area of the hypothetical sphere by a factor of 4 (see Exhibit 1).  With power constant, this increased area reduces intensity by a factor of 4, which, in turn, reduces (I/Iref) by the same factor.  Therefore, for our example, (I/Iref) = 10 at distance 2r and we get LI(2r) = 10 log (10) = 10 dB, a reduction of 6 dB.
     Equating the two expressions for I, above, and rearranging, we get

In this form, it is easy to see that the square of pressure varies with r^2, while power and intensity (i.e. first power) vary with r^2.  Thus the general expression is rewritten for the sound pressure level (LP or SPL) as:

Pref is the reference pressure value; Pref = 2 x 10-5 N/m^2 = 20 μPa, corresponding to the threshold of human hearing at 1000 Hz.  Exhibit 2 provides examples of decibel scale levels and corresponding absolute values of sound power, intensity, and pressure.  The following should be noted in the table:

• Each of the reference values correspond to 0 dB – when the ratio = 1, log (1) = 0.
• Power and intensity are numerically equal at equal dB levels – numerically equal reference values are used.
• All decimal places are shown, but small values of power and intensity are typically expressed in scientific notation (e.g. 10^-12) and small values of pressure are typically expressed in μPa.
  
• Values in the table range from the threshold of human hearing to far beyond the threshold of pain (the range of human hearing will be discussed further in a future installment).
  

     Sound power and intensity levels are useful for acoustics projects – designing sound systems, venues, etc. – but sound pressure levels are most useful in quantifying occupational environments and supporting hearing conservation programs.  Examples of typical sound pressure levels encountered in commercial, recreational, and other settings are shown in Exhibit 3.  The “Noise Navigator,” an extensive database compiled and published by 3M Corporation, is available online.  In it, measurements of numerous sound levels are recorded, providing more useful data for research and planning purposes.

     Thus far, sounds have been treated as if generated by a singular point source in free-field conditions (no interference in spherical transmission).  Realistic soundscapes, however, are comprised of multiple complex sounds from various sources in environments where obstructions and reflective surfaces are ubiquitous.  In the next installment, the “Occupational Soundscapes” series begins to tackle the challenges of real-world conditions, presenting methods for assessing the effects of multiple simultaneous sounds on sound pressure levels.

     For additional guidance or assistance with Safety, Health, and Environmental (SHE) issues, or other Operations challenges, feel free to leave a comment, contact JayWink Solutions, or schedule an appointment.

     For a directory of “Occupational Soundscapes” volumes on “The Third Degree,” see Part 1: An Introduction to Noise-Induced Hearing Loss (26Jul2023).
 
References
[Link] The Noise Manual, 6ed.  D.K. Meinke, E.H. Berger, R.L. Neitzel, D.P. Driscoll, and K. Bright, eds.  The American Industrial Hygiene Association (AIHA); 2022.
[Link] “Noise – Measurement And Its Effects.”  Student Manual, Occupational Hygiene Training Association; January 2009.
[Link] An Introduction to Acoustics.  Robert H. Randall.  Addison-Wesley; 1951.
[Link] “OSHA Technical Manual (OTM) - Section III: Chapter 5 - Noise.”  Occupational Safety and Health Administration; July 6, 2022.
[Link] Noise Control in Industry – A Practical Guide.  Nicholas P. Cheremisinoff.  Noyes Publications, 1996.
[Link] “Noise Navigator Sound Level Database, v1.8.” Elliot H. Berger, Rick Neitzel, and Cynthia A. Kladden.  3M Personal Safety Division; June 26, 2015.
[Link] “Sound Intensity.”  Brüel & Kjaer; Septermber 1993.

Jody W. Phelps, MSc, PMP®, MBA
Principal Consultant
JayWink Solutions, LLC
jody@jaywink.com