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

whence

as claimed. Note that this rule of thumb requires , while the one of the previous section only required .

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University