24 



WAVE ACOUSTICS 



Directivity Characteristics 



So far we have been mainly concerned with the 

 simple point source which gives rise to a spherically 

 symmetric disturbance in the immediate vicinity of 

 the source. It is called a point source because the re- 

 sulting sound field is discontinuous at only one point 

 of space, at the source itself. If the discontinuity is of 

 a more complicated nature, as in the case of a line 

 source, the sound field will not, in general, be spheri- 

 cally symmetric in the neighborhood of the source; 

 that is, the amounts of sound energy radiated into 

 different directions will be different. In this subsec- 

 tion, sources giving rise to sound fields that are not 

 spherically sjonmetric are discussed. 



Double Sources. Suppose there are two point 

 sources. So and <So', one at {xo,yo,Zo) and the other at 

 (xo ,yo' ,Zo') , as indicated in Figure 6. The resulting 



P=Po+Po' 



Also, 



Figure 6. Resultant pressure produced by two sepa- 

 rate sources. 



pressure at any one point P and time t will be the 

 algebraic sum of the pressures that would be pro- 

 duced by each source separately. That is, if / and /' 

 are the two frequencies emitted, e and e' are the two 

 phase constants, and A and A' are the two complex 

 amplitudes, the resulting p(r,t) is given by 



„ = d;g2^i/B-(r/c)] _|_ 4-'g2W/'[(-(r7c)]_ ^^-j 



r r' 



A = 



-ivlft , 



A' = a'e 



-iicif't 



(75) 



according to equation (71). 



We shall restrict our attention to the case of two 

 sound sources situated on the x axis, one at the origin 

 and the other a small distance s away. We assume 

 that these two sources produce initial pressure dis- 

 turbances of equal real amplitude and equal frequency 

 and that the initial disturbances are opposite in phase. 



^x 



Figure 7. 

 source 00'. 



Resultant pressure produced by double 



This case, pictured in Figure 7, is a fairly good ap- 

 proximation to many sources occurring in practice, 

 such as a vibrating diaphragm. Because of these as- 

 sumptions, the following relationships exist among 

 the quantities in equations (74) and (75) : 



/=/'. e = 0; t' = ^y (76) 



a = a 



2/ 



Also, A = a and A' = ae " = —a because of equa- 

 tion (75). 



Of particular interest is the extreme case where the 

 distance between the two sources is very nearly zero, 

 but where the real amplitude a of the individual dis- 

 turbances is so large that the product as is an ap- 

 preciable quantity. Such a combination of two single 

 sources with very small separation and very large 

 individual amplitudes is called a double source. A 

 double source may be described by two quantities: 

 the product as, and its axis, the direction of the line 

 joining the two sources. 



