500 HYDROMECHANICS [HYDRAULICS. 105. Fundamental Differential Equation of Steady Varied Motion. Suppose the equation just found to be applied to an indefinitely short length ds of the stream, limited by the end sections ab, aj) v taken for simplicity normal to the stream bed (fig. 121). For that Fig. 121. short length of stream the fall of surface level, or difference of level of a and a lf may be written dz. Also, if we write u for u , and u + du for u,, the term u ~ u i- becomes - . Hence the eo nation applicable 20 to an indefinitely short length of the stream is &_*? + <& ...... (1).
From this equation some general conclusions may be arrived at as to the form of the longitudinal section of the stream, but, as the investigation is somewhat complicated, it is convenient to simplify it by restricting the conditions of the problem. Modification of the Formula for the Restricted Case of a Stream flowing in a Prismatic Stream Bed of Constant Slope. Let i be the constant slope of the bed. Draw ad parallel to the bed, and ac horizontal. Then dz is sensibly equal to a c. The depths of the stream, h and h + dh, are sensibly equal to ab and a b , and therefore dh = a d. Also cd is the fall of the bed in the distance ds, and is equal to ids. Hence dz=a d = cd -a c=ids-dh ..... (2). Since the motion is steady Q = fl?t = constant. Differentiating, tldu + udn = ; Let x be the width of the stream, then d&=xdh very nearly. Inserting this value, ux , WvU j 7 du= ---dh m (3). Putting the values of du and dz found in (2) and (3) in equation (1), ds _ 2(7 (4). Further Restriction to the Case of a Stream of Rectangular Section and of Indefinite Width. The equation might be discussed in the form just given, but it becomes a little simpler if restricted in the way just stated. For, if the stream is rectangular, xh n, and if x o ji is large compared with h, = = A nearly. Then equation (4) X x becomes dh_ ds 1- (5). gh 106. General Indications as to the Form of Water Surface fur nished by Equation (5). Let AgAj (fig. 122) be the water surface, B Bj the bed in a longitudinal section of the stream, and ab any sec tion at a distance s from B , the depth ab being h. Suppose B B 1( B A taken as rectangular coordinate axes, then is the trigono metric tangent of the angle which the surface of the stream at a makes w^th the axis B B r This tangent will be positive, if the stream is increasing in depth in the direction B^ ; negative, if the stream is diminishing in depth from B towards B x . If -v-"0, the surface of the stream is parallel to the bed, as in cases of uniform motion. But from equation (4) ds 20 .. _-., which is the well-known general equation for uniform motion, based on the same assumptions as the equation for varied steady motion now being considered. The case of uniform motion is therefore a limiting case between two different kinds of varied Tuotion. Fig. 122. Consider the possible changes of value of the fraction i-r Igih 1- As h tends towards the limit 0, and consequently u is large, the numerator tends to the limit - oo . On the other hand if A = oo , in which case u is small, the numerator becomes equal to 1. For a value H of A given by the equation we fall upon the case of uniform motion. The results just stated may be tabulated thus : For A= H >H oo the numerator has the value - oo >0 1 . Next consider the denominator. If h becomes very small, in which case u must be very large, the denominator tends to the limit - oo . As A becomes very large and u consequently very small, the denominator tends to the limit 1. For A = , or u = fg~h, the denominator becomes zero. Hence, tabulating these results as be fore : For A = -^ >^ oo the denominator becomes - oo 9 >0 107. Case 1. Suppose A> , and also A>H, or the depth greater than that corresponding to uniform motion. In this case is ds positive, and the stream increases in depth in the direction of flow. In fig. 123 let B B 1 be the bed, CoCj a line parallel to the bed and Ai Fig. 123. at a height above it equal to H. By hypothesis, the surface A A, of the stream is above C C 1} and it has just been shown that the depth of the stream increases from B towards B 1 . But going up stream
h approaches more and more nearly the value H, and therefore
