Tides in Estuaries 



473 



According to Raylcigh (1914) (see also Lamb, 1932 para. 187 or Forchheimer, 1924, p. 182) equa- 

 tion (XIII. 8) can be derived in the easiest way by means of the artifice of steady motion. If Q denotes 

 the volume per unit width which crosses each section in unit time we have for reasons of continuity 

 Uihi = u z hi = Q. (XIII. 9). If we consider the mass of fluid which is at a given instant contained 



2 hr 3 



Fig. 202. Bore-like rise of the surface in sea-shore rills (Prielen) of German North Sea 

 coast (Watten meer). Coast (Thorade-Schumacher). 



between two cross-sections, one on each side of the transition wave (bore) we see that in unit time it 

 gains momentum to the amount qQ(u 2 — Wi), the second section being supposed to lie to the right 

 of the first. Since the mean pressure over the two cross-sections are Igoh^ and %goh 2 , we have 



and from (XIII. 9) we obtain 



0(« 2 -«i) = igUh-hl). 

 G 2 = ^/z 1 /7 2 ( 1 + / ?2 ). 



(XIII. 10) 

 (XIII. 11) 



If we impress on the entire system a velocity 

 with the velocity of propagation 



-Hi, we get the case of a bore invading still water 



«i 



gfh 



h + hA 

 2lh J 



in the negative direction; the particle velocity in the advancing wave is u x —u 2 in the direction of 

 propagation of the bore. Rayleigh proves also that a positive discontinuity wave (wave above the 

 mean water level) can only continue to progress unchanged if there is a dissipation of energy with 

 the transition from one level to another. To the contrary, a negative discontinuity wave (depression 

 below the mean water level) can progress unchanged when additional energy is supplied. It follows 

 that a negative bore of finite height cannot in any case travel unchanged. 



As pointed out by Jeffreys (1934, p. 157), there does not seem to be an 

 essential difference between a tidal bore in a river and the normal beach surf. 

 Both depend, in the first place, upon the increase of the wave height due to 

 reduction of the cross-section (decrease in depth, reduction of width), secondly 

 upon the circumstance that, when the wave height becomes of the same size 

 as the undisturbed water depth, the wave crests travel more rapidly than the 

 wave troughs, overtake these troughs and break and, in the third place, upon 



