466 Tides in Estuaries 



if h is the water depth in front of the bore. From these two equations we 

 obtain 



2 f(V-cJ and ag+^(K-tf = %, (XIII.6) 



respectively, and substituted into (XIII. 4) when the height of the bore is small, 



V- d = \/(gh) and ( K- c 2 ) = }/[g(h + »?)] 



respectively, which is in excellent agreement with the equation found by Scott 

 Russell for the propagation of wave disturbances with an elevation r\ above 

 the undisturbed water level over a depth h (see (V. 14) on p. 118). With the 

 small current velocities in the river in front of the bore (XIII. 4) becomes 



cx= V-\'{V*-2g<n). (XIII. 7) 



The changes in the form of the waves when they pass from deeper into 

 shallower water have been discussed in more detail in the chapter concerning 

 shallow water waves (see p. 109). With a gradual change of the width b and 

 the depth h of a canal in which a flood wave travels, the amplitude of the 

 wave varies according to Green (1837) according to b~ ll ' 2 h~ lU . When a flood 

 wave approaches the coast, there is no variation in b, but h does vary and, 

 for this reason, the wave amplitude increases approaching the shore. In 

 a river in which b varies also, there is another increase of the amplitude. 

 Besides the decrease of the water depth causes the waves not to travel any 

 more with the same constant velocity of propagation, but with each rise and 

 fall of the surface the wave will travel with a velocity depending on the exist- 

 ing depth. If h is the depth before the wave has arrived and r\ the actual water 

 depth when the wave is there, then this special velocity is c = g 1/2 (3^ 1/2 — 2/i 1/2 ). 



Fig. 197. Progressive changes of wave profile in shallow water. 



For waves with small amplitude r\ is approximately equal to /?, and the ve- 

 locity of propagation is equal to that of the long waves: j (gh). However, 

 if the amplitude becomes comparable with h, parts of the wave crest move 

 faster than those of the wave trough. The flood waves then cannot propagate 

 without changing their profile. With decreasing water depth, the slope of the 

 waves becomes steeper and steeper on the front side and gentler on the rear 

 side, until finally a state develops in which the wave breaks and surges. 

 Equation (V. 21 (p. 122)), which was developed by Airy, gives the curve of Fig. 

 197. It shows a succession of profiles of a single flood wave which travels up 

 a river. The steepening of the front side of the progressing wave adopts 

 finally the form of a bore, as shown in Fig. 196. Fjeldstadt (1941) has 



