32 



TYPES OF ACOUSTIC MEASUREMENTS 



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 EFFECTIVE RADIUS IN WAVE LENGTH (•#} 



Figure 10. Relation between effective radius and direc- 

 tivity index (or beam width) for sonar projector with 

 circular diaphragm. 



called the effective area of the projector 11 A, which is 

 thus defined 



A = 



r 2./, (2M 1 



L 2ka J 



(25) 



Defining the directivity factor 8 by 



A - 10 log 8, 



it is evident from equation (23) that 



I" 2/, (2ka) l 

 L 2ka J 



k-a- 



and from equation (25) that 

 A 2 



A = 



4tt8 



(26) 



In case a exceeds one-half wave length, the term 

 1 - [2./i (2ka)/2ka] is nearly unity, so that A then 

 equals na'-. 



Assuming, furthermore, the transition loss to re- 

 main unchanged, we can simplify equation (24) by 

 introducing a constant K lt 



*i = - (10 log li + 81.9), 



<l The effective area was first defined in these terms by E. E. 

 Teal of the Columbia University Underwater Sound Laboratory 

 at New London in a letter dated February 23, 1943. 



and write 



E P = R T - 10 log A + 20 log \ + K x . (27) 



This expression shows that for constant projector 

 efficiency, the transmitting response varies directly 

 as the effective area, and for any given device (A = 

 constant) increases 6 db per octave. The latter rela- 

 tion is shown by equation (21) to exist also for fixed 

 threshold and transition loss. 



Introducing the expression for the directivity in- 

 dex in equation (20) gives 



£ + T = -10 log A + 10 log ^2— - 194 - 9 



10 log A - 79. 



(28) 



This equation shows that for any given device 

 (A = constant) the threshold pressure is independent 

 of frequency and also indicates that the higher the 

 efficiency of the projector the lower the threshold 

 pressure. For devices of the same type, that is, having 

 the same efficiency, the threshold varies inversely as 

 the size of the unit. 



Substituting the above expression for the directiv- 

 ity index in equation (22) we obtain 



E p = R n - 10 log .4-10 log r + 10 log 



= R, t - 10 log .4-10 log r + 1 15.9. 



P c 10" 



(29) 



From this equation it may be seen that for a given 

 device having fixed size and efficiency, the receiving 

 response is independent of frequency but increases 

 directly with the resistance. Advantage is often taken 

 of this latter fact by using a step-up transformer to 

 increase the receiving response of a low-impedance 

 hydrophone. These relations between receiving re- 

 sponse and impedance are shown by equation (18) to 

 apply also for a fixed threshold. It is also interesting 

 to note that for a given receiving response and effi- 

 ciency the resistance among different projectors varies 

 inversely as their area. 



It is, of course, possible to set up a definition for 

 the receiving efficiency of a hydrophone or a projec- 

 tor. In terms analogous to those used for the projector 

 efficiency given above, this efficiency could be stated 



£*= lOlog^. 



