PX153 - B5.1 - underdamping

x (t) = e^{- α t} (C e^{i ω^{' 2} t} + D e^{- i ω^{' 2} t})

similar to underdamped case, but $e^{- α t}$ implies an exponential decay to the amplitude and a modified "resonance" frequency:

ω^{'} = \sqrt{ω_{0}^{2} - α^{2}} = \sqrt{\frac{k}{m} - \frac{γ^{2}}{4 m^{2}}}

for the initial conditions: $x (0) = x_{0,} v (0) = 0$
$x (0) = C + D = x_{0}$
$(i ω^{'} - α) C - (i ω^{'} D + α) = 0 ⟹ D = \frac{i ω^{'} - α}{2 i ω^{'}} x_{0}, C = \frac{i ω^{'} + α}{2 i ω^{'}} x_{0}$

x (t) = \frac{x_{0}}{2 ω^{'}} e^{- α t} ((ω^{'} + \frac{α}{i}) e^{ω^{'} t} + (ω^{'} - \frac{α}{i}) e^{- ω^{'} t})

∴ x (t) = x_{0} e^{- α t} (\cos ω^{'} t + \frac{α}{ω^{'}} \sin α t)