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 of z/, 

 u = that in the direction of x ; then, if « be the thickness MN at the time t, a + 5 « 



at the time t+8 t, the quantity DG will have been pressed from ^^ to ^3^ 



a+ a 



Therefore the whole DN will be exhibited in y • 



a-\- b a 



and D will descend by a space - y 



=y 



a + O a 

 8a 

 8a 



+ y 



a + 



8a , 



-y — nearly 



hence the velocity of D is 



y da 

 a dt 



(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 wave-jnotion. This amounts to the substitution of 



2ir . 4*77 



/« + asin-^— - (c t—x) + a' sin -^- (cl — x)+ &c. 

 A A 



for z. 



In this formula, k is the original depth ; 

 A the length of a wave ; 

 c the velocity of transmission. 

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

 obtain z=h + asin6, 



2'rr 



6 being equal to 



(c t—x). 



dz 

 dx 



2iri 



cos 



d 



from the supposition that the velocity in the direction parallel to a; is uniform 

 through any vertical section, and that consequently u is a function of a) and t 

 only. 



From this consideration, it follows that -r- is independent of j/ : and conse- 



(*' tXf 



