(2)
p/G + v2/2g = p1/G + v12/2g + (v − v1)2 / 2g.
(3)
This differs from the expression (1), § 29, obtained for cases where no sensible internal work is done, by the last term on the right. That is, (v − v1)2 / 2g has to be added to the total head at CD, which is p1/G + v12/2g, to make it equal to the total head at AB, or (v − v1)2 / 2g is the head lost in shock at the abrupt change of section. But (v − v1) is the relative velocity of the two parts of the stream. Hence, when an abrupt change of section occurs, the head due to the relative velocity is lost in shock, or (v − v1)2/2g foot-pounds of energy is wasted for each pound of fluid. Experiment verifies this result, so that the assumption that p0 = p appears to be admissible.
If there is no shock,
p1/G = p/G + (v2 − v12) / 2g.
If there is shock,
p1/G = p/G − v1 (v1 − v) / g.
Hence the pressure head at CD in the second case is less than in the former by the quantity (v − v1)2 / 2g, or, putting ω1v1 = ωv, by the quantity
(v2/2g) (1 − ω/ω1)2.