APPLICATION TO NATURAL STREAMS. 



237 



characters associated with bending channels. 

 Because of the discovered complexity of the 

 l:;ws affecting capacity it is now apparent that 

 the laboratory formulas can not be applied to 

 streams in general. It is, however, probable 

 that among the great variety of natural streams 

 there is a more or less restricted group which 

 is in such respect similar to the laboratory group 

 that the empiric results of the laboratory or 

 at least the results embodied in exponents 

 may properly be applied to it . 



The criteria of similarity between large and 

 small have been discussed to some extent by 

 others in connection with the investigation of 

 hydraulic problems by means of models. 

 William Froude inferred from theoretic con- 

 siderations that if the speed of a ship and the 

 speed of its miniature model " are proportional 

 to the square roots of the dimensions, their re- 

 sistances at those speeds will be as the cubes 

 of their dimensions," 1 and he afterward veri- 

 fied this result by experiments. T. A. Hear- 

 son, in projecting a model river for the inves- 

 tigation of various hydraulic problems, dis- 

 cussed separately the resistance to flow by the 

 wetted perimeter, the influence of varying sec- 

 tional area, and the influence of bends. He 

 concluded that if the linear dimensions were 

 kept in the same proportion, so that the river 

 channel and its model were similar in the geo- 

 metric sense, the velocities would be related f.s 

 the square roots of the linear dimensions, and 

 the discharges as the 2.5 powers of the linear di- 

 mensions. It would be necessarythat thorough- 

 nesses of the channel surfaces have the same dif- 

 ferences as the linear dimensions, and that the 

 movable debris of the bed r.lso follow the laws 

 of linear dimensions. 2 His deductions were not 

 tested by the construction and use of a model, 

 but they derive a large measure of support from 

 the verification of Froude's analogous theorem. 

 So far as I am aware, r.ll the models actually 

 constructed to represent rivers and tidal basins 

 have been given an exaggerated vertical scale. 3 

 O. Reynolds 4 made a series of models of tidal 

 basins in which the scales of depth and of tidal 



1 These words are quoted from Inst. Naval Arch. Trans., vol. 15, p. 151, 

 1874. I have not seen Froude's original discussion of the subject. 



2 Inst. Civil Eng. I'roc., vol. 146, pp. 21G-222, 1900-1901. 



See Fargue, L., La forme du lit des rivieres a fond mobile, pp. 57, 128, 

 1908. Fargue recommended for a model river a vertical scale of 1:100 

 and a horizontal scale of 1:20, from which he deduced a discharge ratio 

 of 1:3,200 and a velocity ratio of 1:16. 



* British Assoc. Adv. Sci. Repts. 1887, pp. 555-502; 1889, pp. 328-343; 

 I860. pp. 512-534; 1891, pp. 386-404. 



amplitude were greater than the scale of length, 

 the ratios ranging from 31:1 to 105:1. No ad- 

 justment was made as to size of debris, the re- 

 quirements of his investigation being met by 

 any material fine enough to be moved by the 

 currents. A tidal oscillation was communi- 

 cated to water resting on a level bed of sand, 

 with the result that the bed was gradually 

 molded into shapes more or less characteristic 

 of estuaries. From general considerations a 

 "law of kinetic similarity" was deduced: 



= constant 



Lt 



in which p is the tidal period, h the depth of 

 water (proportional to the amplitude of the 

 tide), and L the length of the estuary. Under 

 this law the results were generally consistent, 

 but there was found to be a limit to the range 

 of suitable conditions, and this limit was formu- 

 lated by 



Jt 3 e = constant 



in which e is the exaggeration of the vertical 

 scale. 



Eger, Dix, and Seifert, 5 making a model of 

 a portion of Weser River for the purpose of 

 studying the effect of projected improvements, 

 adopted 1 : 100 as the scale of horizontal dimen- 

 sions and depths, and 1 : 6.7 as the scale of mean 

 diameters of debris particles composing the 

 channel bed. It was then a matter, first of 

 theory and computation but finally of trial, to 

 select scales for discharge and slope. The main 

 condition to be satisfied was that for discharges 

 corresponding to high and low stages the depths 

 of water should be properly related, according 

 to the scale of linear dimensions. For the scale 

 of discharges 1 : 40,000 was finally adopted, and 

 for slopes 650 : 1 . The resulting ratio of veloci- 

 ties was 1:4; and this ratio, combined with the 

 ratio of debris sizes, was found to give a time 

 ratio (for the accomplishment of similar 

 changes in the bed of the stream) of 1:360. 

 The scale of velocities being only 1:4 while the 

 scale of distances was 1:100, there was an 

 exaggeration of velocities in the ratio of 25: 1. 6 



The quantities of debris moved being in the 

 ratio of 1:100 3 , the distances moved in the 



Zeitschr. Bauwesen, vol. 56, pp. 323-344, 1906. 



So stated by the authors. An allowance for the general principle 

 that velocities are proportional to the square root of the hydraulic mean 

 depth, and therefore to the square root of linear dimensions, would 

 indicate 1:10 as the normal ratio of velocities, and give 2.5:1 as the 

 exaggeration. 



