212 Long Waves in Canals and Standing Waves in Closed Basins 



succeed each other. In each cell the wave is rotating and in the centre of each 

 cell the amplitude is reduced to zero. Such star-shaped distributions of the 

 lines of equal phase are called amphidromies. The succession of these am- 

 phidromies in a canal gives the impression that on the left-hand side of the 

 canal a wave is progressing with varying amplitude in the +x-direction and 

 on the right-hand side a similar wave in the — x-direction. The amplitudes 

 are always greatest along the wall of the canal. According to (VI. 103), the 

 horizontal velocity u in the direction of the canal is proportional to rj, and 

 the phases of the horizontal water motion are given by the same lines of 

 equal phase as for r\. Perpendicularly to the canal the velocity is everywhere 

 zero. When the water moves in one direction in the left section of the canal, 

 it moves in the opposite direction in the right section. 



The distribution of the amplitudes shows that nowhere on a line trans- 

 verse to the canal, the amplitude is constantly zero, and this cannot be 

 achieved by selecting another difference of phase for the two waves travelling 

 in opposite directions. In order to reduce to zero the motion caused by the 

 superposition of two Kelvin waves, it is necessary to add a definite disturbance 

 in the canal towards its closed end. This disturbance will depend essentially 

 upon the period of the wave and the dimensions of the canal, especially 

 upon its width. 



The problem of the reflection of a Kelvin wave on a transverse wall was 

 solved by Taylor (1920). First of all, he takes the superposition of two 

 Kelvin waves progressing in opposite direction as a particular solution, as 

 illustrated in Fig. 90. The transverse motion v is equal to zero everywhere, 

 and at a specific cross-section x = x t there is in the canal a current in the 

 longitudinal direction of the canal u = u x . Taylor has succeeded in finding 

 a second particular solution for the transverse motion v in the canal which, 

 according to the boundary conditions, vanishes at the longitudinal walls of 

 the canal, but which at the same time agrees to a current in the direction of 

 the canal giving for the cross-section x = x x exactly the same value u x as 

 the first solution. 



The difference between the two solutions is then the complete solution, 

 because it fulfils the boundary conditions, according to which the transverse 

 velocity is zero everywhere at the longitudinal walls of the canal and at the 

 closed end of the canal (x = x a ) the longitudinal velocity always vanishes. 

 The mathematical difficulties of the problem lie in establishing certain values 

 fixing the cross-section at which the longitudinal currents of both solutions 

 must become equal. Taylor's solution says that in a given canal rotating with 

 a certain angular velocity, a total reflection of a penetrating Kelvin wave at 

 its closed end occurs only then, when its frequency is smaller than a value 

 depending on the dimensions of the canal. At some distance from the closed 

 end, the reflection is practically identical with the superposition of two Kelvin 

 waves moving in opposite directions (Fig. 90); in the inner section of the 



