210 Long Waves in Canals and Standing Waves in Closed Basins 



This means that these Poincare waves are only possible when T < T ( and 

 also T <T q , i.e. the period of the wave must be shorter than half a pendulum 

 day and, at the same time, shorter than the natural period of the transverse 

 oscillations of the canal. This condition greatly reduces the possibility of 

 the occurrence of such waves. Thus, in narrow canals only Kelvin waves 

 with tidal periods are possible. However, it is possible that Poincare waves will 

 play an important part in the development of the tides in ocean sections 

 comparable to wide canals. If the depth of the canal is not constant in its 

 cross-sections, the problem becomes much more difficult mathematically. 

 Proudman (1946, p. 211) gave a thorough discussion of this question. 

 He assumed a frictionless motion of the water-masses for the middle section 

 of the canal, but assumed a dissipation of energy on the two sides where 

 the depth is smaller. This corresponds more to real conditions and permits 

 a more accurate application. 



Another kind of oscillations is still possible in a canal. A solution of the 

 equation (VI. 102), which is also valid for v, is 



v = exp 



, .nn 



—SnX+ij-y 



, if £ = &J-k 2 (VI. 11 2) 



in which nulla > k. As v should vanish for ±a, v can only take the forms: 



(VI. 11 3) 



, nn jot 



v = A„e " x sin^~ y-e'° for even integers n (=2,4, ...) , 



v = A„e s " x cos ^- y • e"" for odd integers n (= 1, 3, ...) 



With the aid of these relations and equation (VI. 101), we can compute easily 

 the corresponding values of r\ and u. These oscillations have the form of 

 standing transverse waves in the canal, with amplitudes decreasing with 

 a power of e along the canal from a point x = 0. They are practically limited 

 to a small section of the canal. This kind of oscillations was introduced by 

 Taylor (1920), in order to fulfil the condition of total reflection of 

 a Kelvin wave in a canal closed at one end. If we wish to compute the 

 oscillations of a rectangular canal closed at both ends (total reflection at 

 both ends of the canal), Poincare waves should be added to the Kelvin waves, 

 so as to fulfil the boundary conditions (p. 216). 



(c) Reflection of a Kelvin Wave. Free Oscillations in a Rotating, Rectangular 

 Basin of Uniform Depth 



The superposition of two Kelvin waves progressing in a rectangular canal 

 in an opposite direction does not result at all times at any cross-section of 

 the canal in a horizontal water-motion zero, where a transverse barrier could 

 be erected which would close the canal without changing the water-motion. 



