26 



WAVE ACOUSTICS 



that this average emission per unit soHd angle is 

 given by 



f (a) = ; = ;— cos^ a {06) 



doi 



PoC^ 



where (ha is an infinitesimal solid angle in the direc- 

 tion a. The maximum value of F{a) occurs in the 

 direction of the x axis, for which 



F(0) = 



2x^ 

 Poc' 



Thus equation (83) can be rewritten as 

 F(ol) = f (0) COS^ a 



(84) 

 (85) 



Figure 8. Rotation of wedge aa about axis. 



and equation (82) as 



7(r,a) = 



F{Q) COS^ a 



(86) 



of energy flow integrated between the angles and tt; 

 that is, 



J'7r/2 

 F{0) cos^ q; • 2ir sin a • da 

 



= '^^. (87) 



3 



It will be remembered that i^(0) is the maximum rate 

 of emission per unit solid angle, by the double source. 

 All sound projectors have pattern functions which 

 describe the distribution of sound energy emit- 

 ted in different directions. A general direction in 

 space can be defined by the two coordinates (9,</)), 

 where 6 is the angle of elevation of the direction OP 

 relative to the horizontal xy plane, and 4> is the polar 

 angle in the xy plane between the x axis and the 

 projection OP', as in Figure 9. Let F{d,4>) be the 



In order to find the total energy emitted by the 

 double source in one second, we calculate the total 

 energy traversing the surface of a sphere of radius r in 

 one second. This is clearly equal to the rate at which 

 the source is putting out power. To get this total 

 energy, it is necessary to integrate the average energy 

 flow (86) over the whole sphere. Such an integral is 

 in general multiple, but in this particular case it can 

 be expressed as a single integral because the energy 

 flow depends only on a. First consider the average 

 rate at which energy is flowing through the small 

 area element intercepted on the sphere by the two 

 cones defined by the angles a and a -{- da, as in 

 Figure 8. This small element of the sphere has the 

 area 2irr^ sin ada; and, therefore, it intercepts a solid 

 angle of 2ir(sin a)da units. By equation (86), the 

 average rate of energy flow through this element is 



F(0) cos^ a • 25r sin a • da. 

 The total emission in one second is this average rate 



Figure 9. Coordinates specifying direction OP. 



emission per unit solid angle in the direction OP, and 

 let Fmax be the emission per unit solid angle in the 

 direction of maximum emission, called the acoustical 

 axis of the projector. Then the pattern function 

 h{0,4>) is defined by 



F{e,<i>) = F^ax h{d,<i>). 



If we take the acoustical axis in the direction (0,0), 

 this becomes 



F{e,<t>) = F{0,0)b{e,<i>). (88) 



The pattern function h clearly depends only on the 

 nature of the projector. 



