Past exam of the mathematical methods I course of the University of Oxford / 2016 resit / 11 / b

Set ${\omega }_d=\sqrt{{\omega }_0^2-{\gamma }^2/4}>0$. The real solution of the damped harmonic oscillator equation is $x-x_0=e^{-\gamma t/2}(A\cos {\omega }_{d}t+B\sin {\omega }_{d}t)$. The initial displacement fixes $A=a-x_0$, and release from rest gives $B=\gamma A/(2{\omega }_d)$. ThereforeThe exponential decay rate of the amplitude is $\gamma /2$, while the weak-damping energy decay rate is $\gamma$.