CHAPTER IV. RELATION OF CAPACITY TO FORM RATIO. 



INTRODUCTION. 



Details of channel form in a natural stream 

 are highly diversified. In connection with the 

 bendings to right and left the current is thrown 

 to one side and the other, with the result that 

 the cross section is not, for the most part, sym- 

 metric about a medial axis but shows greater 

 depth on the side of the swifter flow. In the 

 straight channels of the laboratory there was 

 little departure from bilateral symmetry and 

 the cross section was approximately rectangu- 

 lar. For this reason those relations of traction 

 to form of cross section which are found to ex- 

 ist in the laboratory can not, in general, be in- 

 ferred of natural streams. Nevertheless there 

 is probably an approximate correspondence be- 

 tween the two types when the tractional prop- 

 erties of a broad, shallow channel are compared 

 with those of a narrow, deep channel; and to 



that extent the discussion of form ratio CR= ) 



w' 



is pertinent to the problems of natural streams. 



In connection with the study of the labora- 

 tory data the form ratio is a factor of great im- 

 portance, for not only is capacity for traction 

 directly conditioned by it, but it affects every 

 law of relation between capacity and another 

 condition. 



In the discussion of capacity in relation to 

 slope the effects which might have been referred 

 to form ratio were treated instead as due to 

 width, while small account was taken of the co- 

 ordinate influence of depth. For many pur- 

 poses the choice of viewpoint is indifferent, but 

 when large and small channels are to be com- 

 pared there is decided advantage in taking ac- 

 count of form ratio. The form ratios of labo- 

 ratory channels and river channels, for exam- 

 ple, are of the same order of magnitude, but the 

 widths are not. 



SELECTION OF A FORMULA. 



MAXIMUM. 



When identical discharges are passed through 

 troughs of different width and are loaded with 

 d6bris of the same grade, and the loads are 

 124 



adjusted so as to establish the same slope, it is 

 usually found, not only that the capacity varies 

 with the width, but that some intermediate 

 width determines a greater capacity than do 

 the extreme widths. That is, the curve of 

 capacity in relation to width exhibits a maxi- 

 mum. The form ratio varies inversely with the 

 width; and the same maximum appears when 

 the capacity is compared with form ratio. The 

 curves in figure 38, introduced to illustrate this 

 fact, show data from Tables 12 and 14 for grade 

 (C), with Q= 0.363 ft. 3 /sec. and =1.0 percent. 

 In the upper curve capacities are compared 

 with widths; in the lower one the same capaci- 

 ties are compared with form ratios. 



Width 



O.I 0.2 



Form rati o 



FIGURE 38. Illustration of the relation of capacity to width of channel 

 and to form ratio, when slope and discharge are constant. 



The formula for the discussion of such rela- 

 tions must be one affording a maximum. It 

 must also satisfy various physical conditions, 

 as will presently appear. 



The explanation of the maximum, so far as 

 its main elements are concerned, is not difficult. 

 The phenomenon was in fact anticipated in the 

 planning of the experiments, and certain 

 courses of experimentation were arranged with 

 special regard to the discovery of the form 

 ratio of highest efficiency. 



Conceive a stream of constant discharge and 

 flowing down a constant slope but of variable 

 width. The field of traction is determined by 

 the width, and the evident tendency of this 

 factor is to make the capacity increase as the 

 width increases. The rate of traction for each 

 unit of width is determined by the bed velocity 

 in that unit, and the bed velocity is intimately 



