Proof of Eq. 15-1 in "Open-channel hydraulics," by Ven T. Chow 1

E2/E1 = [(8F12 + 1)3/2 - 4F12 + 1]/[8F12(2 + F12)]

Froude numbers:

• F1 = v1/(gy1)1/2

• F2 = v2/(gy2)1/2

From water continuity:

• v1y1 = v2y2

• v12y12 = v22y22

• F12y13 = F22y23

• F22 = F12/(y2/y1)3

The hydraulic jump equation, Eq. 3-21 (Chow):

• y2/y1 = (1/2)[(1 + 8F12)1/2 - 1]

• N2 = 1 + 8F12

• y2/y1 = (1/2)[N - 1]

• 2(y2/y1) = N - 1

• (y2/y1)3 = (1/8)[N - 1]3

• 2(y2/y1)3 = (1/4)[N - 1]3

• N = (1 + 8F12)1/2

• N3 = (1 + 8F12)3/2

• N2 - 1 = 8F12

• F12 = (N2 - 1)/8

• 4F12 = (N2 - 1)/2

Efficiency of the hydraulic jump:

• E2/E1 = [y2 + v22/(2g)]/[y1 + v12/(2g)]

• E2/E1 = [y2(1 + F22/2)]/[y1(1 + F12/2)]

• E2/E1 = 2 (y2/y1){1 + F12/[2(y2/y1)3]}/(2 + F12)

• E2/E1 = (N - 1){1 + (N2 - 1)/[2(N - 1)3]}/(2 + F12)

• E2/E1 = (N2 - 1)(N - 1){1 + (N2 - 1)/[2(N - 1)3]}/[8F12(2 + F12)]

• E2/E1 = [(N2 - 1)(N - 1) + (1/2)(N + 1)2]/[8F12(2 + F12)]

• E2/E1 = {(N3 - N2 - N - 1) + [(N2/2) + N + (1/2)]}/[8F12(2 + F12)]

• E2/E1 = {(N3 - [(N2 - 1)/2] + 1}/[8F12 (2 + F12)]

• E2/E1 = [(8F12 + 1)3/2 - 4F12 + 1]/[8F12(2 + F12)]

1 Chow, V. T. 1959. "Open-channel hydraulics," McGraw-Hill, New York.

 A hydraulic jump downstream of Tinajones reservoir, Chiclayo, Peru.