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)]