THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

434 15. Underwater Acousticsconsists of two plots that are overlaid on each other. The solid lines constitute anoverlay of a plot of SL versus frequency for a specific passive target or class oftargets. The underlay denoted by dashed lines is a plot of the sum of the parameters(TL + NL − DI + DT) for a specific passive sonar at a number of different ranges.The range and frequency at which the target can be discerned can be determinedby inspection of the plots. For example, the target will be first detected at a range of10 miles according to the line component at frequency f 1 . But suppose it is requiredthat three spectral lines must appear on the display before a detection is called; therange would be reduced to 4 miles and the lines at frequencies f 1 , f 2 , and f 3 wouldbe displayed. This procedure helps to distinguish the target parameter SL fromthe equipment and medium parameters at the location where it is used, while itaccommodates a wide range of frequencies. Targets can therefore be compared forthe same locations or locations can be compared for the same targets, and so on.15.14 Transient Form of Sonar Equations and PulsesSo far, the sonar equations have been expressed in terms of the average acousticpower per unit area or intensity of the sound radiated by the source or received fromthe target. But the time interval implied by the terminology “average” can yieldunreliable results in situations where short transient sources exist or wheneversevere distortion is incurred in sound propagation in the medium during the courseof scattering from the target.We can adopt a more general approach by writing the sonar equations in termsof energy flux density, which is defined at the acoustic energy per unit area ofthe wavefront. Consider a plane acoustic wave that has a time-dependent pressurep(t). The energy flux density of the wave is given byE = 1 ∫ ∞p 2 (t) dt. (15.19)ρ c 0Because the intensity is the rms pressure of the wave divided by the acousticimpedance ρc, averaged over a time interval T , i.e.,I = 1 ∫ Tp 2 (t)T 0 ρc dtit follows thatI = E T . (15.20)The time interval T represents the duration over which the energy flux densityof the sound wave is to be averaged to yield the intensity. In the case of long-pulsedactive sensors, this time interval equals the duration of the emitted pulse and verynearly equals the duration of the echo. But for short transient sonars, the intervalT becomes rather ambiguous, and the duration of the echo can be considerablydifferent from the duration of the transient emitted by the source. Urick (1962,