Yet another meaning for is as follows [20, p. 326]
where the resonator is freely decaying (unexcited).
Proof. The total stored energy at time is equal to the total energy of the remaining response. After an impulse at time 0 , the stored energy in a second-order resonator is
The energy dissipated in the first period is , where
Assuming as before, so that
Assuming further that , we obtain
This is the energy dissipated in one cycle. Dividing this into the total stored energy at time zero, , gives
as claimed. Note that this rule of thumb requires , while the one of the previous section only required .