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250-351 (Cambridge, 1956). 



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DISCUSSION 



J. J. Stoker 



All of you will agree, I am sure, that Professor Lighthill has given us a very 

 stimulating discussion. While it was confined to a very small section of water wave 

 phenomena, it still shows that there is a huge variety in even small portions of that 

 subject, and a variety that stems not only from variety in the physical phenomena, but 

 also from variety with respect to possible ways of mathematical treatment. 



Professor Lighthill has already alluded to the fact that he and I might not agree 

 in all respects, about some of the things that he said concerning wave motions in rivers. 



I have a few slides giving results of calculations which a group of people at our 

 Institute have been making during the last three years. We have been involved in 

 computing flows in large rivers and reservoirs, with the object of making flood predic- 

 tions for lengthy periods of time, and over long stretches of rivers. For example, we 

 have dealt with floods in the Ohio River between Wheeling and Cincinnati; that is a 

 stretch close to 400 miles in length. We have also computed flows through the junction 

 of the Ohio and Mississippi at Cairo, Illinois, taking forty mile lengths of the rivers 

 on both sides of Cairo upstream in the Ohio and Mississippi, and the same distance 

 downstream in the lower Mississippi. Calculations have been made for flows in Ken- 

 tucky Reservoir at the mouth of the Tennessee River. This reservoir lies between two 

 dams, the Kentucky Dam on the downstream side, just about where the Tennessee 

 runs into the Ohio River, and the Pickwick Dam, some 180 miles upstream from there. 



Now it is true that kinematic waves, which -Professor Lighthill described so nicely 

 here, certainly give a correct general picture of what happens in a large river. There 

 is no doubt that the main disturbance is one which goes much slower than the propaga- 

 tion speed of wavelets, and the main disturbance does indeed travel downstream in 

 general, and not both up and down stream, in spite of the fact that the basic theory 

 formulated by the exact differential equations leads always to wave propagation in both 

 directions. 



However, we would be in very great difficulties in the Ohio River if we tried 

 to base our calculations on "kinematic waves," because even the main disturbance 

 sometimes goes up-stream in the Ohio, because the main flow is very much influenced 

 by the flow from tributaries and the local drainage areas. That part of the flow is 

 so large sometimes that the main flood travels in the wrong direction, so to say. We 

 find it necessary to use the full differential equations, both the momentum equation and 

 the continuity equation, and consequently we must put up with the fact that the differ- 

 ential equations have two real characteristics which require us to use small time steps 

 for calculation with the UNIVAC. 



In the Kentucky Reservoir if one wants to make predictions for three weeks, 

 there is ample time for many waves to propagate back and forth from the dams at both 



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