IN CYLINDRICAL TUBES. 235 



SECTION I. 



4. Suppose the tube AB, (fig. I.), open at A, and stopped at B, 

 with some substance possessing any degree of elasticity ; and suppose 

 the vibrations first produced and kept up by a rigid diaphragm, vibrating 

 according to a given law at A, and perfectly excluding the air within 

 the tube from any communication with the external air. We have 

 the usual equations 



v=f{at-x) + F{at + x)] 



(A), 



as=f{at-x)-F{at + x)] 



V denoting the velocity of a particle at distance a; from the origin, 

 and s the condensation at the same point at the time t, and a being 

 the velocity of propagation of an aerial pulse along the tube. 



One of our conditions must necessarily be, that the velocity of the 

 air within the tube and immediately in contact with the diaphragm, 

 must constantly have the same velocity as the diaphragm itself, con- 

 strained to move according to a given law. Let this velocity = <p{at). 

 Then shall we have 



(j){af)=/{ai) + F{at) (1). 



5. To ascertain the nature of the second condition, which must 

 hold at B, where the motion of the wave propagated along the tube 

 is interrupted, we must consider the effect which will be produced on 

 the stop by the action of the air within the tube. The vibratory motion 

 wUl produce alternations of condensation and rarefaction at the ex- 

 tremity B, which will tend to put the substance forming the stop in 

 vibration; and if it will admit of vibrations having the same period 

 as those of the air in the tube, this effect will be produced by the 

 constant reiteration of the cause above-mentioned. If the substance is 

 not susceptible of vibrations of this kind, no appreciable effect will be 

 produced upon it. 



