Viscously damped U-tube oscillation ($\Gamma =\beta /(\rho A)$)

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.