296 TIDAL WAVES. [CHAP. VIII 



asymmetrical class, and so corresponds to the smallest root of (xii), which 

 is 2x^a* = 765577, whence 



_ 1.308 



4. Again, let us suppose that the depth of the canal varies according to 

 the law 



where x now denotes the distance from the middle. Substituting in (1), with 

 b = const., we find 



d (/, x 2 \ dn] o- 2 



, . N 

 -7----^ -r -^ ........................ (xvi). 



ds ^ 



If we put a 2 = n(n + l) ................................ (xvii), 



this is of the same form as the general equation of zonal harmonics, 

 Art. 85 (1). 



In the present problem n is determined by the condition that must be 

 finite for x/a= +1. This requires (Art. 86) that n should be integral; the 

 normal modes are therefore of the types 



(xviii), 



where P n is a Legendre s Function, the value of a- being determined 

 by (xvii). 



In the slowest oscillation (n=l\ the profile of the free surface is a 

 straight line. For a canal of uniform depth A , and of the same length (2a), 

 the corresponding value of &amp;lt;r would be 7rc/2a, where c = (gh$. Hence in the 

 present case the frequency is less, in the ratio 2^/2/vr, or &quot;9003. 



The forced oscillations due to a uniform disturbing force 



JT=/cos (&amp;lt;rt + e) ................................. (xix), 



can be obtained by the rule of Art. 165 (12). The equilibrium form of the 

 free surface is evidently 



(xx), 



and, since the given force is of the normal type ?i=l, we have 



where o- 2 = 2grA /a 8 . 



