MATH TOOLBOX The Case of the NOISY WORKPLACE By Mitch Ricketts Math Toolbox is designed to help readers apply STEM principles to everyday safety issues. Many readers may feel apprehensive about math and science. This series employs various communication strategies to make the learning process easier and more accessible. It has long been known that overexpo- micropascal): is the actual sound pressure measured in sure to loud sounds may cause hearing loss, •One pascal (Pa) is about one hun- the environment. The other sound pres- tinnitus (i.e., ringing in the ears) and other dred-thousandth of standard atmo- sure, p0, is a hypothetical reference sound serious health effects (Basner, Babisch, Da- spheric pressure. Standard atmospheric pressure. In OSH, the reference sound vis, et al., 2014). Hazardous noise levels are pressure is equal to about 14.7 pounds pressure is usually taken to be 20 µPa, or common in many workplaces and the effects per square inch (14.7 psi). This means one alternatively, 0.00002 Pa. This is because on health may surface even among relatively pascal represents a very small pressure of 20 µPa represents the softest sound the av- young workers, as illustrated in Figure 1. about 0.000147 psi (1.47 x 10-4 psi). erage person can distinguish at a range of OSH professionals are often asked to •One micropascal (µPa) is one millionth sound frequencies important to the human identify and control noise hazards. Al- of one pascal, or about 1.47 x 10-10 psi. auditory system. In other words, 20 µPa is though noise control may seem straight- In contrast to the direct measurement of the threshold of hearing, equal to a sound forward, certain concepts are frequently sound pressure in pascals, sound pressure pressure level of 0 decibels. misunderstood. This article explores ter- level (Lp) is an indirect logarithmic mea- In workplaces, sound pressure levels (Lp) minology and calculations to help explain sure based on a ratio of two different sound are normally expressed in units of decibels how sound pressure, sound-pressure level pressures: p and p0. The sound pressure, p, (dB) that reflect loudness increments that and the decibel scale relate to the pressure variations that we call sound. FIGURE 1 WORKPLACE HEARING LOSS, MICHIGAN Sound & Noise Exposure Concepts Noise is sometimes defined as unwanted sound. In air, sound consists of pressure A worker was employed for 13 years in a He also used outdoor power equipment waves emitted from objects that vibrate noisy assembly plant. at home. or move suddenly. Sound-producing ob- jects include guitar strings, tuning forks, clapping hands, vocal cords, rustling leaves, machinery and explosives. Each passing sound wave compresses the air, crowding the molecules together. After the wave’s high-pressure band passes, the air rebounds and expands as molecules move apart again. Thus, sound waves can be imagined as repeating cycles of high and low pressure that spread outward from a source (Figure 2, p. 46). Sound waves may also travel as disturbances of matter in His company provided hearing protectors, Before the age of 40, he had developed liquids and solids, such as water and steel. but he rarely wore them. noise-induced hearing loss that was especially severe at higher frequencies. We perceive sound because pressure waves transmit energy, causing vibra- Audiogram Worker age: Late 30s tions in our eardrums, middle ear bones ) 0 dB Right ear (ossicles), cochlear fluid and inner ear 20 nerve cells. At moderate sound levels, our 40 auditory systems extract useful informa- tion from sound waves. Unfortunately, 60 nerve damage and hearing loss can occur 80 Hearing loss ( Left ear when sound levels are extreme. 100 Sound pressure (p) is the measurable 0.25 0.5 1 2 3 4 6 8 fluctuation in pressure caused by a sound Frequency (kilohertz, kHz) wave. As shown in Figure 2 (p. 46), sound pressure is reflected in the amplitude of a Note. Adapted from “2008 Annual Report on Work-Related Noise-Induced Hearing Loss in Mich- wave. Sound pressure is stated in the inter- igan,” by K.D. Rosenman, A. Krizek, M.J. Reilly, et al., 2009), East Lansing, MI: Michigan State national unit known as the pascal (or more University & Michigan OSHA. commonly, the fractional unit called the assp.org APRIL 2020 PROFESSIONAL SAFETY PSJ 45 MATH TOOLBOX can be detected by average people. For quiet FIGURE 2 sounds, people can detect small changes in loudness; however, for loud sounds, only SOUND PRESSURE WAVES large changes are detected. Near the thresh- old of hearing (0 dB), the average person Sound pressure waves are alternating bands of Sound waves can be depicted as curves, with can detect an increase in sound pressure high and low pressure emitted from a source peaks and troughs corresponding to high and of about 2.24 µPa, which represents an in- of sound. low pressure bands. crease of about 1 dB in a very quiet environ- High pressure bands (blue) ment. On the other hand, if the same person is exposed to a sound pressure of 2,000,000 µPa (100 dB), s/he will not detect a change in loudness unless the sound pressure in- creases by about 244,037 µPa (representing Amplitude equals half the difference between an increase of about 1 dB in this noisier minimum and maximum pressure. Larger environment). We will see in the follow- amplitudes reflect higher sound pressures. ing exercises that an increase of just a few High amplitude Low amplitude decibels in a loud environment represents a serious hazard because the corresponding rise in micropascals is large. In contrast, a small decibel increase is less hazardous in a quiet environment, where the change in Low pressure bands (white) micropascals is small. An important note: OSHA and NIOSH normally express exposure limits as dBA, cals (µPa), or alternatively, pascals (Pa) Step 2. Insert the known values for the which stands for decibels on the A-weight- p = reference sound pressure (thresh- sound pressure measured in the environ- ed scale. Sound pressure levels can be ex- 0 old of hearing; 0 dB); p0 = 20 µPa, or ment (p = 283,000 µPa) and the reference pressed on different frequency-weighting alternatively, 0.00002 Pa sound pressure (p = 20 µPa). Then solve scales. The A-weighted scale emphasizes 0 Note: Be sure to use the same pressure for sound pressure level in decibels (Lp): sound frequencies to which humans are metric (either µPa or Pa) to express vari- most sensitive, while other scales (such ables p and p0 throughout the equation. ,-.,+++ as the C- and Z-weighted scales) do not. " *+ (on the A-weighted scale) = 20 ∙ ,+ = 83.02 When monitoring sound pressure levels Calculating Decibels Note: Most calculators have a LOG in the workplace, we normally set our Based on Sound Pressure button that will provide the correct noise meters to the A-weighted scale if we Today’s noise monitoring devices au- answer, with keystrokes similar to the plan to compare our results with exposure tomatically calculate and display sound following in this case: 20 X LOG (283000 limits established by OSHA and NIOSH. pressure levels in decibels (Figure 3). How- ÷ 20) =. Alternatively, in a spreadsheet, In this article, we will calculate decibels ever, it is easier to interpret these numbers the proper formula for this example is based on sound pressures (p) in work- if we understand the calculations. Fur- = 20*LOG10(283000/20). places. In the next Math Toolbox article, thermore, we may be asked to perform the Step 3. Our calculation indicates we will practice other decibel calculations calculations on certification exams. the sound pressure level is 83.02 dB on based on sound power, in watts per square To begin, imagine a worker’s 8-hour 2 the A-weighted scale (83.02 dBA) as an meter (W/m ). Although these two meth- time-weighted average (TWA) A-weighted 8-hour TWA exposure. ods are based on different characteristics sound pressure exposure is found to be To interpret the measured result, refer of sound, we will obtain the same result in 283,000 µPa. Since our measured sound to the NIOSH recommended exposure decibels regardless of the method we use. pressure is stated in units of µPa, we will limit (REL) of 85 dBA as an 8-hour TWA. use 20 µPa for the reference sound pressure Our calculated daily average noise Sound Pressure Level Equation because this is equivalent to zero decibels. exposure of 83.02 dBA does not exceed To calculate decibels from sound pres- Here is a summary of the information the NIOSH REL of 85 dBA as an 8-hour sure, we may use either micropascals or we will use to calculate the sound pres- TWA. Thus, we conclude that an average pascals as our unit of pressure as long as sure level in decibels: exposure to 83.02 dBA does not trigger we maintain consistency throughout the •The measured A-weighted sound pres- any special NIOSH recommendations. following equation: sure is 283,000 µPa as an 8-hour TWA Alternate example: Calculate the sound exposure for the worker in this environ- pressure level in decibels for a different envi- ment. This is the value of p in the formula. ronmental sound level. In this case, imagine " = 20 ∙ *+ •We are using units of micropascals, so where: + a worker’s 8-hour TWA A-weighted sound Lp = sound pressure level, in decibels the reference sound pressure is 20 µPa. pressure exposure is 1,124,683 µPa, which (dB) This is the value of p0 in the formula. is equivalent to 1.124683 pascals (Pa).