280 DYNAMICAL THEOKY OF SOUND 



assume this source to be distributed uniformly over the cross- 

 section, so that 



Let us suppose for a moment that we have a flux 



2 = A cos nt ........................ (3) 



outwards from the mouth. The pressure at will consist of 

 two components. We have first the part necessary to control 

 the inertia of the 'air near the mouth ; the corresponding part 

 of the velocity-potential just inside is 



6 - q = cos nt, .................. (4) 



CO * ft) 



where a has the same meaning as in 87. Next we have the 

 part which is effective in generating diverging waves outside. 

 On the principles of 88 this is found to be 



7 A cos nt, ..................... (5) 



kA 

 corresponding to 4>-^r~ sm nt > ..................... (6) 



since k = n/c. The total velocity-potential at 0, corresponding 

 to (3), is therefore 



............. (7) 



Generalizing this, we may say that to a flux 



q = Ae int ........................ (8) 



corresponds <f> = o - - e int , . ...(9) 



\ 47T/ ft) 



the expression (7) being in fact the real part of (9), when A 

 is real. The correspondence will hold even if A be complex, 

 since this is merely equivalent to a change in the origin of t. 

 We now assume, for the region of plane waves, 



<t>=r~ {B cos k (I - x) - C sin k (I - x)}e int , . . .(10) 



