filter Q and damping
A second-order filter can behave like a child on a swing. Give it a shove and ask how it settles: a heavily damped swing oozes back to rest with no overshoot, while a lightly damped one swings past, oscillates, and rings for a while. The number that captures this is Q (quality factor), or its inverse cousin the damping factor. Q sets the personality of a second-order section, smooth and sluggish, or sharp and resonant.
Q measures how lightly damped a resonance is. Low Q (high damping) gives a gentle, well-behaved response with no peak; high Q (low damping) gives a tall peak near the cutoff and a sharp, selective response that rings. For a second-order section the damping factor is roughly 1/Q. A Q of 0.5 is critically damped (the fastest settling with no overshoot); Q = 0.707 (the Butterworth value) gives the flattest passband with no peaking; Q = 1 starts to show a small bump; a Q of 10 gives a sharp resonant peak useful for a narrow band-pass. The same Q that sharpens the frequency response also makes the step response overshoot and ring.
Q is the knob you turn to balance sharpness against smoothness. In a band-pass it sets selectivity (Q = f_0 / bandwidth); in a low-pass it sets how flat or peaked the passband is and how the filter rings on transients. The honest caveat is that high Q is touchy: it depends on precise component ratios, so loose tolerances shift the peak and the cutoff, and a too-high Q causes overshoot and ringing that can ruin a signal's shape even while the frequency response looks impressive. This is exactly why the standard approximations, Butterworth, Chebyshev, Bessel, are really just different choices of the Q in each section.
Take a second-order low-pass with a fixed 1 kHz cutoff and vary only its Q. At Q = 0.707 the passband is dead flat right up to cutoff (Butterworth). Raise Q to 2 and a 6 dB bump appears just below 1 kHz, and a square-wave edge through the filter now overshoots and rings instead of settling cleanly.
Same cutoff, different Q: low Q is smooth and flat, high Q peaks and rings, the central frequency/time tradeoff.
High Q is a double-edged sword: it sharpens the frequency response but makes the step response overshoot and ring, and it depends on precise component ratios, so loose tolerances move the peak and cutoff.