PIPES AND RESONATORS 281 



where the constants have been adjusted so as to satisfy (2). 

 Comparing with (8) and (9) we find 



j, -vn- *. 



Hence 



B = A {sin -H ( ka - - ) cos kl\ , 



(12) 



!/ ik 2 (o\ ) 



cos & ( &a ) sin &Z [ . 

 V 47T/ J / 



The latter equation gives A in terms of C. Considering only 

 absolute values we have 



= (cos kl - ka sin kVf + (^)* sin2 kl - -( 13 ) 



Since fcco is usually a small fraction, the emission of energy, 

 which varies as \A*\, will be greatest for a given source (7cosn 

 when 



cos kl = kassmkl, (14) 



nearly, i.e. when the imposed frequency approximates to that 

 of one of the normal modes of the pipe when closed at x = I, 

 as determined by 87 (11). In the case of the reed-pipe, 

 therefore, the tones which are specially reinforced consist of 

 the fundamental and the harmonics of odd order. 

 When (14) is satisfied, we have by (12) 



C 



This determines the relation between the flux outwards at the 

 mouth, and that constituting the source. The former now greatly 

 exceeds the latter in amplitude, and the factor i shews that it 

 differs in phase by a quarter-period. 

 Again, from (10) and (11) we have 



</> = T- \ sin kx + ka cos kx cos kx ! e int . . . .(16) 



