502 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



We have now to solve the problem of finding the motion of fluid in a tube 

 which is continually expanding. The small difference between MP and NQ may, 

 in this investigation, be neglected. 



2. To find the descent of fluid in a tube MQ, where MP is a fixed side, but 

 NQ is moveable by means of the pressure. 



Let D be any point in the fluid; v = the velocity at D in the direction oip, 

 u = that in the direction of w ; then, if « be the thickness MN at the time t, a + 8a 



at the time t+8 t, the quantity DG will have been pressed from 5^ to 5y . 



a 

 a + a 



a+ Oa 



Therefore the whole DN wiU be exhibited in ^. 



and D will descend by a space - 2/. — j- + y 



a + Oa 

 8a, , 



— y — nearly: 

 hence the velocity of D is - • ^ (downwards) 



_ y da. 

 a, dt 



3. This will, however, lead us to no result, except we assume the nature of 

 the motion to be defined. Let us, then, make the hypothesis that the motion is 

 a waw-motion. This amounts to the substitution of 



h + a sin ^r— (c t—x) + a' sin ^^ (c t — x) + &c. 

 A A. 



for z. 



In this formula, h is the original depth ; 



^ the length of a wave ; 



a the velocity of transmission. 

 Now, if we retain only the first term in the variable part of this expression, we 

 obtain z=h + asa\e, 



e being equal to ^ {c t-x). 



dz 2ira n 



—-= ^— cost), 



ax A 



from the supposition that the velocity in the direction parallel to an is uniform 

 through any vertical section, and that consequently u is a fimction of x and t 

 only. 



From this consideration, it follows that ^ is independent of;/ : and conse- 



