BELATION OF CAPACITY XO FORM KATIO. 



125 



associated with the mean velocity. Velocity 

 varies directly with depth, and, inasmuch as 

 increase of width causes (in a stream of con- 

 stant discharge) decrease of depth, the tend- 

 ency of this factor is to make capacity 

 decrease as width increases. Velocity is also 

 affected by lateral resistance, the retarding 

 influence of the side walls of the channel. The 

 retardation is greater as the wall surface is 

 greater, therefore as the depth is greater, and 

 therefore as the width is less. As capacity 

 varies inversely with the retardation, and as the 

 retardation varies inversely with width, it fol- 

 lows that the tendency of this factor is to make 

 capacity increase as width increases. Thus the 

 influence of width on capacity is threefold : Its 

 increase (1) enlarges capacity by broadening 

 the field of traction, (2) reduces capacity by 

 reducing depth, and (3) enlarges capacity by 

 reducing the field of side-wall resistance. Now, 

 without inquiring as to the laws which affect 

 the several factors, it is evident that when the 

 width is greatly . increased a condition is in- 

 evitably reached in which the depth is so small 

 that the velocity is no longer competent and 

 capacity is nil. It is equally evident that when 

 the width is gradually and greatly reduced 

 the field of traction must become so narrow 

 that the capacity is very small, and eventually 

 the current must be so retarded by side-wall 

 friction that its bed velocity is no longer 

 competent and capacity is nil. For all widths 

 between these limits capacity exists, and some- 

 where between them it attains a maximum. 



The forms of algebraic function which afford 

 a maximum are many; but no general examina- 

 tion of them is necessary, because the physical 

 conditions of the problem serve to indicate the 

 appropriate type. As just observed, the varia- 

 tion of form ratio (when discharge and slope are 

 constant) involves simultaneous variations of 

 width and depth. To develop an expression 

 for the relation of capacity to form ratio, it is 

 convenient first to determine separately the 

 relations of capacity to width and to depth, 

 and then to combine the two functions. 



CAPACITY AND WIDTH. 



To consider separately the response of 

 capacity for traction to variation of width it is 

 necessary to relinquish, for the time being, the 

 assumption of constant discharge and variable 

 depth, and substitute for it the assumption of 



constant depth and slope, with variable dis- 

 charge. That is, we are to conceive a stream 

 of constant slope, of which the width is pro- 

 gressively increased or diminished and of 

 which the discharge is varied in such way as to 

 maintain a constant depth. Figure 39 repre- 

 sents the cross section of such a stream, whether 

 natural or of the laboratory type. 



Near the sides the current is retarded by side 

 friction. Also, the freedom of its internal 

 movements is restricted by the sides, just as it 

 is everywhere restricted by the upper surface 

 and the bed. These lateral influences diminish 

 with distance from the sides and finally cease to 

 be perceptible. We may thus recognize, in a 

 broad stream, two lateral portions, AB, in 

 which capacity is affected by the sides, and a 

 medial portion, AA, in which capacity is not 

 thus affected. In the medial portion total 

 capacity is strictly proportional to the distance 

 AA; or, in other words, the capacity per unit 



FIGURE 39. Cross sections of stream channels; to illustrate the rela- 

 tion of capacity to width. 



distance, C t , is uniform. In a lateral portion 

 the capacity per unit distance diminishes as 

 the side is approached. Whatever the law of 

 diminution, the total capacity of a lateral por- 

 tion is equivalent to the capacity per unit dis- 

 tance in the medial portion, multiplied by some 

 distance AD, less than AB. Therefore the 

 total capacity for the whole stream is 



2DB) . 



-(48) 



C=C l (AA + 2AD) = C l > 



It is evident that for a shallow stream the 

 distances AB and DB are less than for a deep 

 stream; and while the assumption may not be 

 strictly accurate, it must be approximately 

 true that DB is proportional to the depth. 

 Making that assumption and introducing the 

 numerical constant , I replace 2DB by ad, 

 and write 



As we are here concerned only with the law of 

 variation of C, we may conveniently replace 

 this by the proportion 



C*xw ad. 



