Fluid dynamics of a liquid in a U-shaped tank

Using the unsteady Bernoulli equation, we find solutions for the oscillatory motion of an incompressible and non-viscous liquid in a U-shaped tank. Alternative expressions for height-vs-time relation and the period are obtained through a transformation formula of the elliptic integral of the second kind. From the viewpoint of the mechanical energy conservation, the motion of the free surface can be interpreted as the motion of one particle in a restoring effective potential. This article provides an instructive example of the application of unsteady Bernoulli equation at an undergraduate level.