Energy ratio between equal-phase damped crossings ($E(t+T)/E(t)=e^{-4\pi \gamma /{\omega }_d}$)

A linearly damped oscillator $m\ddot{x}+b\dot{x}+kx=0$ has $x=A{e}^{-\gamma t}\cos ({\omega }_{d}t+\varphi )$, with $\gamma =b/(2m)$ and ${\omega }_d^2=k/m-{\gamma }^2>0$. After one damped period $T=2\pi /{\omega }_d$, both displacement and velocity are multiplied by $e^{-\gamma T}$. Their quadratic mechanical energy is therefore multiplied exactly by $e^{-2\gamma T}$, regardless of the starting phase. In particular successive same-direction equilibrium crossings have this ratio even though potential energy is zero at each. A cosine phase maximum is not exactly a displacement turning point when damping is nonzero.