SHORT-RANGE PROPAGATION IN DEEP WATER 



207 



SHOT 26 RANGE 580 YARDS 



SHOT 27 RANGE 710 YARDS 



SHOT 29 RANGE 940 YARDS 



Figure 12. Samples of irregular pressure-time curves. 

 Source: No. 8 detonating cap at depth 50 feet and 

 ranges indicated. Hydrophone at depth 54 feet. Date: 

 Apr. 3, 1942. 



elucidating the physical mechanisms operative in the 

 propagation of sound in the sea and in predicting the 

 response of resonant or band-pass receiving systems 

 to explosive sound. 



The representation of an arbitrary disturbance as a 

 superposition of sine waves is described mathemati- 

 cally by Fourier's theorem. The most useful form of 

 this theorem for our present purpose is the integral 



form, which states that if p{t) is any function of an 



\p\^dt con- 



- CD 



verges, then for all values of t except points at which 

 p is discontinuous 



where 



Pit) = r <t>{f)e^''"df 



c/ — CO 



p(t)e-^'''''dt. 



(1) 

 (2) 



It can also be shown that 



\p\^dt= \<i>\Mf. (3) 



-a, ^ _ OD 



When these mathematical theorems are applied to 

 the pressure p{t) in a pulse of sound, the physical 

 interpretation of the results is simple. The integral 

 on the left of equation (3), when divided by pc, 

 represents the total energy in the pulse per unit area. 

 The quantity / represents frequency, measured for 

 example in cycles per second, and so the integrand 

 |<^|^ on the right of equation (3), when divided by pc, 

 represents the energy per unit area per unit frequency 

 range. The spectrum level of the pulse at frequency/, 

 as measured for example by the energy received from 

 it by a receiving system sensitive only to a narrow 

 band of frequencies in the neighborhood of /, is 



f/(/) = 101ogiok(/)P (4) 



and U will be in decibels per cycle above 1 dyne per 

 sq cm, if p was measured in dynes per sq cm and / 

 in cycles per second. 



In evaluating the expression (2) for an experi- 

 mentally obtained pressure-time curve it is of course 

 not possible to extend the upper limit of integration 

 to infinity; in UCDWR work described in reference 9, 

 for example, the oscillographic record obtained only 

 lasted for a few hundred microseconds, and over the 

 latter part of this range it was hard to estimate the 

 position of the zero line accurately. The integrals 

 which were actually evaluated were therefore 



and 



I p(t) sin 2Trftdt 

 I p(t) cos 2irftdt, 



where the origin of time is taken as the time of arrival 

 of the first perceptible pressure and where ii is a few 

 hundred microseconds, i.e., is of the order of the 

 duration of the traces which were shown in Figures 

 7, 10, and 12. Such a curtailment of the upper limit 



