144 Long Waves in Canals and Standing Waves in Closed Basins 



direction of .v-positive. In the same way the second term of equation (VI. 9) 

 represents a progressive wave travelling with velocity c in the direction of 

 x-negative and it appears, since equation (VI. 9) is the complete solution 

 of equation (VI. 8), that any motion whatever of the fluid, which is subject 

 to these conditions, may be regarded as made up of waves of these two kinds. 

 If in any case of waves travelling in one direction only, without change 

 of form, we impress on the entire water mass a velocity — c, equal in force 

 and in opposite direction to the velocity of progation, the motion becomes 

 steady, whilst the forces acting on any particle remain the same as before. 

 Bernoulli's theorem for the free surface gives then 



— + gn + 2 (" - c) 2 = constant . (VI. 10) 



{?0 



If b is the width of the canal, a water-mass b(h + rj)(u — c) flows through 

 the cross-section bih + rj) to the left, whereas for a cross-section not yet 

 reached by the wave the mass flown through is — bhc. If the slope of the 

 wave-profile be everywhere gradual, and the depth h small compared with 

 the wave-length, the horizontal velocity may be taken to be uniform throughout 

 the depth and approximately equal to the velocity. The equation of con- 

 tinuity then gives 



« = rr-1 (VI. 11) 



or, if r] is small compared to h, u = c/htj. If we substitute in (VI. 10), we obtain 



+ g*l + f (l - ff = constant . (VI. 1 2) 



With the condition for a free surface p = const, and by developing into 

 series, we obtain the relation 



2n J r ^ c % r Lj r = c 2 - 

 gr] I 2 C ^ 2 -h... y 



If if is small compared to h\ we get for c the Lagrangian formula c = j (gh) ; 

 sl closer approximation gives 



c = \(gh)-[\ + Hv/h)]. (VI. 13) 



This equation by St. Venant (1870) states that the wave velocity in this 

 second approximation is also independent of the wave length, but it varies 

 with the height, and consequently, such a wave cannot be propagated without 

 change of profile. Airy (1842) has shown, in applying the method of suc- 

 cessive approximations, what variations occur; a complete solution for this 

 has been given by Fjelstad (1941). In an advancing wave system, the front 

 slopes become steeper, the rear slopes flatter, until finally a point is reached 



