PIPES AND RESONATORS 263 



( 7), it expresses the momentum (which may be symbolized by 

 pfa) in terms of the velocity q. The dynamical equation 



c 2 s=<i ........................... (5) 



of 70 (3) may in like manner be interpreted as expressing 

 the relation between change of momentum and force. If the 

 zero of q correspond to the equilibrium state, we have 



s = -q/Q ......................... (6) 



Eliminating s and <, between (4), (5), and (6), we obtain 



The motion is therefore of the type 



q = Ccos(nt + e), ..................... (8) 



provided n* = Kc?/Q ............ . ............ (9) 



If we write n = kc, this gives 



t? = K/Q, X = 2nV(Q/#) ................ (10) 



The wave-length depends, as we should expect, solely on the linear 

 dimensions of the resonator and its aperture. For resonators 

 which are geometrically similar in all respects, it varies directly 

 as the linear dimension. This is in accordance with a general 

 principle which may be inferred from the differential equation 

 (2) of 76, or otherwise. The formula (9) indicates further that 

 the pitch of the resonator is lowered by contracting or partially 

 obstructing the aperture, whilst it is raised by diminishing the 

 internal capacity. 



The kinetic energy, being mainly resident in the neighbour- 

 hood of the mouth, may be calculated from the principles 

 applicable to an incompressible fluid. If the actual motion 

 were generated instantaneously from rest, the work required 

 would be the sum of half the products of the impulses into 

 the corresponding velocities. The equations (9) of 69 shew 

 that the requisite impulsive pressure is p^ hence 



The potential energy is, by 70 (8), 



F=4 / >cVQ = |( / >cVQ).3 2 ............. (12) 



The coefficients in these expressions being known, the speed 

 n of the oscillations can be inferred at once by the general 



