St Venant equations for unsteady open-channel flow consist of continuity and momentum equations. These equations are used to describe the flow of water in open channels and are essential in hydraulic engineering.

Continuity Equation:
The continuity equation states that the rate of flow of water in an open channel is equal to the product of the cross-sectional area of the channel and the average velocity of the water flowing through it. This equation is derived from the principle of conservation of mass.

Momentum Equation:
The momentum equation describes the motion of water in an open channel and is derived from the principle of conservation of momentum. It relates the change in momentum of the water to the external forces acting on it. The momentum equation can be written in two different forms, namely:

1. The local form, which relates the rate of change of momentum at a point to the pressure gradient and the shear stress at that point.
2. The integrated form, which relates the total change in momentum over a length of the channel to the total external forces acting on the water in that length.

Energy Equation:
The energy equation is derived from the principle of conservation of energy and relates the total energy of the water in an open channel to the external forces acting on it. It takes into account the potential energy, kinetic energy, and pressure energy of the water.

In conclusion, the St Venant equations for unsteady open-channel flow consist of continuity and momentum equations, which are essential in hydraulic engineering. While the momentum equation can be written in two different forms, the energy equation is derived from the principle of conservation of energy.