150 Long Waves in Canals and Standing Waves in Closed Basins 



Z= A 

 g 



Atl-cjt y 2 16tt 4 7 4 y 4 



i+ + + .. 



6 a 2 216 a 4 



V 



Z and V = 



# 2.T:a j 

 -A — - 

 c i a 



47i 2 a? y 2 



(VI.26) 



Thus if a canal has the following dimensions 200 km wide (a = 100 km) h = 100 m and A = 100 

 in the middle of the canal, with a wave period T = 12 h 25 min (tidal wave period), the distribu- 

 tion of wave heights and of the ratio V:U across the canal (Table 18c) the wave velocity becomes 



Table 18r 



c = 0-9966 \'{gh) and one recognizes the good approximation with (VI. 25) when the expression 

 between brackets is ignored. In a parabolical canal a wave travels with rectilinear crests and 

 constant velocity when the wave height is almost constant. There is no doubt that, when the cross- 

 sections of the canal are more complicated, the influence of the bottom configuration in the trans- 

 versal direction on the formation of the waves is equally small as in the case under consideration 

 as far as the wave crests and the wave velocity are concerned. 



(c) Wave Motion in a Canal of Variable Section in the Direction of Propagation 

 If we introduce in the equation of continuity (VI. 14) which is also valid 

 in this case, the surface of the cross-section S = bh, in which h is the mean 

 depth over the width b, we get 



1 8 



Tj = 



b ox 



(W) , 



(VI. 27) 



where h and b are functions of x. The dynamical equation has the same 

 form (VI. 5) as before. By eliminating I we obtain 



8hi 



b 8x 



g ° [hb 8,] 



8x 



(VI. 28) 



The laws of propagation of waves in a canal of gradually varying rect- 

 angular section were first investigated by Green (1837). Rayleigh (1876) 

 showed (see p. 121) that, considering the principle of energy, it follows from 

 the variation in the cross-sections that the amplitude of the wave in such 

 a canal is inversely proportional to the square root of the width and the 

 fourth root of the depth. 



If we take a solution of the equation (VI. 28) similar to that with a constant /;, we can assume 



rj = rj coso(t—(p) , (VI. 29) 



whereby we substitute the unkown function <p(x) for x\c. If we introduce this expression into (VI. 28) 



