208 



TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 



40 60 80 100 



TIME IN MICROSECONDS 



120 



5 10 



FREQUENCY IN KILOCYCLES 



50 



100 



Figure 13. Influence of time of rise on spectrum of a single pressure pulse. 



of integration results, unfortunately, in omission of 

 the surface-reflected pulse when its time of arrival 

 exceeds h. As a result the computed will rise to a 

 maximum value as the frequency approaches zero, 

 whereas if the surface reflection were included <^ would 

 approach a small value or zero. Even if there were no 

 surface reflection we should expect the computed 

 value of <l> to be considerably in error at very low 

 frequencies through neglect of the tail of the shock 

 wave, which has been discussed in Section 8.5. At 

 frequencies large compared with l/k the neglect of 

 the surface reflection is unimportant, although it may 

 result in the absence of some small-scale ripples from 

 the curve of spectrum level against frequency, which 

 would be present if the surface reflection were in- 

 cluded. When the surface reflection arrives well within 

 the time <i, of course, the error due to cutting off the 

 integration at this time is usually negligible, although 

 of course if there are bottom reflections arriving after 

 time ti the computed spectral distribution will be 

 that which would apply in the absence of a bottom. 

 At very high frequencies the spectrum level de- 

 pends primarily upon the time of rise, and the com- 

 puted value may be in error if the response time of 

 the hydrophone and recording system does not per- 

 mit faithful reproduction of the rising portion of the 

 pulse. The extent to which the time of rise affects the 

 spectral distribution is shown by some sample calcu- 

 lations given in reference 9, for hypothetical pulses, 

 which are presented in Figure 13. Note that pulse I, 

 which has a vertical rise, has the highest spectrum 



level at high frequencies. It has been shown by 

 mathematicians that the Fourier transform of a finite 

 but discontinuous function p is of order 1// at large 

 values of the frequency /, and we should therefore 

 expect the spectrum level U for pulse I to decrease 

 at 6 db per octave at high frequencies. Similarly, it 

 can be shown that a pulse which is continuous but 

 has discontinuities in slope, such as IV in the figure, 

 should have a spectrum level which decreases at 

 12 db per octave at sufficiently high frequencies. The 

 spectrum of a perfectly smooth pulse, such as III, 

 should decrease still more rapidly. 



Since actual oscillograms usually show irregularities 

 in the tails of the pressiu-e pulses, and since these ir- 

 regularities are usually not very reproducible from 

 shot to shot, it is pertinent to inquire how much in- 

 fluence they have on the spectral distribution. Sample 

 calculations for hypothetical pulses have shown that 

 the principal effect of a fairly smooth "satellite" peak 

 is to introduce irregularities into the curve of spec- 

 trum level against frequency, without much altera- 

 tion of its general trend. ^ 



Figure 14 gives curves of spectrum level U against 

 frequency /, computed from the oscillograms of 

 Figure 7. In these curves the irregularities which ap- 

 pear in the curves directly computed from (4) and (2) 

 have been arbitrarily smoothed out; from what has 

 been said in the last paragraph these irregularities 

 probably have little significance, and efiminating 

 them makes it easier to follow the changes in the 

 spectrum as the range of the shot is increased. All the 



