Underwater Acoustics

Whilst the decibel scale makes multiplication and division easier, it complicates addition and subtraction. Where this needs to be done, for example when there are two or more sources contributing to a sound pressure, it is necessary to first convert the data to a form which is additive.

The linear equivalent of the decibel sound pressure level (SPL) is the sound pressure, or acoustic pressure. So if two SPLs need to be added, they should first be converted to sound pressures in Pa (Pascals).

We have to first distinguish correlated and uncorrelated sounds. Correlated sound can be made by a coordinated array of sources, all activated by the same electrical signal. The table below predicts the sound pressures generated at a point by two such sources, positioned so that their sounds will be in phase and reinforce each other. Their pressures (Pa) can then be added. By changing the pressures due to Source #2 whilst keeping Source #1 constant, we see that the true total SPL values differ dramatically from the erroneous addition of contributory SPL values.

Source #1 only		Source #2 only		Two correlated sources
SPL (dB re 1 μPa)	Sound Pressure (Pa) rms	SPL (dB re 1 μPa)	Sound pressure (Pa) rms	Sound pressure (Pa) rms	SPL (dB re 1 μPa) correlated
140	10	140	10	20	146
140	10	130	3.16	13.16	142.4
140	10	120	1	11	140.8
140	10	110	0.316	10.316	140.3

The more commonly encountered example, still somewhat idealised, has two noise sources emitting random noise within the same defined measurement bandwidth. Now it is the mean square pressures in band (Pa²) which are additive, as shown below, reducing the totals by up to 3dB.

Source #1 only		Source #2 only		Two uncorrelated sources
SPL (dB re 1 μPa)	Mean sq Pressure (Pa²)	SPL (dB re 1 μPa)	Mean sq Pressure (Pa²)	Mean sq Pressure (Pa²)	SPL (dB re 1 μPa) uncorrelated
140	100	140	100	200	143
140	100	130	10	110	140.4
140	100	120	1	101	140.04
140	100	110	0.1	100.1	140.004