338 



TIDAL WAVES. 



[CHAP, viii 



can be readily traced by means of the tables of the functions J 8 (z\ I 8 (z\ and 

 its intersections with the parabola 



will give, by their ordinates, the values of o-/2?i. The constant /3, on which 

 the positions of the roots depend, is equal to the square of the ratio 

 2na/(gh)% which the period of a wave travelling round a circular canal of 

 depth h and perimeter 2?ra bears to the half-period (ir/'n) of the rotation of 

 the water. 



The accompanying figures indicate the relative magnitudes of the lower 

 roots, in the cases s = l and s = 2, when ft has the values 2, 6, 40, respectively*. 



y 



With the help of these figures we can trace, in a general way, the changes 

 in the character of the free modes as /3 increases from zero. The results may 

 be interpreted as due either to a continuous increase of w, or to a continuous 

 diminution of h. We will use the terms * positive ' and ' negative ' to distin- 

 guish waves which travel, relatively to the water, in the same direction as the 

 rotation and the opposite. 



When ft is infinitely small, the values of x are given by J s ' (#*) = 0; these 

 correspond to the vertical asymptotes of the curve (iv). The values of a- 

 then occur in pairs of equal and oppositely-signed quantities, indicating that 

 there is now no difference between the velocity of positive and negative waves. 

 The case is, in fact, that of Art. 187 (13). 



* For clearness the scale of y has been taken to be 10 times that of x. 



