282 DYNAMICAL THEOEY OF SOUND 



When (14) holds, this reduces to 



x) -r sin kl cos kx e int 



'kcoSmkir 47T 



{ l"i 



,*"? 011 \ cos A; (Z - a) - V- sin M cos ^ r e * n '- (17) 

 ) 2 sm 2 ^[ y 4?r j 



The real part gives 



<f> = = -T =-; -^ cos k(l x) sin w^ : sin &Z cos ^a? cos w^ [ , 

 A^<w 2 sm 2 A;Z ( 4?r j 



(18) 



corresponding to a source Ccosnt. The variable part of the 

 pressure at the source (x = I) is 



-r sin H cos 



The first part of this is by far the more considerable ; it is, 

 moreover, the only part which is effective in doing work. The 

 mean rate of work done at the source, i.e. the mean value of 

 pCcosnt, is 



ZTTpcC* 



"Jfc"Bin"' 



It may easily be verified that this is equal to the work spent in 

 generating waves at the mouth, where, by (15), 



tf ................ (21) 



7 



k*o> sin kl 



It appears further from (19) that the maximum of pres- 

 sure at xl synchronises almost with the maximum influx 

 of air, following it however by a short interval. There is 

 therefore a tendency slightly to lower the pitch of the reed, 

 which is, in the instruments here referred to, of the " in- 

 beating type," i.e. the passage is opened when the reed swings 

 inwards, towards the wind -chest. The fact that the resultant 

 force on the reed is approximately in the same phase with the 

 displacement indicates that the reed is vibrating with a 

 frequency somewhat less than that natural to it ( 12). 



