Sound basics for noise control

Every decision in a hearing conservation program rests on a few numbers: the sound level at a workstation, the daily exposure of a worker, the attenuation of a protector, the reduction an enclosure provides. Those numbers are all decibels, and decibels behave in ways that surprise people who are used to ordinary arithmetic. Two machines at 90 dBA do not make 180 dBA. Halving the distance to a source does not double the level. A reduction of 3 dB sounds small but halves the acoustic energy. This lesson gives you the working knowledge of sound you need to read noise surveys, check calculations and avoid the common mistakes. It deliberately stays practical; FILM-310 Sound for Film, lesson 2, covers acoustics in more depth.

What sound is

Sound is a pressure wave travelling through air, or through another medium such as water or steel. A vibrating surface, such as a saw blade, a loudspeaker cone or the panel of a machine, pushes and pulls on the air next to it, creating tiny alternating regions of higher and lower pressure that spread outward. In air at room temperature these waves travel at about 343 metres per second.

Two properties describe a simple sound. Its amplitude, the size of the pressure fluctuation, determines how loud it is. Its frequency, the number of pressure cycles per second, measured in hertz (Hz), determines its pitch. Young, healthy ears can hear roughly 20 Hz to 20,000 Hz. Most real noises contain a broad mix of frequencies at once, and noise control engineers break them down into frequency bands to understand them.

The pressure fluctuations involved are tiny compared with atmospheric pressure, which is about 101,000 pascals. The quietest sound a young person can hear at 1,000 Hz is a fluctuation of about 20 micropascals, or 0.00002 Pa. A sound loud enough to cause pain is about 20 Pa, a million times larger. That enormous range is the reason we use a logarithmic scale.

The decibel

The sound pressure level, in decibels, compares a sound pressure with the reference pressure of 20 micropascals:

On this scale the threshold of hearing is 0 dB and the threshold of pain is about 120 dB. The logarithm compresses a range of a million to one into a range of 0 to 120, which is easier to work with and closer to the way loudness is perceived. It also changes the rules of arithmetic. A few landmarks are worth memorizing.

The first row explains a theme that runs through this course. Hearing damage depends on acoustic energy, not perceived loudness. A 3 dB increase doubles the energy reaching the ear, even though most people can barely hear the difference. That is why Canadian jurisdictions halve the allowable exposure time for every 3 dB increase in level, as Lesson 3 explains.

Adding and subtracting levels

Because decibels are logarithmic, levels from separate sources cannot be added directly. Instead, each level is converted back to a relative energy, the energies are added, and the total is converted back to decibels:

This formula applies to sources that are independent, such as two different machines, which is the usual case in workplaces. Several useful rules follow from it.

Two machines at 90 dBA together produce 93 dBA, not 180. Ten identical machines produce 10 dB more than one. A quiet machine next to a loud one barely changes the total: a 92 dBA press beside an 80 dBA conveyor gives 92.3 dBA. The practical lesson is powerful. Noise control effort should go to the loudest sources first, because quietening the smaller ones does almost nothing to the total.

The same logic works in reverse. If the total level with all machines running is 95 dBA and it drops to 92 dBA when one machine is switched off, that machine was contributing about 92 dBA on its own, because removing it halved the energy. This kind of switch-off test is a simple way to rank sources during a survey.

import math

def add_levels(levels):
    """Combine independent sound levels in dB."""
    return 10 * math.log10(sum(10 ** (L / 10) for L in levels))

def subtract_level(total, background):
    """Level of one source, given the total and the level without it."""
    return 10 * math.log10(10 ** (total / 10) - 10 ** (background / 10))

print("Two machines at 90 dBA:", round(add_levels([90, 90]), 1), "dBA")
print("Ten machines at 90 dBA:", round(add_levels([90] * 10), 1), "dBA")
print("Press 92 + conveyor 80:", round(add_levels([92, 80]), 1), "dBA")

shop = {"press": 94, "grinder": 91, "compressor": 88, "fans": 82, "conveyor": 79}
total = add_levels(shop.values())
print(f"Shop total: {total:.1f} dBA")
for name, L in sorted(shop.items(), key=lambda kv: -kv[1]):
    others = add_levels([v for k, v in shop.items() if k != name])
    print(f"  without {name:<10} {others:5.1f} dBA  (reduction {total - others:4.1f} dB)")

print("Source level from a switch-off test:", round(subtract_level(95, 92), 1), "dBA")

Two machines at 90 dBA: 93.0 dBA
Ten machines at 90 dBA: 100.0 dBA
Press 92 + conveyor 80: 92.3 dBA
Shop total: 96.7 dBA
  without press       93.3 dBA  (reduction  3.4 dB)
  without grinder     95.3 dBA  (reduction  1.4 dB)
  without compressor  96.0 dBA  (reduction  0.6 dB)
  without fans        96.5 dBA  (reduction  0.2 dB)
  without conveyor    96.6 dBA  (reduction  0.1 dB)
Source level from a switch-off test: 92.0 dBA

The output shows the ranking clearly. Removing the press would lower the shop level by more than any other single change, while removing the conveyor would make almost no difference. Even silencing everything except the press would leave the level above 94 dBA.

Distance and the inverse square law

Outdoors, or in a very large room with sound-absorbing surfaces, sound from a small source spreads over an ever larger sphere. The sound intensity falls with the square of distance, so each doubling of distance reduces the level by about 6 dB. A compressor measuring 96 dBA at 1 metre would measure about 90 dBA at 2 metres, 84 dBA at 4 metres and 78 dBA at 8 metres in these free-field conditions.

Indoors, things are different. Sound reflects from walls, floors, ceilings and machines, building up a reverberant field that does not fall off with distance. Close to a source, the direct sound dominates and the 6 dB rule roughly applies. Farther away, the reverberant sound dominates and the level levels off. In a hard-walled factory, moving a worker from 2 to 8 metres from a machine may reduce the level by only 3 or 4 dB instead of 12. A long line of machines, such as a conveyor or a busy road, behaves more like a line source and falls off by about 3 dB per doubling of distance.

This is why simply moving workers away from a machine is often less effective indoors than people expect, and why sound-absorbing treatment of the room, discussed in Lesson 6, can help workers who are not right next to the source.

Frequency and octave bands

Real noises contain many frequencies. To describe and control them, acousticians divide the audible range into octave bands, each spanning a doubling of frequency, identified by their centre frequency: 31.5, 63, 125, 250, 500, 1,000, 2,000, 4,000, 8,000 and 16,000 Hz. Finer one-third octave bands are used for detailed work. An octave band analysis shows where a noise's energy lies, which matters for two reasons. First, low-frequency noise is much harder to block with barriers and protectors than high-frequency noise. Second, the ear is more sensitive to some frequencies than others, so the same sound pressure at different frequencies carries different risk.

Frequency weighting

The ear is less sensitive to low frequencies at moderate levels. To produce a single number that reflects hearing risk, sound level meters apply a frequency weighting, an electronic filter that reduces some bands before combining them. Three are common in occupational work.

Comparing dBC and dBA readings gives a quick clue about low-frequency content. If the C-weighted level is much higher than the A-weighted level, say by more than about 10 dB, the noise has a lot of low-frequency energy. That matters when choosing hearing protectors, because many protectors give less attenuation at low frequencies, and when designing barriers and enclosures.

Applying the weighting to octave band data is straightforward: add the correction for each band, then combine the corrected bands with the decibel addition formula.

import math

bands = [63, 125, 250, 500, 1000, 2000, 4000, 8000]
a_corr = [-26.2, -16.1, -8.6, -3.2, 0.0, 1.2, 1.0, -1.1]
c_corr = [-0.8, -0.2, 0.0, 0.0, 0.0, -0.2, -0.8, -3.0]

def combine(levels):
    return 10 * math.log10(sum(10 ** (L / 10) for L in levels))

# Unweighted octave band levels measured beside a large fan (illustrative)
fan = [101, 99, 95, 90, 86, 82, 78, 72]

a_bands = [L + c for L, c in zip(fan, a_corr)]
c_bands = [L + c for L, c in zip(fan, c_corr)]
print("Band (Hz)  dBZ   dBA band")
for f, L, La in zip(bands, fan, a_bands):
    print(f"{f:>8}  {L:5.1f}  {La:6.1f}")
LA, LC = combine(a_bands), combine(c_bands)
print(f"Overall: {LA:.1f} dBA, {LC:.1f} dBC, difference {LC - LA:.1f} dB")

Band (Hz)  dBZ   dBA band
      63  101.0    74.8
     125   99.0    82.9
     250   95.0    86.4
     500   90.0    86.8
    1000   86.0    86.0
    2000   82.0    83.2
    4000   78.0    79.0
    8000   72.0    70.9
Overall: 92.6 dBA, 103.6 dBC, difference 10.9 dB

The fan's overall A-weighted level is far below its C-weighted level because most of its energy sits in the low-frequency bands that A-weighting discounts. The large difference tells the program to check protector performance at low frequencies and to expect that barriers will be less effective than for a high-pitched source.

Time weighting and averaging

Sound levels fluctuate constantly, so instruments also need rules for time. Sound level meters offer slow (one second) and fast (one-eighth second) time weightings, which smooth the reading. For exposure assessment, though, the key quantity is the equivalent continuous level, written Leq or LAeq when A-weighted. It is the steady level that would deliver the same acoustic energy as the fluctuating noise over the measurement period. Lesson 5 shows how equivalent levels are combined into a daily exposure.

Impulse noise, such as hammering, nail guns and gunfire, rises and falls too fast for slow or fast weightings to follow. Peak levels are measured with a separate peak detector, usually C-weighted, and many jurisdictions set a separate ceiling of 140 dB peak. Lesson 11 covers impulse noise in detail.