Small Vibratory Motions of Elastic Fluids. 299 



II. Application to the Vibrations in Musical Tubes- 



12. I proceed to say a few words on the application of 

 the theory to sounds produced in musical tubes. Conceive a 

 series of waves of the primary type to be generated at the open 

 end .(4 of a tube closed, at the other end B, and to be propa- 

 gated from J towards B. When they arrive at B they will be 

 reflected back, will return to A, and will be there alFected just 

 as they would be, if the end B were removed, the tube were 

 prolonged to A', so that BA' = BA, and the waves proceeded to 

 an open extremity A'. Hence a tube closed at one end, and 

 a tube of double the length of the other, open at both ends, 

 comport themselves alike in respect to the propagation of waves. 

 After reflection two waves, indicated by the two curves cb, c'b', 

 (Fig. 12.) exactly equal, will meet ; and in consequence at some 

 point m the velocity will be always equal to o. Let t be reckoned 

 from the time at which c, c', were simultaneously at m. Then 

 cm = at; and if 



mp = X, p^ =y, y = m\ sin - .{x + at). 



A. 



Also \i pr=r.y', y' =m\ sm -.{at — x). 

 Hence, v =y - y' = m\ ]sin^(x + at) + sin -{x - at)\ 



_ . . ttX irat 



= 2m\ sin — cos , 



X \ 



as = y + y' = m\ sin -{x + at) — sin- {x — at) 

 X X 



„ , nrX . wat 



= 2 »iX COS — - Sin — — . 

 X X 



PP2 



