Underwater Acoustics

The sonar equations were devised to formalise the relationship between various decibel levels found useful in design calculations for sonar equipment. This work was given great impetus by its military significance in the Cold War.

The decibel levels used were assigned symbols such as SL for source level, and the equations typically add or subtract them as a means of performing the fundamental multiplication or division of the underlying physical parameters.

The sound pressure level SPL or PL is then given by:

where TL is the transmission loss. Since the source level concept is defined for a non-reverberant spherically spreading field, the main term in the transmission loss is:

This equation thus represents the underlying physics where division of the measured P·r source output by the range r predicts the sound pressure P.

This equation can be modified to allow for sound absorption by the water, given as a coefficient α, the loss in dB/km. The modified sonar equation then becomes:

The decibel form is particularly convenient here, including in this form, both the spreading loss and the exponential decay of sound energy with range, as it is converted to heat.

It is also possible to modify the equation to show how the transmission may be influenced by the environment. In some circumstances the sound can be channelled within a layer to give what is called "cylindrical spreading". A simply defined example is a surface duct formed in isothermal water, where the sound is refracted by the increase of the sound speed with depth.

In this case the sound initially spreads spherically, but this changes to "cylindrical spreading" at a range r' which depends on the duct geometry. The sonar equation when r >> r' then becomes:

This acoustic pressure level PL, is that measured as incoherent sound over a finite bandwidth. Narrow band (tonal) signals tend to fluctuate rapidly as the reverberation causes interference.

The product r·r' has replaced the r² of the simple spherical spreading rule, but this still gives a correct unitary analysis. When the duct is formed from isothermal water with a uniform sound speed depth gradient, the sound paths are gently curved upwards with radius R, typically 90km. If the sound source is close to the reflecting surface, and the duct depth is H, then r' = √(R·H/8).