Sound Level

The range of sound pressure varies from 20x10^-6 Pascals at the threshold of hearing , up to 200 Pascals in the region of pain level or instantaneous damage. The sound pressure is small when comparing with the normal atmospheric pressure of 10⁵.
The large orders of magnitude involved is inconvenient for quantitative analysis. Besides, the perception of sound is also not directly proportional to the sound pressure. A more meaningful scale is needed. The response of human ear to the sound pressure is in logarithmic scale. And the threshold of hearing, 20µPa at 1000Hz frequency is used as zero of the scale.

Decibels, dB

Decibel is a logarithmic scale used to measure the sound pressure level (SPL) in practice. This is not an absolute scale, it is comparative scale with a reference sound pressure of average threshold of hearing, 20µPa at 1000Hz.
The sound pressure level is defined as

where

L_p is the sound pressure level
P is the sound pressure
P_o is the reference pressure, 20µPa

Noise Addition

The global level of the total sound level is the algebraic sum of the acoustic energies,

Therefore for random sounds, as there is no interference, the superimposing of sounds at each point in space is only the summation of all energy sum. The resultant acoustic energy is the sum of individual acoustic source energies, not the sound levels of sources.

But for related sound sources, the resultant acoustic pressure field due to acoustic sources is the algebraic sum of the waveforms of each sources. The sum of acoustic pressure is time and space dependent. The resultant sound may have a larger amplitude with constructive interference or a smaller amplitude with destructive interference.

For uncorrelated or incoherent random noise, the acoustic pressures radiated from two sources A and B:

and

where

L_pA is sound level of source A
P_A is sound pressure of source A
L_pB is sound level of source B
P_B is sound pressure of source B

Since the sound pressure ratio square in term of sound level is:

.

The total sound level at a point in space by superimposing is:

.

Or:

.

For example:
Two sources 90dB and 90dB

.

Multi sources 90dB, 100dB, 73dB and 93dB

.

Noise Level Subtraction

When measuring the noise radiated by a noise source, the background noise from other sources is also included in the measurement reading by superimposing.

.

In order to determine the actual noise level of a source, the background noise level should be measured before running the noise. Let A is the new noise source and B be the background noise. That is

and

Therefore, the noise level of the new noise source:

.

Or:

.

For example:
Measured total noise level 93dB and background noise 90dB

.

Measured total noise level 90.05dB and background noise 90dB

.

Noise Control Example

The noise level of 10 identical machines operating in a room is 54dB. In order to reduce the noise level to 50dB, the maximum number of machines can be used is?

Since ten identical machines run independently from each others, the noise from the ten identical machines can be considered as uncorrelated sound sources. The total noise level at a point in space generated by n identical machines can be determined by superimposing. imply

.

Therefore the sound pressure generated by one machine is

The maximum number of machines can be used to generate the maximum sound level 50dB is

let n be 4, the total sound level is

Since the total sound level only exceed 0.02dB, using maximum 4 machines are acceptable.