116 Shallow Water Wave Transformation through External Factors 



Gaillard (1924) derived from 533 waves of very different amplitudes that 

 Stokes's equation (II. 10) is valid. He also pointed out that, with decreasing 

 depth the wave length decreases, which causes a decrease in the wave velocity. 

 If the bottom slopes up very slightly, he found that the empirical formula 



2 r K 



is applicable, in which c x and c 2 represent the velocity of the same wave 

 corresponding to a depth h x and to the shallower depth h 2 . We can see that 

 in shallow water waves the cnoidal waves represent quite well the actual 

 waves. 



2. The Solitary Wave of Scott Russell 



Scott Russell (1845, p. 311) during his experiments in wave tanks 

 (610 m long, 30-5 cm wide, rectangular basin), found a particular type 

 of wave, which he called the solitary wave. The wave was generated on one 

 end of the tank, ran through it and was reflected on the other end. He re- 

 peated this reflection 60 times, and he could, in this way, observe the wave 

 over a length of 360 m. This wave consists of a single elevation, of a height 

 not necessarily small as compared with the depth and which, if properly 

 started, may travel for a considerable distance along a uniform canal with 

 little or no change of type. The velocity of propagation of this wave is 

 constant and is given by 



c=\ / [g{h + a)} (V.4) 



in which a is the height of the wave crest (the maximum elevation above the 

 undisturbed level). Scott Russell considers the wave profile a trochoid. 



It has been tried to generate waves which consisted only of a wave trough, 

 having the same amplitude as the solitary crest. However, the experiments 

 were unsuccessful and the wave thus generated always broke up in shorter 

 waves after a relatively short time. 



Bazin (1865) confirmed the results obtained by Scott Russell by ex- 

 perimenting in two long canals of rectangular shape, one 450 m long, 1 99 m 

 wide, 1 95 m deep, and also in another canal 500 m long, 6-5 m wide and 

 2-4 m deep. He also found that the form of the wave profile was quite 

 permanent and that the wave velocity was expressed correctly by equa- 

 tion (V.4). Even if the water is in motion, equation (V.4) is valid, if we take the 

 wave velocity relative to the velocity of the water. If we have waves travelling 

 in a direction opposite to the current, we get a change in the wave profile, 

 and finally the wave breaks up. According to Scott Russell, this would happen 

 when h = a, but Bazin found that it happens a little earlier. 



Boussinesq(1871, 1877) and Rayleigh(1876,p. 257) have developed inde- 

 pendent theories on the solitary wave in which they considered the vertical 



