ADJUSTMENT OF OBSERVATIONS. 



67 



tion of capacity to slope and discharge jointly 

 being tentatively represented by 



C cc S n Q m (16) 



the values of n and m were computed for many 

 different conditions, and it was found that on 



atv\ 



the average = 0.34. There is, however, con- 



71- 



siderable variation in the ratio, and it is rela- 

 tively large when C is small. On plotting the 

 values of the ratio in relation to C and graphic- 

 ally extrapolating, it was found that when 

 (7=0 the ratio is about 0.5. The condition of 

 zero capacity is that of competent bed velocity 

 and competent slope. 



For any particular value of C in (16) the 

 product of S n by Q m is constant, or 



whence 



But, as we have just seen, when <7has the par- 



M 



ticular value 0=0, = 0.5. Therefore compe- 

 tent slope varies inversely as Q- 5 . Its assumed 

 representative, a, is assumed to vary inversely 

 with the square root of discharge. 1 



Turning now to the relation of a to width, 

 w, and examining Table 7, we see that a is not 

 exclusively either an increasing or a decreas- 

 ing function of w. Where the smallest widths 

 are concerned, as with Q = 0.093 ft. 3 /sec. and 

 Q = 0.182 ft. 3 /sec., for grades (B) and (C), the 

 function is decreasing. Where the greater 

 widths are concerned it is for the most part in- 

 creasing. There is a rational explanation for 

 such double relationship in the case of compe- 

 tent slope. 



Figure 23 represents two troughs in cross 

 section. Each has a bed of debris, and they 

 are assumed to be carrying the same total 

 discharge. In the wider there is less discharge 

 for each unit of width, and the tendency of the 

 smaller discharge is to reduce velocity. There- 

 fore to maintain a particular velocity namely, 



i The preliminary discussion on which is based a cc -^-. did not have 



y.8 



the advantage of the formula using 9. A rediscussion, to be found in 

 Chapter V, yields a oc gj^t but it was not practicable to give the adjust- 



ment the benefit of this later work without repeating the greater part 

 of the computations of the paper. It is not believed that the advantage 

 to the results would be commensurate with the labor involved. 



the competent velocity the tendency is to 

 produce a relatively steep slope. So far as 

 this factor is concerned, competent slope is 

 relatively steep for a wider trough. But there 

 is another factor; the velocity is influenced by 

 the resistance of the sides of the trough. This 

 resistance is greater where the water is deeper, 

 because the surface of contact is broader; and 

 the water is deeper in the narrower trough. 

 The tendency of the resistance is to reduce 

 velocities and therefore to make the slope for 

 competent velocity steeper in the narrower 

 trough. 



Water 



FIGURE 23. Diagrammatic sections of laboratory troughs, illustrating 

 relation of current depth to trough width. 



In very wide troughs the influence of the 

 sides is of minor importance, the influence of 

 discharge per unit width dominates, and the 

 competent slope varies directly with the width. 

 In very narrow troughs the influence of the 

 sides dominates, and the competent slope 

 varies inversely with the width. For some 

 intermediate width the two tendencies are 

 balanced, and the competent slope has its 

 minimum value. In figure 24 abscissas 

 measured from represent width of trough, 

 and ordinates represent competent slope. The 

 curve sketched, while not quantitative, has 

 adequate basis for its broader features and 



FIGURE 24. Ideal curve of competent slope ( C. S.) in relation to width 

 of trough (W). 



shows the general character of the relation of 

 competent slope to width. From its minimum 

 it ascends gradually on the side of greater 

 width, and on the side of lesser width rises with 

 relative rapidity toward a vertical asymptote 

 near the axis of competent slope. These 

 various characters are approximately paralleled 

 by the variations of a as shown in the table. 



The relation of a to fineness does not come 

 out very clearly in Table 7, and a different 



