The intensity level of the sound radiated by a high-frequency sonar at distance

from the sonar and angle

measured with respect to the sonar transmit-receive array axis is

,

where

is the projector source intensity,

is the projector beam pattern on a power scale, and

is absorption in dB per unit length. The received target echo from a scattering object located at distance

from the sonar is

,

where

is the receive beam pattern and

is the decibel (dB) target strength of the reflecting object.

When the sonar projector-receiver array axis is pointed directly at the target, then

and both the projector and receiver beam pattern functions are unity. In this case the target echo level on a decibel scale reduces to

,

where

is the projector source level in dB at unit distance from the source and

is the transmission loss. In isovelocity water, sound spreads spherically and the transmission loss is

*.* The signal-to-noise ratio for a high-frequency sonar that transmits a short continuous wave (CW) pulse and employs incoherent signal processing can be written in the form

,

where

is the sonar pulse length,

is the directivity factor of the sonar transmit-receive array and

is the ambient noise spectral intensity measured in units of power per Hz. The numerator in this equation is the signal intensity and the denominator the mask intensity. The mask is a measure of the amount of unwanted sound that is received by the sonar. The specific terms in the mask are the bottom, surface, and volume reverberation intensities and the receiver noise intensity. The combined effect of these four terms is often just referred to as the noise. Under normal circumstances, reliable detection can be achieved whenever

(dB) exceeds about 12 dB.

When

is the dominant reverberation component,

becomes

,

where

is the bottom back-scatter function,

is the speed of sound, and

is the effective azimuthal beam width of the sonar in radians. If the projector source level is sufficiently large, then the reverberation will dominate the noise and

.

In this case, no further gains in performance can be achieved by using a larger source level. Active sonars are usually operated in a reverberation-limited mode in order to maximize sonar detection range.

[1] R. J. Urick,

*Principles of Underwater Sound*, New York: McGraw–Hill, 1983.

[2] C. Clay and H. Medwin,

*Acoustical Oceanography*, New York: John Wiley & Sons, 1977.