115 



sin^ x= — (8/37r) (sin x- 1/5 sin 3a:- 1/35 sin 5x— 1/105 sin 7/x. ..) (123) 



It similarly may be shown that lor values of x between 2ir and Sir 

 the expression for sin^x is given by equation (120); for values of x 

 between Sir and 47r, by equation (123) and so on. 



226. Therefore, when v=B sin (at-\-^) the value of F is represented 

 by the continuous function: 



F=(8/37r){ByC'r)[sm (a^+/3)-l/5 sin 3(a^+)S) 



-1/35 sin 5 (a^+/3) . . .] (124) 



The friction term is then the resultant of a principal component, 



Fr={8/ST)(B'/C'r) sin (Gi+|3) = (8/3 x) Bv/C'r ■ (125) 



with the speed of the velocity, and minor components whose speeds 

 are 3, 5, 7, etc., times the speed of the principal component. The 

 correspondence between the principal component and the complete 

 value of F is shown in figure 36. 



The derivation of a mathematically continuous expression for F 

 when the velocity is the resultant of two or more harmonic components 

 would be difficult, if not impossible. 



SURFACE, VELOCITY, ACCELERATION, AND FRICTION HEADS 



227. It is the generally accepted practice to apply the formulas for 

 steady flow to sections or reaches of a channel of considerable length, 

 even though the velocity is not entirely uniform throughout such 

 reaches because of a variation of successive cross sections of the water 

 prism in the channel. For the computation of the friction term, the 

 velocity throughout the reach is taken as the average velocity, as 

 determined usually by the discharge through the average cross sec- 

 tion. The error introduced by the assumption, as well as the error 

 introduced by considering the velocity at any point in the channel 

 as the mean velocity in the cross section, is generally small in com- 

 parison with the uncertainty in the selection of the proper coefficient 

 of roughness to derive the value of C. The equation for varying flow 

 may similarly be applied to sections of channel of considerable length, 

 so long as the velocity and the slope at any instant are tolerably con- 

 stant throughout the section. In deep channels these conditions are 

 fulfilled in sections several miles in length. Denoting the length of 

 such a section by I, equation (112) establishes the relationship: 



ldy/dx+l(v/g)dv/dxi'(l/g)dv/dt±lv'/C'r=0. (126) 



228. The end of the section from which distances in the section 

 extend in the positive direction may be designated the initial end. 



