Derive the relationship between upstream and downstream depths in a hydraulic jump.

Hydraulic jump is an abrupt transition from supercritical (Fr>1) to subcritical (Fr<1) flow in open channels, with significant energy dissipation. Applying momentum equation to control volume across jump (ignoring friction): ρQ(V₂-V₁) = ½ρg(y₁²-y₂²)b, where y is depth and b is channel width. Using continuity Q = V₁y₁b = V₂y₂b and Froude number Fr₁ = V₁/√(gy₁): y₂/y₁ = ½(√(1+8Fr₁²)-1). This conjugate depth relationship shows downstream depth increases with upstream Froude number. Energy loss: ΔE = (y₂-y₁)³/(4y₁y₂). Jump classification by Fr₁: weak (1.7-2.5), oscillating (2.5-4.5), steady (4.5-9), strong (>9). Applications: stilling basins, energy dissipators downstream of spillways, flow measurement.