Vibrations of a Rod of Varying Cross- Section. 105 



where a numerical factor has been incorporated in the 

 constant. If the canal be supposed to communicate at a — a 

 with an open sea, in which a tidal oscillation ?7 = Acos (crt + e) 

 is maintained, we have on determining the constant in (6), 



A3 ^±^i\2^ m x ' ) 3 >!±H^{2^n a : )• (7) 



2— w I 2 — w 



' m + n-1 I m + n -i 



x 2~ J a 2 



It is now of interest to consider some special cases of (7). 



(1) Let ??i = 0, n = l. The depth now is constant and the 

 breadth varies as the distance from the end x = 0. Equation 

 (7) then gives us 



A Jft(fC,v) . ' 



"= A jfc) cos ^ +e )- 



(2) m=l, n = 0. The breadth is constant and the depth 

 varies as x. We then have 



^ =A Jo(2^A) cos( ^ + e)< 

 J^Ki/a) 



(3) m = i, n = ^. The breadth and the depth both vary 

 as y/x. 



Hence ?? = A ^f— r4 , cos (at-\- e). 



(4) ?n = 0, n = 2. The depth is constant but the breadth 

 varies as x 2 . We then have 



. Ji(fcx) f Ji(/ca) / . , \ 



But we know that JiC^xAi 7 ^)™ sin 0. Hence the above 

 expression can be written 



A sin kx /si 

 7? = A /- 



KX I 



sin *ra f v 



. cos {at + 6) . 



tea 



This shows that rj vanishes for kx = stt, where 5 is a positive 

 integer. 



(5) m=l, n = ^. The depth now varies as x and the 

 breadth as \/x. Hence we have 



, =A M^)/£fe^). cos((7 , +e ). 



