For people not versed in acoustics, the decibel scale that is commonly used to describe sound can be very strange to behold. Consider the following decibel math and human hearing facts:

A doubling (x2) of sound pressure level always corresponds to an increase of 6 decibels

Combining two sound sources (that are not coherent in phase) of the same decibel magnitude results in a 3 decibel increase

A 10 decibel increase in sound level corresponds to a doubling of the perceived loudness of a sound

This decibel “funny” math and human hearing perception makes the acoustic engineering of a product very challenging.

Decibel Scale

The decibel (dB) scale is used for describing acoustic level. The scale shows differing decibel levels and how they are perceived by the human ear as shown in Figure 1.

Figure 1: Decibel Scale for Acoustics

A level of 10 dB at the human ear is barely audible. A 120 dB level at the human ear is painfully loud. Each decibel level actually corresponds to a specific sound pressure level.

Decibel and Sound Pressure

What are decibels? A decibel is a format that is commonly used to look at sound pressure (typically measured in Pascals).

The human ear picks up changes in pressure on the ear drum. These changes in pressure can be very small numbers as small as 0.00063 Pascals, 0.00034 Pascals, 1.014 Pascals, etc. The decibel scale makes it easier to understand these pressure changes.

Decibels are calculated by converting sound pressure to a decibel value via Equation 1:

Equation 1: Equation relating sound pressure to decibels

The ‘measured pressure’ is a RMS quantity of sound pressure recorded by a microphone (or human ear) at some location in space. The ‘reference pressure’ is 0.00002 Pascals, which is the lowest detectable pressure for human hearing at 1000 Hz.

Doubling sound pressure is 6 dB?

Using Equation 1, different sound pressures have been converted into decibels as shown in Table 1:

Table 1: Sound pressure values and their corresponding decibel level

Something interesting happens in the table – as the sound pressure doubles, the decibel value always increases by exactly 6 decibels!

Suppose you are an acoustic engineer tasked with making a product quieter. You reduce the sound pressure that your product produces in half, and are eager to report the results. Imagine telling a boss: “I reduced the sound by a factor of 2, the product went from 66 dB to 60 dB”.

This could be dangerous for your acoustic career if your boss does not understand how the decibel scale works! For a casual observer, the numbers 66 and 60 do not appear to be related by a factor of two in any way whatsoever!

Human Hearing and Decibels

Even more interesting is that a doubling of sound pressure is not perceived as twice as loud by the human ear. Instead, a 10 decibel increase is typically perceived as twice as loud. A 10 dB change corresponds to a 3.16 times change in the linear amplitude of sound pressure.

The doubling perception of 10 dB is a “rule of thumb”, not an absolute fact. This rule of thumb is primarily based on the range of frequencies from 1000 Hz to 4000 Hz, where the human ear is most sensitive.

Another common “rule of thumb” in acoustics is that 3 dB is the average perceivable change. This means that one must alter the sound level of a product by at least 3 dB for the average person to notice any difference.

Decibel “Funny” Math: Sound Source Addition

Confused? Just wait!

Let’s say you have two independent sound sources, each producing 100 dB of sound pressure. You put these sound sources together and get a combined dB reading. What is the sum of 100 dB + 100 dB?

If you guessed that the sum of the sound sources is 200 dB, you are wrong (Figure 2)! Decibel values cannot be added directly.

Figure 2: 100 dB + 100 dB is not 200 dB!

Instead, the decibel values must be converted back to linear sound pressure values before math operations are performed. The following steps are to be done (Figure 3):

Decibel values are converted to their corresponding linear sound pressure values in Pascals

Math is performed (like subtraction and addition) on the linear sound pressure values

The resulting total sound pressure is then converted back to dB

Decibels should not be directly added and subtracted without converting to their corresponding linear values first as shown in Figure 3.

Figure 3: To add decibels, first convert back to linear data to perform summation. Be warned that this is not usually representative of two sound sources in the real world

Think you have the answer? The decibel quantity of 100 dB converts to 2 Pascals. So you just have to add 2 Pascals plus 2 Pascals. The total is 4 Pascals, or 106 dB!

Guess what, you are probably wrong again! The total will not be 106 dB! In the real world, it is more likely the total sound of two independent 100 dB sources will be 103 dB!

Decibel “Funny” Math: Sound Source Addition with Phasing

Shouldn’t 100 dB + 100 dB = 106 dB? Why doesn’t a 2 Pascal and 2 Pascal sound source add up to 4 Pascals?

This is due to the fact that the sound pressure is not a constant value, but rather a dynamic pressure fluctuation in the air. In other words, there is phasing between the sources that comes into play.

There is a good chance that in the real world, two independent sound sources would operate at different frequencies. For example, one source might operate at 100 Hz while the other operates at 120 Hz as shown in Figure 4.

Figure 4: 100 Hz Sound Source [Top], 120 Hz Sound Source [Middle], Sum [Bottom]. Amplitude of sum varies greatly over time as waves cancel and add due to phase differences

Over a period of time, the 100 and 120 Hertz sources will operate in and out of phase with each other. When in phase, the total pressure is 4 Pascal. When out of phase, the total is 0 Pascal.

Most of the time, the level is neither 4 Pascals nor is it 0 Pascals, it is somewhere in between. The phase of the two signals is constantly changing over time, thus making the combined magnitude constantly vary between 0 and 4 Pascals over time.

A Root Mean Square (RMS) Summation of the two sources is the best representation of the combined magnitude over time. For a RMS sum, the level would be the sum of '2 Pascals squared' plus '2 Pascals squared' for a sum total of 8 Pascals. The sum would then be square rooted, resulting in 2.82 Pascals (or 103 dB) as shown in Figure 5.

Figure 5: The RMS summation of 2 Pa sound sources adds to 2.82 Pascals total.

Only if the sound sources were always acting completely in phase would the result be 106 dB. While not impossible, the chances are typically higher that two independent sound sources would not be operating completely in phase.

The previous example was for two sine waves of 100 and 120 Hz. The same math and principals also holds true for two broadband random sources of 100 dB each (Figure 6). For random sound sources, there is no fixed phase relationship at any frequency.

Figure 6: Summing random sound sources of identical amplitude sum to 3 dB higher

Combining both 100 dB random sound sources together would result in a 3 dB increase in total sound level.

Sound Source Issues

Decibel math also makes acoustic engineering very challenging. Why is it so difficult to reduce sound levels? A lot has to do with decibel math.

Suppose there are three closely spaced sound sources that each produce 100 dB at a far away microphone (Figure 7).

Figure 7: Three sound sources and a far away microphone

At this distant microphone, the three sources would combine to produce a total of 104.7 dB as shown in Figure 8. This assumes that they are not making sounds completely in phase of course.

Figure 8: Sum of three 100 dB sound sources equals 104.7 dB

The total is NOT 300 dB! So 100 dB + 100 dB + 100 dB does not equal 300 dB! In fact, 104.7 is not much higher than 100 dB. Without an understanding of dB math, this is not intuitive.

Suppose one of the three 100 dB sources is entirely knocked out via some fancy engineering. Entirely eliminated. Think the total goes down by one third? Not even close!

The total is reduced from 104.7 dB to 103 dB as shown in Figure 9. This is a 1.7 dB decrease.

Figure 9: One sound source is entirely eliminated, total is reduced from 104.7 dB to 103 dB

What if two out of three sources were eliminated entirely? The total is 100 dB. By eliminating two 100 dB sources we have reduced the total from 104.7 to 100 dB as shown in Figure 10.

Figure 10: Two sound sources are entirely eliminated, total is reduced from 104.7 dB to 100 dB

Eliminating two 100 dB sound sources in this example has only reduced the total from 104.7 dB to 100 dB. This is not a very big decrease, and illustrates how difficult is to reduce the sound level of a product.

A better alternative?

If one could reduce each 100 dB source by just 10 dB, a greater overall reduction would be achieved. It would be greater than even eliminating one or two sources entirely.

After reducing the 100 dB sound source to 90 dB, the sound source Pascal level is about 0.64. Taking a RMS sum of the three 0.64 Pascal sources, one gets about 1.1 Pascals, which is 94.8 dB (Figure 11). This is well less than 100 dB.

Figure 11: Reducing each source by 10 dB reduces total more than eliminating two sources entirely

Completely eliminating two of three 100 dB sources only reduced the total by 4.7 dB. Reducing each source by about 10 dB (only 30 dB out of 300 dB) gives slightly a larger net reduction (about 10 dB total reduction rather than 4.7 dB reduction).

Conclusion

Welcome to the wacky world of acoustic decibel engineering! Hopefully this overview gives a good introduction to decibel math and human hearing.