EXPLICIT AND TACIT ASSUMPTIONS 



265 



gin with, that the instantaneous voltage across the 

 terminals of the receiving circuit is exactly propor- 

 tional to the instantaneous pressure in any plane 

 wave which is incident on the transducer. This as- 

 sumption is equivalent to assuming that the re- 

 ceiver introduces no phase shifts, or in other words, 

 that it behaves like a pure resistance or like an ideal 

 infinitely wide band-pass filter. Of course, actual re- 

 ceivers never behave in this way, but it is convenient 

 to postpone temporarily consideration of the effects 

 caused by departure from ideal response in the re- 

 ceiver circuit. Now suppose a plane wave of pressure 

 p is incident on the transducer from a direction de- 

 fined by the angles {d,<j>) of Figure 2. Then the voltage 

 E across the receiver terminal is 



E = f'?'{d,4>)v, 



(56) 



where 0'{d,<t>) is the "pressure pattern function" of 

 the receiver, and /' is the voltage across the receiver 

 terminals when a plane wave of unit pressure is 

 incident on the receiver in the direction of its maxi- 

 mum response. Since there is no phase distortion, all 

 the quantities in equation (56) may be assumed real. 

 The rms power output resulting from E, in watts 

 across the receiver terminals, will be 



£2 y'2^'2 _ 



(57) 



where the receiver is assumed terminated in the pure 

 resistance Z, and the bar indicates a time average 

 over many cycles of the wave. 



Now, in a plane wave, the relation between the 

 sound pressure p and the average sound intensity /, 

 from Section 2.4.3, is just 



poc 



(58) 



where po is the density of water, c is the velocity of 

 sound, and the bar again indicates a time average 

 over many cycles of the wave. Equation (58) re- 

 mains valid even if the plane wave is being refracted 

 by velocity gradients in the ocean. From equations 

 (57) and (58), we see that the rms power output 

 across the terminals of our ideal receiver, caused by a 

 plane wave incident on the transducer, is proportional 

 to the average sound intensity in the water. Further- 

 more, by comparing equations (57) and (58) with the 

 definition of F' and b'{9,(t>) in Section 12.2, it is 

 evident that 



^'=-^',&'(M)=^'^(M. 



However, the scattered sound which produces re- 

 verberation reaches the transducer from all direc- 

 tions, and therefore cannot be regarded as a plane 

 wave. For this scattered sound the pressure in the 

 water at any instant is 



= T,Pi, 



(60) 



where pi is the pressure in the ith plane wave which 

 arrives at 0. The voltage generated across the re- 

 ceiver terminals, by equation (60), is 



E = /'Z^'(e.-,0.)P.- (61) 



i 



where the angles di,<j)i define the direction from which 

 the ith plane wave reaches the transducer. The rms 

 intensity resulting from equation (61) is given by 



E^ 

 Z 



(59) 



= ~[jl0'(ei,<t,i)pij]^Zfi'(ej,,t>i)pq 



(62) 



where the double sum includes terms for all values 

 of i and j except i equal to j. 



12.5.1 Average Reverberation 

 Intensity 



One of the basic assumptions made in Section 12.1 

 was that the average reverberation intensity is the 

 sum of the average intensities of the individual 

 scattered waves reaching the transducer. Because 

 equation (62) represents the average reverberation 

 intensity, while equation (57) represents the in- 

 tensity of the individual scattered wave, it is clear 

 that this assumption will be strictly valid only if the 

 double sum on the right-hand side of equation (62) 

 vanishes. 



The average in equation (62) is the average over a 

 large number of cycles. All the waves reaching the 

 transducer have very nearly the same frequency 

 when ordinary single-frequency (CW) pings are 

 used. Thus, the value of the double sum on the right 

 of equation (&2) depends on the relative phases of the 

 various waves arriving at the transducer. On any one 

 ping the value of this double sum may be positive or 

 negative, and its absolute value may be appreciable 

 or near zero, depending on the phases. Thus, if the 

 expression (62) is averaged over a number of pings, 

 and if the phases vary in a random way from ping 

 to ping, the double sum in equation (62) can be neg- 



