22 



TYPES OF ACOUSTIC MEASUREMENTS 



Figure 4. Planar directivity pattern for a circular plate. 

 Frequency = 25 kc, diameter of plate = 15 in. 



is 24 db below the peak. Therefore 10 log I a /I = 

 -24 or IJI = 0.004. The sine of 25 degrees is j).2fj 

 Consequently /„//„ sin a = 0.00104. The values of 

 I a /Io sin a are computed for all angles and plotted on 

 rectilinear graph paper against the angle a expressed 

 in radians (1 radian = 57.3 degrees). This is illus- 

 trated in Figure 5. The area under the curve is meas- 

 ured with a planimeter and is found to be 5 square 

 inches. In this particular case the scales were chosen 

 so that 1 inch on the abscissa represents 0.1 radians 

 and 1 inch on the ordinate represents an intensity 

 ratio Ia/Io = 0-0'- Thus, 1 square inch represents a 

 contribution to the integral of 0.001. Hence, the total 

 area gives 



- 



- 



There are several cases for which a directivity in- 

 dex can be obtained in a relatively simple manner. 14 

 If the directivity pattern of the transducer has rota- 

 tional symmetry about the acoustic axis, one may 

 make use of the following formula: 38 



A = 10 log )0 



/ — sin a da \. 



JO l J 



(6) 



h 



sin a da = 0.001 x 5.00 = 0.005. 



Equation (6) includes a factor of i/ 2 in front of the 

 integral. Thus the directivity factor is 0.0025 and the 

 corresponding directivity index A is 



A = 10 log 0.0025 = -26 db. 



In this formula, a represents the angle from the 

 acoustic axis, I a the intensity at this angle, and I the 

 intensity on the axis. The above formula is valid, for 

 example, in the case of a circular diaphragm vibrat- 

 ing symmetrically about this normal axis. In the case 

 of a line source, the acoustic axis is usually taken 

 perpendicular to the line. If the directional pattern 

 of the line is symmetrical about the line itself, the 

 directivity index is given by the formula: 38 



A = 101og 10 



'■la. , , , 



-r~ COS a da 

 r ■/» 



(7) 



where a is now the angle measured from the acoustic 

 axis (normal to the line) in a plane including the 

 acoustic axis and the line itself. 



To indicate the use of these formulas, consider the 

 pattern shown in Figure 4, representing a measured 

 pattern for a circular piston in a plane including the 

 acoustic axis. To find the directivity index for the 

 pattern, equation (6) above is applied. The integral 

 is evaluated graphically by means of a planimeter. 

 This requires obtaining for different angles the 

 ratio I a /I . For instance, at 25 degrees the response 



The procedure in the case of a line is analogous to 

 the one described above, except that cos a is used in 

 all cases in place of sin a, as indicated by comparison 

 of equations (7) and (6). 



Sometimes the pattern as obtained experimentally 

 is not exactly symmetrical. In that case, it is usually 

 sufficiently accurate to use the average values of the 

 two halves of the pattern obtained experimentally. 



This computation is quite straightforward but 

 somewhat tedious. Figure 6 shows a chart which has 

 been prepared to reduce the amount of algebraic 

 computation involved. This chart shows a family 

 of curves, each curve corresponding to a particular 

 value of lath sin <*■ The chart is used in conjunction 

 with the directivity pattern, plotted on polar coordi- 

 nate paper, of the instrument whose directivity index 

 is to be obtained. The use of the chart is as follows: 

 The transparent chart is laid over the directivity pat- 

 tern of the instrument so that the coordinate systems 

 on the two charts coincide. Then, to find the value 

 of I a lh sin a for any angle a, one proceeds along the 

 radial line corresponding to the angle a until one 

 reaches the intersection of that line with the direc- 

 tivity pattern of the instrument. The point of inter- 

 section of the pattern and the line will then fall on 



