182] 



WAVE-MOTION IN CONTRACTING CHANNELS. 



295 



The annexed diagram of the curve y=^o(V^)> where, for clearness, the 

 scale adopted for y is 200 times that of x, shews how the amplitude continually 

 increases, and the wave-length diminishes, as we travel up the canal. 



These examples may serve to illustrate the exaggeration of oceanic tides 

 which takes place in shallow seas and in estuaries. 



We add one or two simple problems of free oscillations. 



3. Let us take the case of a canal of uniform breadth, of length 2a, whose 

 bed slopes uniformly from either end to the middle. If we take the origin at 

 one end, the motion in the first half of the canal will be determined, as 

 above, by 



(xi), 



where K = a- 2 a/gk , as before, h denoting the depth at the middle. 



It is evident that the normal modes will fall into two classes. In the first 

 of these ?/ will have opposite values at corresponding points of the two halves 

 of the canal, and will therefore vanish at the centre (x=a). The values of <r 

 are then determined by 



e7 (2^a*)=0 ................................. (xii), 



viz. K being any root of this, we have 



.(xiii). 



In the second class, the value of ?/ is symmetrical with respect to the 

 centre, so that drj/dx=Q at the middle. This gives 



(xiv). 



Some account of Bessel's Functions will be given presently, in connection 

 with another problem. It appears that the slowest oscillation is of the 



