This equation is satisfied if H0 = H1, which corresponds to the case of uniform motion. Dividing by H0 − H1, the equation becomes

(H1/H0) (H0 + H1) = 2u02 / g;

(3)

∴ H1 = √ (2u02H0 / g + 1⁄4 H02) − 1⁄2 H0.

(4)

In Bidone’s experiment u0 = 5.54, and H0 = 0.2. Hence H1 = 0.52, which agrees very well with the observed height.

Fig. 127.

§ 122. A standing wave is frequently produced at the foot of a weir. Thus in the ogee falls originally constructed on the Ganges canal a standing wave was observed as shown in fig. 127. The water falling over the weir crest A acquired a very high velocity on the steep slope AB, and the section of the stream at B became very small. It easily happened, therefore, that at B the depth h < u2/g. In flowing along the rough apron of the weir the velocity u diminished and the depth h increased. At a point C, where h became equal to u2/g, the conditions for producing the standing wave occurred. Beyond C the free surface abruptly rose to the level corresponding to uniform motion with the assigned slope of the lower reach of the canal.

Fig. 128.

A standing wave is sometimes formed on the down stream side of bridges the piers of which obstruct the flow of the water. Some interesting cases of this kind are described in a paper on the “Floods in the Nerbudda Valley” in the Proc. Inst. Civ. Eng. vol. xxvii. p. 222, by A. C. Howden. Fig. 128 is compiled from the data given in that paper. It represents the section of the stream at pier 8 of the Towah Viaduct, during the flood of 1865. The ground level is not exactly given by Howden, but has been inferred from data given on another drawing. The velocity of the stream was not observed, but the author states it was probably the same as at the Gunjal river during a similar flood, that is 16.58 ft. per second. Now, taking the depth on the down stream face of the pier at 26 ft., the velocity necessary for the production of a standing wave would be u = √ (gh) = √ (32.2 × 26) = 29 ft. per second nearly. But the velocity at this point was probably from Howden’s statements 16.58 × 40⁄26 = 25.5 ft. per second, an agreement as close as the approximate character of the data would lead us to expect.