Long Waves in Canals and Standing Waves in Closed Basins 153 



when the depth changed rapidly. He determined the shape of a long wave, 

 travelling from deep ocean (depth h ) over a concave, convex or flat slope 

 into a shallow sea (depth h x ). The dimensions of the bottom profile cor- 

 responded approximately to those of the continental shelf, which slopes 

 down from the shelf to the deep sea. His method consisted in the splitting 

 up of a progressive wave 



t] = t] O cos (at — xx) 



into two standing waves 



V — r) x cosot ■}- fysmat , 



whose behaviour was examined separately. Their superposition then re- 

 presents the progressive wave. Figs. 68 and 69 are examples, namely for 

 h = 3200 m and h x = 200 m, for a slope 100 km long, which is defined 

 by the bottom h = h (l±x 2 /a 2 ) and a = 115-5 km. The wave length over 

 the deep water is A . The dotted wave trains represent the partial waves rj x 

 and r} 2 and are at the same time wave profiles for at = 0° and 90°. For 

 at = 180° and 270° the wave profiles are obtained by inverting v\ x and rj z and 

 it can be seen that when the wave passes over the slope its wave length de- 

 creases with its amplitude increasing. For X = a and smaller values Green's 

 formula still applies with fairly close approximation, but for greater wave 

 lengths the two partial waves do not fit any longer and are displaced in such 

 a way that there are important deviations from this equation. The variation 

 in amplitude (full line) does not agree any longer with this law 



but follows the simpler one 



\h : \h x = 2 , 



j//* : y'h! = 4 



Therefore, the ratio shows a considerably higher value than Green's equations. 



Table 19. Travel time of high water in minutes over a slope 

 100 km long and rising from 3200 m to 200 m depth 



(a= 200: j/3 = 115-5 km) 



