338 



TIDAL WAVES. 



[CHAP, viii 



can be readily traced by means of the tables of the functions J s (0), I a (z\ and 

 its intersections with the parabola 



3/ 2 =l+*//3 (v), 



will give, by their ordinates, the values of &amp;lt;r/2i. The constant /3, on which 

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

 2na/(ffh)t which the period of a wave travelling round a circular canal of 

 depth h and perimeter 2na bears to the half-period (irjri) of the rotation of 

 the water. 



The accompanying figures indicate the relative magnitudes of the lower 

 roots, in the cases s = I and s = 2, when /3 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 n, 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 |3 is infinitely small, the values of x are given by //(#*) = 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, 



