Occupational Soundscapes – Part 4: Sound Math

     Occupational soundscapes, as outlined in Part 1, are comprised of many sounds.  Each has a unique source and set of defining characteristics.  For some purposes, treating all sounds in combination may be appropriate.  For others, the ability to isolate sounds is integral to the purpose of measuring sound levels.
     Of particular importance to a hearing loss prevention program (HLPP) is the ability to add, subtract, and average contributions to the sound pressure level (LP, SPL) in a workplace.  The ratios and logarithms used to calculate SPLs, presented in Part 3, complicate the arithmetic, but only moderately.  This installment of the “Occupational Soundscapes” series introduces the mathematics of sound, enabling readers to evaluate multiple sound sources present in workers’ environs.

     As mentioned in Part 3, sound pressure is influenced by the environment.  The number of sources, the sound power generated by each, and one’s location relative to each source contribute to the sound pressure level to which a person is subjected.

SPL Addition
     A representative example of a typical application will be used to place SPL addition in context.  This should make it easier to understand the process and its value to hearing conservation efforts.  Consider a manufacturing operation with several machines running in one department.  The company’s industrial hygienist has tasked the department manager with reducing the noise to which operators are exposed.  With a capital budget insufficient to replace machines or make significant modifications to the facility, the manager concludes that the only feasible option is to schedule work within the department such that, at all times, some machines remain idle (i.e. quiet).  To determine a machine schedule that will yield acceptable noise exposures while meeting production demands, SPLs generated by each machine are added in various combinations.
     A baseline for comparison must be established to evaluate sound-level reduction results.  The baseline SPL includes sounds from all sources and can be established by either the formula method or the table method.
     Using the formula method, the total (i.e. combined) SPL generated by n sources is calculated with the following equation:

where LPt is the total SPL and LPi is the SPL generated by the ith source.  The calculation of total SPL for our example, which includes five machines, is tabulated in Exhibit 1, where it is found to be 96.8 dB.

     To use the table method, first sort the source SPLs in descending order.  Compare the two highest SPLs and determine the difference.  Find this value in the left column of the table in Exhibit 2 and the corresponding value in the right column.  Interpolation may be necessary, as only integer differences are tabulated.  Add the value from the right column to the higher SPL to obtain the combined SPL to be used in subsequent iterations.

     Compare the combined SPL to the next source in the sorted list, repeating the process described until all source SPLs have been added or they no longer contribute to the total SPL.  When adding a large number sources, sorting SPLs first may allow the process to be abbreviated; once the difference between the combined SPL and the next source exceeds 10 dB, the remainder of the list need not be considered.
     A pictorial representation of the cascading calculations performed in the table method of SPL addition is provided in Exhibit 3, where the total SPL for our example is found to be 96.5 dB.  This result differs slightly from that attained by the more-precise formula method, but this need not be a concern.  The reduced complexity of computation often justifies the sacrifice of accuracy.  A 0.3 dB difference, like that found in our example, is imperceptible to humans and is, therefore, inconsequential.  While some circumstances warrant use of the formula method, the table method of SPL addition provides a convenient estimate without the need for a calculator.

     The department manager proposes running the machines in two groups – machines 1, 3, and 5 will run simultaneously, alternating with machines 2 and 4.  Total SPL calculations for each machine grouping, using the formula method and the table method, are shown in Exhibit 4 and Exhibit 5, respectively.

     Total SPL results are the same for both methods – 93.3 dB for machine group (1, 3, 5) and 94.3 dB for machine group (2, 4).  This represents 3.5 dB and 2.5 dB reductions, respectively, from the baseline SPL (all machines running).  These are consequential reductions in noise exposure; the proposal is accepted and the new machine operation schedule is implemented.  The total SPL remains high, however, and further improvements should be sought.

SPL Subtraction
     To determine the SPL contribution of a single source, the “background” SPL is subtracted from the total.  Like SPL addition, there are two methods.
     Consider the 5-machine department presented in the SPL addition example; for this example, the SPL attributed to machine 3 is unknown.  With machine 3 turned off, the total SPL in the department is 95.5 dB; this is considered the background level with respect to machine 3.  Recall that the total SPL with all machines running is 96.8 dB.
     To subtract the background SPL by the formula method, use the following equation:

where LPi is the SPL of the source of interest, LPt is the total SPL (all machines running), and LPb is the background SPL (source of concern turned off).  In our example, LP3 = 10 log (10^9.68 – 10^9.55) = 90.9 dB.  The result can be verified using SPL addition:  LPt = 10 log (10^9.55 + 10^9.09) = 96.8 dB.
     To determine the SPL attributed to machine 3 by the table method, find the difference between the total and background SPLs (96.8 dB – 95.5 dB = 1.3 dB) in the left column of the table in Exhibit 6.  Subtracting the corresponding value in the right column of the table (~ 6.0 dB) from the total SPL gives the machine 3 SPL (96.8 dB – 6.0 dB = 90.8 dB).  Again, interpolation causes a small variance in the results that remains inconsequential.

SPL Averaging
     Measurements may be repeated across time or varying conditions.  In our 5-machine department example, this may be to document different machine combinations or sounds generated during specific operations.  In the latter scenario, an average SPL may be a useful, though simplified, characterization of the environment.
     SPLs are averaged using the following formula:

where n is the number of measurements to be averaged and LPi is an individual measurement.
     As an example, the SPLs of the 5-machine department example will be reinterpreted as multiple measurements in a single location.  Averaging the five SPL values (88.0, 92.5, 91.0, 89.5, and 83.5 dB) using the equation above gives LPavg = 88.9 dB.  When the range of SPLs to be averaged is small (e.g. < 5 dB), the arithmetic average can be used to approximate the decibel average.  The arithmetic and decibel average calculations for this example are shown in Exhibit 7.  Arithmetic averaging provides a convenient estimation method, but the decibel average should be calculated for any “official” purpose, as the two rapidly diverge.

     In the coming installments of the “Occupational Soundscapes” series, the connections between previous topics and hearing conservation begin to strengthen.  The discussion of audiometry brings together the physiological functioning of the ear (Part 2), speech intelligibility (introduced in Part 2), and the decibel scale (Part 3) to lay the foundation of a hearing loss prevention program.

     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.

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