296 TIDAL WAVES. [CHAP. VIII 



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

 is 2K*a* = -76557r, whence 



= 1-306 



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

 the law 



(xv), 



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

 b = const., we find 



d / # 2 \ dr ' 



If we put <r*=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 xja 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 (/& = !), 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 o- would be ?rc/2a, where c = (gh^f. Hence in the 

 present case the frequency is less, in the ratio 2^/2/Tr, or '9003. 



The forced oscillations due to a uniform disturbing force 



X=fcos (<rt + f) ................................. (xix), 



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

 free surface is evidently 



H-f) .............................. (xx), 



and, since the given force is of the normal type n = l t we have 



where o- 2 



