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

15.14 Transient Form of Sonar Equations and Pulses 435Figure 15.12. Time stretching: equivalent source level (SL) in short-pulse sonar.1983) demonstrated that under these conditions that the intensity form of the sonarequations can be applied, providing the source level is defined asSL = 10 log E − 10 log τ e (15.21)where E is the energy flux density of the source referred to 1 yard and measuredin units of the energy flux density of a 1-μPa plane wave taken over an intervalof 1 s, and τ e represents the duration of the echoes expressed in seconds for anactive sonar. As an example, E can be determined for explosives by measurementsfor a given charge weight, the depth where the explosion occurs, and the type ofexplosive.Consider a pulsed sonar emitting a rectangular pulse of constant source level SL ′over a time interval τ 0 . The energy density of a pulse equals the average intensityH duration,10 log E = SL ′ + 10 log τ 0 . (15.22)Combining Equations (15.21) and (15.22), the effective source level (SL) to beused accordingly in sonar equations isSL = SL ′ + 10 log τ 0(15.23)τ ewhere τ 0 denotes the duration of the emitted pulse of source level, SL ′ , and τ e theduration of the echo. For long-pulsed sonars, τ 0 is equal to τ e , and therefore SL =SL ′ . In the case of short-pulsed sonars, τ 0 >τ e , and thus the effective SL is lessthan the actual SL ′ by the amount 10 log (τ 0 /τ e ). This effect of time stretching isdepicted in Figure 15.12. A short pulse duration τ 0 at source level SL ′ is replacedin a sonar calculation by a longer pulse of duration τ e at a lower SL. The energyflux density source levels must be the same for these two source levels, i.e.,SL + 10 log τ e = SL ′ + 10 log τ 0which is merely a rearrangement of Equation (15.23). A pulse emitted by thesource stretches out in time and becomes reduced in level by the multipath effectof propagation and by the mechanism of target reflection. The appropriate values