PROPERTIES OF SOUND WAVES 



25 



With these assumptions, equation (74) becomes 



p = ?g2r.y[(- (r/c)] _ «g2W/[(-(r7c)] ^77^^^ 



r r' 



where 



r2 = x2 + 2/- + Z-; r'2 = (x - s)^ + y"- + z^. (77b) 



If F{r) is an arbitrary function of r, and if r and r' 

 are very nearly equal, we have from simple calculus 



dF 



F{r)-Fir') « (r-r')-- 



The quantity r — r' may be calculated as a function 

 of a;,s,r as follows : 



r' = 



r + r' 



r- — r ' 

 2r 



because r 



which equals sx/r from equation (77b). The quan- 

 tity dF/dr may also be calculated: 



dF dF dx dFdy dFdz 



— I" r 



dr dx dr dy dr dz dr 



As the origin changes from to 0' on the x axis, 

 thereby changing r to r', the coordinates y and s of 

 all points in space are unchanged. Thus, for all 

 changes in r defined in this manner, 



dy dz 

 dr dr 

 so that 



dF _ dFdx _ dFr 

 dr dx dr dx x 



because of equation (36). Using these values of 

 r — r' and dF/dr, 







F{r) 



dF 



F{r') = s— 

 dx 



and equation (77) maj' be rewritten as 



.d\ 



dxLr 



J ig-2,ri/(r/c) . 



tLr J 



By calculating out the derivative of the bracket with 

 respect to x, and by remembering that dr/dx = x/r, 

 this equation becomes 



p = ase^"f'e-^"^^^''^ 



r^\ c r/ 



If a is the angle between the x axis and the radius 

 vector OP, x/r = cos a, and the preceding equation 

 becomes 



p = ase'''-^'e-2"^<'-/'=' 



cos a, 



(¥+;)■ 



If r is very large compared with c/f, the second term 

 ill the brackets may be neglected, and as a result 



2ir/ast 



P 



cos ae 



,2f/C(- (r/c)] 



cr 



Replacing the factor as by b, we obtain the final 

 result 



p = '^-^ cos ae^'^*-(^/^'l 

 cr 



(78) 



By comparing equation (78) with equation (73), we 

 see that the pressure changes produced at great dis- 

 tances by the double source are identical with the 

 pressure changes produced by a single source, which 

 is situated at the same place, vibrates with the same 

 frequency, and has the following complex amplitude. 



2TfU 

 A = cos a. 



(79) 



It is clear from equation (75) that the real ampli- 

 tude of the vibration is equal to the absolute value 

 of the complex amplitude. From algebra, we know 

 that the absolute value of a complex number A is 

 just V AA, where A is the conjugate complex of A. 

 Let a be the real amphtude corresponding to equa- 

 tion (79) . Then a is given by 



2irfb 



cos a. 



(80) 



With this definition of a, the actual pressure dis- 

 tribution defined by equation (78) is 



p{r,a,t) = a cos 2 



'<'-;)■ 



(81) 



Since p is harmonic, its square averaged over a com- 

 plete period is one-half the square of its amplitude 

 (80) ; from equation (58) we have for the intensity at 

 the distance r and angle a : 



1/2^/6 Y 1 



I{'>;a) = -[ cos a I — - 



r'\ c / 2poC 



2Tr'-f¥ 



COS^ a. 



(82) 



Thus, the sound intensity caused by a double sound 

 source is directly proportional to the square of b and 

 to the square of the cosine of the angle a of emission 

 and is inversely proportional to the square of the 

 distance from the sound source. 



Let F(a} denote the average rate at which energy 

 is emitted in the direction a. It is clear from Figure 8 



