276 DYNAMICAL THEORY OF SOUND 



and the complete expression for the disturbing velocity -potential 

 near the mouth must be 



kK \ . 



nt --^ cosnt \ ....... (28) 



In the problem as it actually presents itself the value of </> 2 

 at the mouth is prescribed, say 



</> 2 = Jcos(nJ-e); ............... (29) 



and in order to identify this with (28) we must have 



TcK nC r . /- n 2 \ nC 

 _._, / sin = (l-_j)^. (30) 



Hence 



This determines C in terms of J. If r denote the modulus of 

 decay of free vibrations, as given by 86 (18), the formula may 

 also be written 



7"2 1 ( f n 2\2 A M 2^ 



^_ 2 = J_|h _M +JL.1LI (32) 



Except in the case of approximate synchronism the second 

 term within the brackets will be small compared with the 

 first. Hence for a given value of J, the value of nC (which is 

 the amplitude of the flux q) will be greatest when n = n , 

 approximately. Moreover, for a given deviation of the ratio n/n 

 from unity the intensity of the resonance falls short of the 

 maximum in a greater proportion the greater the value of W O T, 

 i.e. the greater the ratio of the modulus of decay to the free 

 period. In other words, the smaller the damping of free 

 vibrations, the more sharply defined is the pitch of maximum 

 resonance. This is in accordance with the general theory 

 of 13. 



The vibrations of a resonator under the influence of an 

 internal source of sound are discussed in 90 with special 

 reference to the theory of reed-pipes. 



89. Mode of Action of an Organ Pipe. Vibrations 

 caused by Heat. 



Although the loss of energy in a single period may be small, 

 the free vibrations of the column of air contained in an organ 



