State Bernoulli's equation for unsteady motion of an irrotational, incompressible, inviscid fluid subject to a conservative body force $-\nabla \chi$.

A long vertical U-tube of uniform cross section contains an inviscid, incompressible fluid whose surface, in equilibrium, is at height $h$ above the base. Derive the equation

$$h \frac{d^{2} \zeta}{d t^{2}}+g \zeta=0$$

governing the displacement $\zeta$ of the surface on one side of the U-tube, where $t$ is time and $g$ is the acceleration due to gravity.

$$\begin{aligned} & \frac{\partial \eta}{\partial t}+U \frac{\partial \eta}{\partial x}=\frac{\partial \phi}{\partial y}, \quad \frac{\partial \phi}{\partial t}+U \frac{\partial \phi}{\partial x}+g \eta=0 \quad \text { on } \quad y=0 \\ & \frac{\partial \phi}{\partial y}=0 \quad \text { on } \quad y=-h, \end{aligned}$$