U-tube liquid oscillation (${\omega }_0=\sqrt{2g/l}$)

In a uniform U-tube containing incompressible liquid of density $\rho$, cross section $A$ and total moving length $l$, let one surface fall by $y$ while the other rises by $y$. The height difference is $2y$, and potential energy relative to equilibrium is $U=\rho gA{y}^2$. The whole moving mass is $\rho Al$, so $K=\rho Al{\dot{y}}^2/2$. Newton's second law or energy conservation gives $\ddot{y}+(2g/l)y=0$ and ${\omega }_0=\sqrt{2g/l}$. This assumes small excursions that keep the free surfaces in the uniform vertical limbs.

If resisting force per unit moving length is $\beta v$, the total resistance is $\beta l\dot{y}$ and $\ddot{y}+\Gamma \dot{y}+{\omega }_0^{2}y=0$, where $\Gamma =\beta /(\rho A)$. For $\Gamma <2{\omega }_0$, an initial velocity $v_0$ from zero displacement gives $y=(v_0/{\omega }_d)e^{-\Gamma t/2}\sin {\omega }_{d}t$, with ${\omega }_d^2={\omega }_0^2-{\Gamma }^2/4$. Its quality parameter is $Q={\omega }_0/\Gamma$, and the amplitude envelope decays at half the energy-decay exponent. The effective damping model is an assumption, not a derivation of detailed viscous flow near the wall.