and also the flows from tributaries, which contributed as much as 30 per cent of the 

 flow sometimes. By integrating the differential equations we find the results that you 

 see on the slide. Again computed and observed values are very close — the difference 

 between the observed and the calculated stages is a matter of inches. 



G. K. Morikawa 



I wish to make a brief comment on Prof. Lighthill's interesting paper, in par- 

 ticular, about his "kinematic waves" (Lighthill and Whitham, 1955). An unexpected 

 feature of his proposed theory of flood movement is that shock (or bore) formation is 

 included as an essential part of the theory. This approximate theory might appear rather 

 strange at first glance since, in river flows, resistance plays such an important role. (Of 

 course, in flow with no, or with small, resistance we know that shocks are always 

 imminent.) For the kinematic wave approximation, all derivative terms (inertia plus 

 gravity-surface slope) in the momentum eqn. (20) (see Ref.) are ignored, leaving only 

 the gravity-bottom slope to balance the resistance force in the momentum equation. 

 The neglect of the restoring force provided by the gravity-surface slope term allows 

 bore formation. 



It is well known that bores are formed only under very severe conditions such 

 as those encountered in very steep rivers or at large tidal estuaries. In slow, long rivers, 

 bores are not formed even under the extreme conditions of flood stage during which 

 time heavy tributary flows into the main stream cause relatively (compared to the 

 bottom slope) large surface gradients. These steep gradients generally are damped as 

 they travel downstream. However. I hope that the "kinematic wave" approach, which 

 seems quite attractive from the viewpoint of its possible simplicity, will be pursued 

 further in the study of flood movement and other physical problems. 



Reference: 



M. J. Lighthill and G. B. Whitham. "On Kinematic Waves. I. Flood Movement in Long 

 Rivers," Proceedings of the Royal Society, A. Volume 229, pp. 281-316, 1955. 



M. J. Lighthill 



In reply to Dr. Morikawa, there are different kinds of "shock waves." One is 

 the dynamic shock wave, which we call a bore, and the other is the kinematic shock 

 wave, which is very extensive, covering many miles. You could see, from one of the 

 curves Professor Stoker showed, this rapid rise at the beginning of the flood. 



Now, of course, my answer to Professor Stoker is that I agree with him. On 

 the other hand, he has the advantage of having come to the end of his programme, 

 and being able to show the final results. The kind of thing that Whitham and I have 

 suggested hasn't yet been applied, and of course I wasn't able in the lecture to discuss 

 all the modifications we propose. When run-off, for example, is present, as Professor 

 Stoker was describing, we incorporate it in the theory in a perfectly straightforward 

 manner (described in our paper). We also attach importance to the inclusion of second- 

 derivative terms which in our paper we called diffusion (rather like diffusion in a 

 gaseous shock wave, which increases its thickness) but which in my lecture I treated 

 only in a brief section. 



This theory, as it were, hasn't been fed into the data on which Professor Stoker 

 worked. 



Now, I am not suggesting that it would give such good results as the kind of 

 thing Professor Stoker does, because obviously if you spend a lot of money, if you use 

 very big machines, and very long and complicated models, you are bound to get better 

 results. To a large extent it is a matter of compromise. By applying kinematic waves, 

 and then improving the theory, you may be able to get rough results which have some 



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