RELATIONS BETWEEN MEASUREMENTS 



31 



3. There exists, in addition, a reciprocal relation'' 

 between the transmitting response and the re- 

 ceiving response of a projector. This relation 

 has the following form: 



R T = R B + 201ogf+ 10 log ^ 



- 10 log r + 20 log fj£- 



= R R + 20 log / + 10 log ^ - 10 log r 4- 94.2. 



(19) 



4. From these three equations, it is possible to de- 

 rive a relation between the projector efficiency 

 E p and the threshold T: 



I'- 



ll) 1 



E„ + T = A + 20 log/ + lOlog^— - -194.9 



= A + 20 log/ + 171.6. 



(20) 



5. Furthermore, by combining equations (18) and 

 (19) or (8) and (20), a relation can be obtained 

 between the threshold T and the transmitting 

 response of the unit R T . This relation is as fol- 

 lows: 



R T + T = 10 log f 4- 20 log/ 



+ 20 log ^ - 194.9 



Pi 



10 log iji 4- 20 log/- 100.7. (21) 



"a 



the study of measured data. For the sake of simplicity 

 the discussion is confined to circular pistons. 



In the case of a circular piston moving rigidly in an 

 infinite baffle, there exists a simple relation between 

 its radius a and the directivity index A. 3 * 



A = - 10 log 



k 2 a? 



r 2/ 1 (2An) 1 

 L 2ka J 



(23) 



where k = 2ir/A, A being the wave length = c/f, and 

 /, = first order Bessel function. 



The directivity of a physical projector usually is 

 less than that of a theoretical circular piston of the 

 same geometrical size. It is, however, possible to de- 

 fine the effective or acoustic radius of the physical 

 projector to be the radius of the theoretical piston 

 having the same directivity index as the projector. On 

 Figure 10 the directivity index and beam width are 

 plotted against effective radius in wave lengths (fl/A) 

 for a theoretical piston. 



A number of interesting relations are obtained by 

 introducing this expression in the above equations in 

 which the directivity index occurs. These are the 

 equations which include the projector efficiency. For 

 instance, substituting the above expression for the 

 directivity index in the relation between projector 

 efficiency and transmitting response, equation (8) 

 gives: 



E„ = R T - 10 log 



7T«" 



r 2/. (2Aa) 1 

 L 2ka J 



10 log p- 



+ 20 log A - 10 log ^j^ 



6. Finally, by combining equations (8) and (19) a 

 relation can be found between the projector 

 efficiency and the receiving response of the de- 

 vice: 



£ p = R R 4- A 4- 20 log / - 10 log r 4- 10 log 



p -k\{)'' 



= R, 



10 log 



r 2/ t (2ka) i 



L 2ka J 



4- 20 log A 



10 log 



P, 



81.9. (24) 



7T«" 



R B 4- A + 20 log/ - 10 log r + 23.3. (22) In the term 



The following discussion has for its purpose the ex- 

 ploration of the meaning of the above relations and 

 the indication of their usefulness in connection with 



« The reciprocity theorem and the conditions under which it 

 applies are stated in Chapter 3. 



[2/, (2ka)l i' 1 th's equation, na 2 is 

 1 2ka~~ J 



the area of the theoretical piston of radius a. In the 

 case of an actual projector, in accordance with the 

 above discussion, a is the effective radius and can be 

 found by means of the chart on Figure 10. The ex- 

 pression in the bracket is a pure numeric, so that the 

 whole term has the dimensions of an area. Let it be 



