237-2.39] TRANSVERSE OSCILLATIONS IN A CANAL. 429 



238. An interesting problem which presents itself in this 

 connection is that of the transversal oscillations of water contained 

 in a canal of circular section. This has not yet been solved, but 

 it may be worth while to point out that an approximate determi- 

 nation of the frequency of the slowest mode, in the case where the 

 free surface is at the level of the axis, can be effected by Lord 

 Rayleigh's method, explained in Art. 165. 



If we assume as an ' approximate type ' that in which the free 

 surface remains always plane, making a small angle 6 (say) with 

 the horizontal, it appears, from Art. 72, 3, that the kinetic energy 

 T is given by 



where a is the radius, whilst for the potential energy V we have 



2F = f#>a0 2 ........................ (2). 



If we assume that <x cos (<rt + e), this gives 



whence <r = 1169 (>/)*. 



In the case of a rectangular section of breadth 2a, and depth 

 a, the speed is given by Art. 236 (8), where we must put k = ?r/2a 

 from Art. 186, and h = a. This gives 



o- 2 = \TT tanh \TT . ........................ (4), 



or <r = 1'200 (g/a)t The frequency in the actual problem is less, 

 since the kinetic energy due to a given motion of the surface 

 is greater, whilst the potential energy for a given deformation 

 is the same. Cf. Art. 45. 



239. We may next consider the free oscillations of the water 

 included between two transverse partitions in a uniform horizontal 

 canal. It will be worth while, before proceeding to particular 

 cases, to examine for a moment the nature of the analytical 

 problem, with the view of clearing up some misunderstandings 

 which have arisen as to the general question of wave-propagation 

 in a uniform canal of unlimited length. 



